2.1 Expanded pairs

Let (Y, D) be a smooth pair. The associated bubble component is the following ${\mathbb P}^1$ -bundle over D:

\begin{align*}\Delta := {\mathbb P}_D ({\mathcal O}_D \oplus N_{D/Y}) .\end{align*}

It comes with two canonical sections $D_0 = {\mathbb P}_D ( {\mathcal O}_D )$ and $D_\infty = {\mathbb P}_D (N_{D/Y})$ .

For an integer $k\geqslant 0$ , let

\begin{align*}Y[k]_0 := Y \cup_D \underbrace{\Delta \cup_D \cdots \cup_D \Delta}_{k \text{-times}},\end{align*}

where the divisor $D_\infty$ of the ith $\Delta$ is glued to the $D_0$ of the $(i+1)$ th $\Delta$ . The normal bundle of $D_0$ and that of $D_\infty$ in $\Delta$ are inverse to each other, making the singularity of $Y[k]_0$ smoothable in the first order.

Let $D[k]_0$ be the $D_\infty$ of the last bubble component $\Delta$ . Then

\begin{align*}(Y[k]_0, D[k]_0)\end{align*}

is a locally smooth pair, which we call the expanded pair of length k.

There is a projective map

(2.1) \begin{equation}p: Y[k]_0 \to Y, \end{equation}

which contracts all the bubble components, and maps the divisor $D[k]_0$ isomorphically to D.

In [Reference LiLi01], for each k, a family of expanded pairs is explicitly constructed whose properties are given below. In particular, the generic fiber is Y and the most special fiber is $Y[k]_0$ . This family is called the standard family.

Proposition 2.2 ([Reference LiLi01, Reference Li and WuLW15], [Reference ZhouZho18, Proposition 2.16]). There is a flat family

\begin{align*}\pi_k : Y[k] \to {\mathbb A}^k, \end{align*}

together with a smooth connected divisor $D[k] \subset Y[k]$ , and a projective map $p_k: Y[k] \to Y \times {\mathbb A}^k$ which satisfy the following properties.

1. (1) The pair (Y[k], D[k]) is a smooth pair, with Y[k] a smooth quasi-projective variety of dimension $\dim Y + k$ . There is an isomorphism $D[k] \cong D \times {\mathbb A}^k$ under which $\pi_k |_{D[k]}$ coincides with a projection to ${\mathbb A}^k$ .

2. (2) Let $t = (t_1, \ldots, t_k) \in {\mathbb A}^k$ be a closed point. Let $l = \sharp \{ i \mid t_i =0 \}$ . The fiber of $\pi_k$ over t is isomorphic to $Y[l]_0$ . The restriction of $p_k$ to this fiber is the contraction map p (2.1).

3. (3) There is a $({\mathbb C}^*)^k$ -action on Y[k], such that $\pi_k$ and $p_k$ are equivariant, where $({\mathbb C}^*)^k$ acts on ${\mathbb A}^k$ in the standard way. For a closed point $t\in {\mathbb A}^k$ as above, the stabilizer group $\operatorname{Stab}_t ({\mathbb A}^k) \cong ({\mathbb C}^*)^l$ acts on the fiber $Y[l]_0$ by scaling the l bubble components $\Delta$ .

There is a smooth equivalence relation $\sim$ on both Y[k] and ${\mathbb A}^k$ [Reference ZhouZho18, § 2.5], induced by the $({\mathbb C}^*)^k$ -action above and permutations of the coordinates of ${\mathbb A}^k$ off diagonal. Moreover, for $l\leqslant k$ , the family $Y[l] \to {\mathbb A}^l$ has standard embeddings into $Y[k] \to {\mathbb A}^k$ . The direct limit of the families, up to the equivalence relations, define the stacks of interest.

${\mathcal A}$ is a smooth Artin stack of dimension 0, and ${\mathcal Y}$ is a smooth Artin stack of dimension $\dim Y$ . There is a representable, projective map

\begin{align*}p: {\mathcal Y} \to Y \times {\mathcal A}, \end{align*}

whose composition with the projection to ${\mathcal A}$ gives the family map

\begin{align*}\pi : {\mathcal Y} \to {\mathcal A}, \end{align*}

which is flat and also representable. The universal relative divisor ${\mathcal D}$ satisfies

\begin{align*}{\mathcal D}\cong D \times {\mathcal A}, \end{align*}

such that the restriction of p to ${\mathcal D}$ coincides with the base change of $D \hookrightarrow Y$ over ${\mathcal A}$ . For a scheme S, a family of expanded pairs over S is a map $S \to {\mathcal A}$ . Pulling back the universal family $\pi$ gives a family

\begin{align*}\pi_S : {\mathcal Y}_S \to S, \end{align*}

endowed with a divisor ${\mathcal D}_S\subset {\mathcal Y}_S$ obtained by pulling back the universal divisor ${\mathcal D}$ . We also refer to $({\mathcal Y}_S,{\mathcal D}_S)$ as a family of expanded pairs over S.

• – $({\mathcal Y}_S, {\mathcal D}_S)$ is log Calabi–Yau (log CY) over S if $\omega_{{\mathcal Y}_S / S} ({\mathcal D}_S) \cong {\mathcal O}_{{\mathcal Y}_S}$ ;

• – in particular, if $S = \operatorname{{Spec}} \mathbb{C}$ , where $({\mathcal Y}_S, {\mathcal D}_S) = (Y[k]_0, D[k]_0)$ for some $k\geqslant 0$ , we say that $(Y[k]_0, D[k]_0)$ is log CY if it is log CY over $\operatorname{{Spec}} \mathbb{C}$ .

If (Y, D) is log CY, the adjunction formula implies that $\omega_D \cong {\mathcal O}_D$ .

Proof. Since ${\mathcal Y}_S \to S$ is flat, the log CY property is preserved under arbitrary base change. It suffices to prove the log CY property for the standard families (Y[k], D[k]) over ${\mathbb A}^k$ . Obviously,

\begin{align*}\omega_{Y[k] / {\mathbb A}^k} \cong \omega_{Y[k]}. \end{align*}

Hence it suffices to prove (Y[k], D[k]) is log CY. We prove this by induction on k. Recall (see [Reference LiLi01, p. 521]) that Y[k] is constructed as

\begin{align*}Y[k]=\operatorname{Bl}_{D[k-1] \times \{0\} } (Y[k-1] \times {\mathbb A}^1), \quad k\geqslant 1. \end{align*}

When $k=1$ , $Y[1] = \operatorname{Bl}_{D \times \{0\} } (Y \times {\mathbb A}^1)$ . The map $p: Y[1] \to Y \times {\mathbb A}^1$ is the blow-up, and D[1] is the proper transform of $D\times {\mathbb A}^1$ . Let E be the exceptional divisor. We have

\begin{eqnarray*}\omega_{Y[1]} &\cong& p^* \omega_{Y\times {\mathbb A}^1} \otimes {\mathcal O} (E) \\&\cong& p^* {\mathcal O} (-D \times {\mathbb A}^1 ) \otimes {\mathcal O} (E) \qquad\quad \text{(since }(\textit{Y, D})\text{ is log CY)} \\&\cong& {\mathcal O} (-D[1]) \otimes {\mathcal O} (-E) \otimes {\mathcal O} (E) \\&\cong& {\mathcal O} (-D[1]) .\end{eqnarray*}

Therefore, (Y[1], D[1]) is log CY and the proposition follows. The inductive process is obtained by replacing Y by $Y[k-1]$ in the $k=1$ case above.

2.2 Expanded degenerations

Definition 2.6. Let X be a smooth quasi-projective variety over ${\mathbb C}$ . A map

\begin{align*}\pi : X \to {\mathbb A}^1\end{align*}

is a simple degeneration if it is flat, with smooth fiber $X_c=\pi^{-1}(c)$ for $c\neq 0$ , and central fiber

\begin{align*}X_0 = Y_- \cup_D Y_+,\end{align*}

where $(Y_\pm, D)$ are smooth pairs (Definition 2.1), and $Y_\pm$ intersect transversally at the singularFootnote ⁴ divisor $D=Y_- \cap Y_+ $ in the sense that étale locally around $0\in {\mathbb A}^1$ , the map $\pi$ is of the form

\begin{align*}\operatorname{{Spec}} {\mathbb C} [x, y, t, \vec z\,] / (xy -t) \to \operatorname{{Spec}} {\mathbb C} [t], \end{align*}

where $Y_\pm$ are cut out by the equations $(x=0)$ and $(y=0)$ respectively.

By definition, we automatically have

\begin{align*}N_{D / Y_+} \otimes N_{D / Y_-} \cong {\mathcal O}_D. \end{align*}

For $k\geqslant 0$ , consider the following expanded degeneration of length k:

\begin{align*}X[k]_0 := Y_- \cup_D \underbrace{\Delta \cup_D \cdots \cup_D \Delta}_{k \text{-times}} \cup_D Y_+.\end{align*}

Similarly, these arise as central fibers of standard families. There is a projective map

(2.2) \begin{equation}p: X[k]_0 \to X_0,\end{equation}

which contracts all the $\Delta$ .

Define the multiplication map

(2.3) \begin{equation}m: {\mathbb A}^{k+1} \to {\mathbb A}^1, \quad (t_0, \ldots, t_k) \mapsto t_0 \cdots t_k. \end{equation}

Let $({\mathbb C}^*)^k$ act on ${\mathbb A}^1$ trivially and on ${\mathbb A}^{k+1}$ such that m is equivariant.

Proposition 2.7 ([Reference LiLi01, Reference Li and WuLW15], [Reference ZhouZho18, Proposition 2.13]). There is a standard family

\begin{align*}\pi_k: X[k]\to {\mathbb A}^{k+1}, \end{align*}

which satisfies the following properties.

1. (1) The total space X[k] is a smooth quasi-projective variety of dimension $(\dim X + k)$ . There is a projective map $p_k: X[k] \to X \times_{{\mathbb A}^1} {\mathbb A}^{k+1}$ .

2. (2) Let $t = (t_0, \ldots, t_k) \in {\mathbb A}^{k+1}$ be a closed point. If $c = t_0 \cdots t_k\neq 0$ , then the fiber of $\pi_k$ over t is isomorphic to $X_c$ , to which the restriction of $p_k$ is the identity map; if $t_0 \cdots t_k = 0$ , and hence $l := \sharp \{i \mid t_i = 0 \} \geqslant 1$ , then the fiber of $\pi_k$ over t is isomorphic to $X[l-1]_0$ , to which the restriction of $p_k$ is the contraction map p (2.2).

3. (3) There is a $({\mathbb C}^*)^k$ -action on X[k], such that $\pi_k$ and $p_k$ are equivariant. For a closed point $t\in {\mathbb A}^{k+1}$ as above with $t_0 \cdots t_k = 0$ , the stabilizer group $\operatorname{Stab}_t ({\mathbb A}^{k+1} ) \cong ({\mathbb C}^*)^{l-1}$ acts on the fiber $X[l-1]_0$ by scaling $(l-1)$ bubble components.

An expanded degeneration can be obtained by gluing two expanded pairs:

\begin{align*}X[k]_0 = Y_-[i] \cup_D Y_+[k-i],\end{align*}

for any $0\leqslant i\leqslant k$ . The following proposition is a description of this fact in terms of standard families.

Proposition 2.8 ([Reference LiLi01, Reference Li and WuLW15], [Reference ZhouZho18, Proposition 2.18]). The composition $\operatorname{pr}_i \circ\, \pi_k: X[k] \to {\mathbb A}^1$ is a simple degeneration, with singular divisor $D_i[k]$ and central fiber $X[k] \times_{{\mathbb A}^1} 0 = X[k]_-^i \cup_{D_i[k]} X[k]_+^i$ , where ^{Footnote 5}

\begin{align*}D_i[k] \cong {\mathbb A}^i \times D \times {\mathbb A}^{k-i} , \quad X[k]_-^i \cong Y_-[i] \times {\mathbb A}^{k-i}, \quad X[k]_+^i \cong {\mathbb A}^i \times Y_+ [k-i]^\circ.\end{align*}

As before, one can define the stack of expanded degenerations by quotienting out the $({\mathbb C}^*)^k$ together with the discrete symmetries, and then taking the direct limit.

${\mathcal C}$ is a smooth Artin stack of dimension 1, and ${\mathcal X}$ is a smooth Artin stack of dimension $\dim X$ . They both admit a map to ${\mathbb A}^1$ . There is a representable and projective map

\begin{align*}p: {\mathcal X} \to X \times_{{\mathbb A}^1} {\mathcal C}, \end{align*}

whose composition with the projection to ${\mathcal C}$ is the family map

\begin{align*}\pi: {\mathcal X} \to {\mathcal C}, \end{align*}

which is flat and also representable.

In general, let $S \to {\mathbb A}^1$ be a scheme over ${\mathbb A}^1$ . Given a ${\mathbb A}^1$ -morphism $S \to {\mathcal C}$ , the pullback of the universal family $\pi$ gives a family

\begin{align*}\pi_S : {\mathcal X}_S \to S, \end{align*}

which we call a family of expanded degenerations over S.

• – $\pi_S: {\mathcal X}_S \to S$ is called Calabi–Yau (CY), if the relative canonical bundle is trivial: $\omega_{{\mathcal X}_S / S} \cong {\mathcal O}_{{\mathcal X}_S}$ ;

• – in particular, $\pi: X \to {\mathbb A}^1$ is called Calabi–Yau (CY) if it is CY over ${\mathbb A}^1$ ; any expanded degeneration $X[k]_0$ is CY if it is CY in the usual sense, i.e. $\omega_{X[k]_0} \cong {\mathcal O}_{X[k]_0}$ .

1. (1) the smooth pairs $(Y_\pm, D)$ in Definition 2.6 are log CY;

2. (2) any family ${\mathcal X}_S \to S$ of expanded degenerations is CY. In particular, the standard families $\pi_k: X[k] \to {\mathbb A}^{k+1}$ are CY; the universal family $\pi: {\mathcal X} \to {\mathcal C}$ is CY; any expanded degeneration $X[k]_0$ is CY.

Proof. For (1), it is straightforward to see that

\begin{align*}\omega_X \cong \omega_{X / {\mathbb A}^1} = {\mathcal O}_X, \quad {\mathcal O}_X (Y_-) \otimes {\mathcal O}_X (Y_+) \cong {\mathcal O}_X. \end{align*}

By the definition of simple degeneration, we have

\begin{align*}{\mathcal O}_X (Y_-) |_{Y_+} \cong {\mathcal O}_{Y_+} (D). \end{align*}

The adjunction formula then implies that

\begin{align*}\omega_{Y_+} \cong \omega_X |_{Y_+} \otimes {\mathcal O}_X (Y_+) |_{Y_+}\cong {\mathcal O}_X (-Y_-) |_{Y_+}\cong {\mathcal O}_{Y_+} (-D).\end{align*}

The case for $Y_-$ is similar.

For (2), as in the proof of Proposition 2.5, it suffices to show that the standard family X[k] is CY. Recall (see [Reference LiLi01]) that X[k] is constructed inductively from $X[k-1]$ . We prove this by induction on k. When $k=1$ , X[1] is constructed in two steps. First, one takes the blow-up $\operatorname{Bl}_{D \times \{0\} } (X \times_{{\mathbb A}^1} {\mathbb A}^2)$ , where the map ${\mathbb A}^2 \to {\mathbb A}^1$ is the multiplication. The exceptional divisor is $\Delta \times_D \Delta$ , with normal bundle ${\mathcal O}_{\Delta \times_D \Delta} (-1, -1)$ . One then blows down a factor of ${\mathbb P}^1$ and obtains X[1]. The resulting map

\begin{align*}p: X[1] \to X \times_{{\mathbb A}^1} {\mathbb A}^2\end{align*}

is a resolution of singularities, with exceptional divisor $\Delta$ , whose normal bundle is ${\mathcal O}_\Delta ({-}1)^{\oplus 2}$ . Therefore, it is a crepant resolution, and X[1] is CY. Replacing X by $X[k-1]$ in the $k=1$ case gives the inductive process.

Next we describe the central fiber of the universal family ${\mathcal X}$ . The multiplication map (2.3) gives a map

\begin{align*}{\mathcal C}\to {\mathbb A}^1, \end{align*}

whose fiber over 0 is denoted by

(2.4) \begin{equation}{\mathcal C}_0 := {\mathcal C} \times_{{\mathbb A}^1} 0. \end{equation}

By construction, this is a limit of the standard smooth charts $[{\mathbb A}^{k+1} \times_{{\mathbb A}^1} 0 / \sim] = [\,\bigcup_{i=1}^{k+1} H_i / \sim]$ , where $H_i = (t_i=0) \subset {\mathbb A}^{k+1}$ is the ith standard hyperplane. Let

(2.5) \begin{equation}{\mathcal C}_0^\dagger \to {\mathcal C}_0 \end{equation}

be the normalization, which means that ${\mathcal C}_0^\dagger := \underrightarrow\lim\, [\,\coprod_{i=1}^{k+1} H_i / \sim]$ . Let

(2.6) \begin{equation}{\mathcal X}_0^\dagger := {\mathcal X} \times_{\mathcal C} {\mathcal C}_0^\dagger \end{equation}

be the fiber product, which by definition is $\underrightarrow\lim\, [\,\coprod_{i=1}^{k+1} X[k] |_{H_i} / \sim]$ .

A geometric interpretation of ${\mathcal X}^\dagger_0$ and ${\mathcal C}^\dagger_0$ can be obtained from Proposition 2.8, which implies that ${\mathcal C}_0^\dagger$ classifies expanded degenerations with a marked nodal divisor, and ${\mathcal X}^\dagger_0$ is the universal divisor. This marking then makes it possible to split the expanded degeneration at the marked nodal divisor, resulting in two expanded pairs.

Corollary 2.12. There is a canonical isomorphism ${\mathcal C}_0^\dagger \cong {\mathcal A} \times {\mathcal A}$ , such thatFootnote ⁶

\begin{align*}{\mathcal X}_0^\dagger \cong ({\mathcal Y}_- \times {\mathcal A} ) \cup_{{\mathcal A} \times D \times {\mathcal A} } ({\mathcal A} \times {\mathcal Y}_+^\circ).\end{align*}

2.3 Topological data

Let B be a scheme and $\pi: V \to B$ be a quasi-projective scheme over B. Let $K_c (V / B)$ be the fiberwise compactly supported K-group, i.e. the Grothendieck group of coherent sheaves F on V, whose supports $\mathop{\rm Supp}\nolimits F$ are proper over B. In particular, there is a well-defined K-theoretic pushforward map

(2.7) \begin{equation}\pi_*^K: K_c (V / B) \to K_0 (B)\end{equation}

to the Grothendieck group $K_0 (B)$ of coherent sheaves on B.

Definition 2.13. $F_1$ , $F_2 \in K_c (V / B)$ are called fiberwise numerically equivalent and are denoted by $F_1 \sim_{\mathrm{num}} F_2$ , if

\begin{align*}\pi_*^K (F_1\otimes E) = \pi_*^K (F_2\otimes E)\in K_0 (B), \end{align*}

for any locally free sheaf E on V. Define the fiberwise compactly supported numerical K-theory as

\begin{align*}K_c^\mathrm{{num}} (V /B):= K_c(V /B) / \sim_{\mathrm{num}}.\end{align*}

When $B = \text{pt}$ , we simply denote it by $K_c^\mathrm{{num}} (V)$ .

Let $\pi: X\to {\mathbb A}^1$ be a simple degeneration, with central fiber $X_0 = Y_- \cup_D Y_+$ and smooth fibers $X_t$ (for any $t\neq 0$ ). Denote the closed embeddings as follows:

\begin{align*}\iota_{D,\pm}: D\hookrightarrow Y_{\pm}, \quad \iota_D: D \hookrightarrow X_0, \quad \iota_\pm: Y_\pm \hookrightarrow X_0, \quad i_t: X_t \hookrightarrow X, \quad t\in \mathbb{A}^1.\end{align*}

Lemma 2.14. The K-theoretic Gysin pullback [Reference FultonFul98, Example 15.1.8]

\begin{align*}i_t^*: K_c(X/ {\mathbb A}^1)\to K_c(X_t), \quad \iota_{D,\pm}^*: K_c(Y_\pm)\to K_c(D)\end{align*}

descends to the corresponding numerical K-theories

(2.8) \begin{equation}i_t^*: K^\mathrm{{num}}_c(X/ {\mathbb A}^1)\to K^\mathrm{{num}}_c(X_t), \quad \iota_{D,\pm}^*: K^\mathrm{{num}}_c(Y_\pm)\to K^\mathrm{{num}}_c(D).\end{equation}

Proof. We prove the case for $i_t^*$ ; the other proof is similar. Take $[F_i]\in K_c(X/ {\mathbb A}^1)$ ( $i=1,2$ ) such that

(2.9) \begin{equation}\pi_*^K (F_1\otimes E) = \pi_*^K (F_2\otimes E)\in K_0 (B) \end{equation}

for any locally free E on X. Then for any locally free sheaf V on $X_t$ ,

\begin{align*}\chi(X_t,i_t^*F_i\otimes V)=\chi(X,i_{t*}(i_t^*F_i \otimes V))=\chi(X,F_i\otimes i_{t*} V), \quad i=1,2. \end{align*}

Since X is smooth, $i_{t*} V$ has a resolution by a perfect complex $E^\bullet$ of locally free sheaves. By (2.9) and a proper pushforward to a point, we know that $\chi(X,F_1\otimes i_{t*}V)=\chi(X,F_2\otimes i_{t*} V)$ .

Using the above Gysin pullback, we define the following.

Definition 2.15. Given $P_0\in K_c^{\mathrm{num}} (X_0)$ , a splitting datum for $P_0$ is a pair $(P_-, P_+)$ , with $P_\pm \in K_c^\mathrm{{num}} (Y_\pm)$ , such that

(2.10) \begin{equation} P_+ |_D = P_- |_D, \quad \iota_{+*} P_+ + \iota_{-*} P_- - \iota_{D*}(P_+ |_D) = P_0,\end{equation}

where $(-)|_D$ denotes the K-theoretic Gysin pullback from $Y_\pm$ to D (2.8). We denote the set of all splitting data of $P_0$ by $\Lambda^{P_0}_{\rm spl}$ .

Recall that a K-group has a topological filtration

\begin{align*}K_{0, \leqslant i} (V) \subset K_0 (V), \end{align*}

which consists of classes [F] whose support has dimension at most i. The filtration restricts to give corresponding compactly supported classes and descends to numerical K-theory classes:

(2.11) \begin{equation} K_{c, \leqslant i} (V), \quad K_{c, \leqslant i}^{\mathrm{num}} (V). \end{equation}

2.4 Relative Hilbert stacks

Let (Y, D) be a smooth pair and $\pi: X \to {\mathbb A}^1$ be a simple degeneration. Let Z be a properly supported closed subscheme on the expanded pair $Y[k]_0$ or the expanded degeneration $X[k]_0$ . We treat Z as an object in the Hilbert scheme by considering the quotient sheaf $ {\mathcal O}_{Y[k]_0} \twoheadrightarrow {\mathcal O}_Z$ . An automorphism of Z is understood as an automorphism induced by the $({\mathbb C}^*)^k$ -action.

• – ${\mathcal O}_Z$ is normal to every singular divisor D in $Y[k]_0$ or $X[k]_0$ , as well as the relative divisor $D[k]_0$ (in the case of $Y[k]_0$ ); and

• – the automorphism group $\operatorname{Aut} ({\mathcal O}_{Y[k]_0} \twoheadrightarrow {\mathcal O}_Z)$ or $\operatorname{Aut} ({\mathcal O}_{X[k]_0} \twoheadrightarrow {\mathcal O}_Z)$ is finite.

The same definition can be made for families of expanded pairs and expanded degenerations, where we require the stable condition on geometric fibers and the flatness of Z over the base. In particular, one can consider stable subschemes on the universal families ${\mathcal Y} \to {\mathcal A}$ and ${\mathcal X} \to {\mathcal C}$ .

1. (1) Given a numerical K-theory class $P \in K_c^{\mathrm{num}} (Y)$ , let $\operatorname{{Hilb}}^P (Y, D)$ be the substack parameterizing stable subschemes Z such that $p^K_*[{\mathcal O}_Z] = P$ . Here $p^K_*$ is the K-theoretic proper pushforward (2.7) of the contraction map $p: Y[k]_0 \to Y$ .

2. (2) Suppose that we are given numerical K-theory classes $P_t \in K_c^\mathrm{num} (X_t)$ for all $t\in {\mathbb A}^1$ . Recall that the standard families $X[k] \to {\mathbb A}^{k+1}$ have fibers either of the form $X[k]_0$ ( $k\geqslant 0$ ), or isomorphic to $X_t$ with $t\neq 0$ . Let $\operatorname{{Hilb}}^{P_t} (X/{\mathbb A}^1)$ be the substack parameterizing stable subschemes Z such that the following hold.

3. – If Z is a subscheme in a fiber of the form $X[k]_0$ ( $k\geqslant 0$ ), then $p^K_* [{\mathcal O}_Z] =P_0$ . Here $p^K_*$ is the K-theoretic proper pushforward (2.7) of the contraction map $p: X[k]_0 \to X_0$ .

4. – If Z is a subscheme in a fiber isomorphic to $X_t$ with $t\in {\mathbb A}^1 \backslash \{0\}$ , then $[{\mathcal O}_Z] = P_t$ .

For any geometric point $s\in S$ , the fiber ${\mathcal X}_s$ satisfies

\begin{align*}{\mathcal X}_s\cong\Bigg\{\begin{array}{l@{\quad \ }l}X[k]_0 \ \mathrm{for}\ \mathrm{some}\ k\geqslant 0 &\mathrm{if} \ s\in (m \circ \xi)^{-1} (0), \\[3pt] X_t & \mathrm{if} \ s\in (m \circ \xi)^{-1} (t), t\neq 0.\end{array}\end{align*}

Let ${\mathcal Z}_s$ be the corresponding stable subscheme in the fiber ${\mathcal X}_s$ . It suffices to prove that for all $s\in (m \circ \xi)^{-1} (0)$ (respectively $s\in (m \circ \xi)^{-1} (t)$ ), the numerical K-theory class $p^K_* [{\mathcal O}_{{\mathcal Z}_s}] \in K_c^\mathrm{num} (X_0)$ (respectively $[{\mathcal O}_{{\mathcal Z}_s}] \in K_c^\mathrm{num} (X_t)$ ) is locally constant in s. By definition, it suffices to prove that for any locally free sheaf E on ${\mathcal X}_s \cong X [k]_0 $ (respectively ${\mathcal X}_s \cong X_t$ ), the function

\begin{align*}\chi (X_0 , p_* {\mathcal O}_{{\mathcal Z}_s} \otimes E ) = \chi (X_0, p_* ( {\mathcal O}_{{\mathcal Z}_s} \otimes p^* E ) ) = \chi (X[k]_0 , {\mathcal O}_{{\mathcal Z}_s} \otimes p^* E ) \quad (\text{respectively } \chi (X_t, {\mathcal O}_{{\mathcal Z}_s} \otimes E ) )\end{align*}

is locally constant in s. But this follows from the flatness of ${\mathcal Z}_S$ over S.

1. (1) The relative Hilbert stack $\operatorname{{Hilb}} (Y, D)$ (respectively $\operatorname{{Hilb}} (X/{\mathbb A}^1))$ is a Deligne–Mumford stack, locally of finite type (respectively locally of finite type over ${\mathbb A}^1)$ .

2. (2) If Y is proper (respectively $\pi: X\to {\mathbb A}^1$ is proper), then $\operatorname{{Hilb}}^P (Y, D)$ (respectively $\operatorname{{Hilb}}^{P_t} (X/{\mathbb A}^1))$ is proper (respectively proper over ${\mathbb A}^1)$ .

• – local curves (see Equation (6.12)); and

• – toric varieties (see e.g. [Reference Cao and KoolCK18, Reference Cao and KoolCK20] for discussions on toric Calabi–Yau 4-folds).

The standard way out in such situations is to find a torus T acting on (Y,D) (respectively T acting on X, trivially on ${\mathbb A}^1$ and $\pi: X\to {\mathbb A}^1$ being equivariant), such that the fixed loci $\operatorname{{Hilb}}^P (Y, D)^T$ (respectively $\operatorname{{Hilb}}^{P_t} (X/{\mathbb A}^1)^T$ ) are proper. Enumerative invariants can then be defined via T-equivariant localization. This is indeed the case in the two situations mentioned above. The reason is that one can then find a natural smooth T-equivariant compactification $\bar Y$ of Y, where the closure $\bar D$ of D is still smooth, such that the T-fixed loci $\operatorname{{Hilb}}^P (Y, D)^T$ form a union of connected components of the T-fixed locus $\operatorname{{Hilb}}^P (\bar Y, \bar D)^T$ , which is proper by Proposition 2.22, and similarly for the $X/\mathbb{A}^1$ case.

For example, in the case of local curves, we can take T to be the torus which acts on fibers, or its Calabi–Yau subtorus.

Several variants are needed for applications in the present paper. For $P \in K_c^{\mathrm{num}} (Y)$ , there is a stack ${\mathcal A}^P$ parameterizing expanded pairs with a weight structure, whose total continuous weight is P. A weight structure on an expanded pair assigns each irreducible component and each singular divisor an element in $K_c^\mathrm{{num}}(Y)$ , so that the summation of all weights is the fixed K-theory class P. The assignment is subject to a continuity condition. We refer an interested reader to [Reference ZhouZho18, Definition 7.2] and [Reference Li and WuLW15, Definition 2.14] for details. Similarly, one has the stack ${\mathcal C}^{P_t}$ . There are étale maps forgetting the weight structure:

\begin{align*}{\mathcal A}^P\to {\mathcal A}, \quad {\mathcal C}^{P_t}\to {\mathcal C}. \end{align*}

Denote the universal families over ${\mathcal A}^P$ and ${\mathcal C}^{P_t}$ by ${\mathcal Y}^P$ and ${\mathcal X}^{P_t}$ , with family maps

(2.12) \begin{equation} {\mathcal X}^{P_t} \to {\mathcal C}^{P_t}, \quad {\mathcal Y}^P \to {\mathcal A}^P. \end{equation}

The Hilbert stack $\operatorname{{Hilb}}^P (Y, D)$ in Definition 2.18 can also be described as the stack parameterizing families of stable subschemes Z in the universal family ${\mathcal Y}^P / {\mathcal A}^P$ satisfying $p^K_*[{\mathcal O}_Z] = P$ . This is because objects in $\operatorname{{Hilb}}^P (Y, D)$ have fixed topological data P, so $\operatorname{{Hilb}}^P (Y, D)\to {\mathcal A}$ factors through ${\mathcal A}^P$ . The argument is similar for $\operatorname{{Hilb}}^{P_t} (X/{\mathbb A}^1)$ .

We also have the weighted versions of (2.4), (2.5), and (2.6):

\begin{align*}{\mathcal C}_0^{P_0} := {\mathcal C}^{P_t} \times_{{\mathbb A}^1} 0, \quad {\mathcal C}_0^{\dagger,P_0}\to {\mathcal C}_0^{P_0}, \quad {\mathcal X}_0^{\dagger,P_0} := {\mathcal X}^{P_t} \times_{{\mathcal C}^{P_t}} {\mathcal C}_0^{\dagger,P_0}. \end{align*}

For each splitting datum $(P_-, P_+)$ of $P_0$ in Definition 2.15, there is an open and closed substack ${\mathcal C}_0^{\dagger, (P_-, P_+)} \subset {\mathcal C}_0^{\dagger, P_0}$ , and we let

\begin{align*}{\mathcal X}_0^{\dagger, (P_-, P_+)}= {\mathcal X}^{P_t} \times_{{\mathcal C}^{P_t}} {\mathcal C}_0^{\dagger, (P_-, P_+)}. \end{align*}

Corollary 2.12 can be refined as follows.

Corollary 2.24 ([Reference Li and WuLW15, Proposition 2.13], [Reference ZhouZho18, Proposition 7.4]). Given a splitting datum $(P_-, P_+)$ of $P_0$ , there is a canonical isomorphism ${\mathcal C}_0^{\dagger, (P_-, P_+)} \cong {\mathcal A}^{P_-} \times {\mathcal A}^{P_+}$ , such that

(2.13) \begin{equation}{\mathcal X}_0^{\dagger, (P_-, P_+)} \cong ({\mathcal Y}_-^{P_-}\times {\mathcal A}^{P_+} ) \cup_{{\mathcal A}^{P_-} \times D \times{\mathcal A}^{P_+} } ({\mathcal A}^{P_-} \times {\mathcal Y}_+^{\circ, P_+})._{\vphantom{\big(}}\end{equation}

Denote by $\Lambda^P_{\rm spl}$ the set of all splitting data of P.

• – we have

  \begin{align*}\bigotimes_{(P_-, P_+) \in \Lambda^{P_0}_{\rm spl} } L_{P_-, P_+} \cong {\mathcal O}_{{\mathcal C}^{P_t}}, \quad \prod_{(P_-, P_+) \in \Lambda^{P_0}_{\rm spl} } s_{P_-, P_+} \cong \pi^{*} t,\end{align*}
  where $\pi: \mathcal{C}^{P_t} \to {\mathbb A}^1 = \operatorname{{Spec}} {\mathbb C}[t]$ is the canonical map; and

• – $\mathcal{C}_0^{\dagger, (P_-, P_+)}$ is the closed substack in $\mathcal{C}^{P_t}$ defined by $s_{P_-, P_+} = 0$ .

The lemma leads to a similar decomposition of Hilbert stacks. More precisely, the Hilbert stack

\begin{align*}\operatorname{{Hilb}}^{P_0} (X/{\mathbb A}^1)_0 := \operatorname{{Hilb}}^{P_t} (X/{\mathbb A}^1) \times_{{\mathbb A}^1} 0\end{align*}

parameterizes stable closed subschemes on singular expanded degenerations. We can consider its normalization $\operatorname{{Hilb}}^{P_0} (X/{\mathbb A}^1 )_0^\dagger$ , which parameterizes stable closed subschemes in the family ${\mathcal X}_0^{\dagger, P_0} \to {\mathcal C}_0^{\dagger, P_0}$ . There are open and closed stacks

\begin{align*}\operatorname{{Hilb}}^{(P_-, P_+)} (X/{\mathbb A}^1)_0^\dagger \subset \operatorname{{Hilb}}^{P_0} (X/{\mathbb A}^1 )_0^\dagger, \end{align*}

indexed by all possible splitting data of $P_0$ , whose union is $\operatorname{{Hilb}}^{P_0} (X/{\mathbb A}^1 )_0^\dagger$ . Since $\operatorname{{Hilb}}^{P_t} (X/{\mathbb A}^1)$ is of finite type, so is $\operatorname{{Hilb}}^{P_0} (X/{\mathbb A}^1 )_0^\dagger$ , and therefore the above union is a finite union.

For each splitting datum, we have the following decomposition.

Proposition 2.26 ([Reference Li and WuLW15, Proposition 2.20], [Reference ZhouZho18, Proposition 7.4]). The decomposition in Corollary 2.24 induces the following commutative diagram.

An alternative viewpoint for the Hilbert scheme is to consider it as a moduli of ideal sheaves. The following lemma shows that this also works in the relative setting.

Now consider F on $X[k]_0$ . For simplicity, we may assume that $k=0$ , i.e. $X[k]_0 = X_0 = Y_- \cup_D Y_+$ . The case for a general k is similar. Since F is normal to D, we have a short exact sequence

\begin{align*}0 \to F \to F |_{Y_-} \oplus F |_{Y_+} \to F |_D \to 0.\end{align*}

Now by [Reference ZhouZho18, Lemma 3.4], both $F |_{Y_\pm }$ are torsion-free (and also of rank 1 with trivial determinant). Therefore, there exists $Z_\pm \subset Y_\pm$ , such that $F |_{Y_\pm} \cong I_{Z_\pm}$ . It is straightforward to see that the normality of $Z_\pm$ to D is equivalent to that of $I_{Z_\pm}$ to D. The $Z_\pm$ then glue up to $Z \subset X_0$ , and we have $F \cong I_Z$ .

3.1 Shifted symplectic structure on $\mathbf{Hilb}(X/{\mathbb A}^1)/\mathcal{C}$

Let $\pi : X \to {\mathbb A}^1$ be a Calabi–Yau simple degeneration (Definitions 2.6, 2.10) and

\begin{align*}\pi: {\mathcal X}\to {\mathcal C}\end{align*}

be the induced map from the universal expanded degeneration ${\mathcal X}$ to the stack ${\mathcal C}$ of expanded degenerations. By Proposition 2.11, we know that

\begin{align*}\omega_{{\mathcal X}/{\mathcal C}}\cong \mathcal{O}_{{\mathcal X}}. \end{align*}

Let $\textbf{RPerf}$ be the derived (classifying) stack of perfect complexes of quasi-coherent sheaves, and $\Omega_{\textbf{RPerf}}$ be the natural 2-shifted symplectic form on $\textbf{RPerf}$ [Reference Pantev, Toën, Vaquie and VezzosiPTV⁺13, Theorem 0.3]. Consider the derived mapping stack over ${\mathcal C}$ :

\begin{align*}\textbf{RPerf}({\mathcal X}/{\mathcal C}):=\textbf{Map}_{\mathrm{dst}/{\mathcal C}}({\mathcal X},\textbf{RPerf}\times {\mathcal C}), \end{align*}

where we treat classical stacks ${\mathcal X}$ and ${\mathcal C}$ as derived stacks by the natural inclusion functor.

There is a determinant map (e.g. [Reference Schürg, Toën and VezzosiSTV15, § 3.1])

\begin{equation*}\det: \textbf{RPerf}({\mathcal X}/{\mathcal C})\to \textbf{RPic}({\mathcal X}/{\mathcal C}) \end{equation*}

to the derived moduli stack of line bundles on ${\mathcal X}/{\mathcal C}$ . The homotopy pullback diagram

(3.1)

defines the derived moduli stack $\textbf{RPerf}({\mathcal X}/{\mathcal C})_{0}$ of perfect complexes on ${\mathcal X}/{\mathcal C}$ with trivial determinant. Let $\textbf{RPerf}({\mathcal X}/{\mathcal C})_{0, \mathop{\rm rk}\nolimits=1}$ be the open and closed substack of perfect complexes with (virtual) rank 1.

Lemma 3.1. The classical truncation

\begin{align*}\mathrm{RPerf}({\mathcal X}/{\mathcal C})_{0,\mathop{\rm rk}\nolimits=1}:=t_0(\mathbf{RPerf}({\mathcal X}/{\mathcal C})_{0, \mathop{\rm rk}\nolimits=1} )\end{align*}

contains $\operatorname{{Hilb}} (X/{\mathbb A}^1)$ as an open substack.

Proof. Let $\mathrm{Perf}({\mathcal X} / {\mathcal C})^{\geqslant 0}_{0, \mathop{\rm rk}\nolimits=1}$ be the open substack of $\mathrm{RPerf}({\mathcal X}/{\mathcal C})_{0, \mathop{\rm rk}\nolimits=1}$ consisting of perfect complexes F, such that $\mathrm{Ext}^i (F, F) = 0$ for $i\lt0$ . We will prove that $\mathrm{Perf}({\mathcal X} / {\mathcal C})^{\geqslant 0}_{0, \mathop{\rm rk}\nolimits=1}$ contains $\operatorname{{Hilb}} (X/{\mathbb A}^1)$ as an open substack. Let ${\mathcal X} \to S$ be a family of expanded degenerations over S, and $Z \subset {\mathcal X}$ be a family of stable subschemes, which gives an object in $\operatorname{{Hilb}} (X/{\mathbb A}^1) (S)$ . According to [Reference WuWu07a, Corollary 5.9] (see also [Reference WuWu07b, Corollary 2.9]), the ideal sheaf $I_Z$ is a fiberwise perfect complex. Since we have assumed that $\operatorname{codim} Z \geqslant 2$ , we obtain a map

\begin{align*}\operatorname{{Hilb}} (X/{\mathbb A}^1) \to \mathrm{Perf}({\mathcal X} / {\mathcal C})^{\geqslant 0}_{0, \mathop{\rm rk}\nolimits=1},\end{align*}

sending Z to $I_Z$ . By Lemma 2.27, an ideal sheaf $I_Z$ in $\operatorname{{Hilb}} (X/{\mathbb A}^1)$ can be characterized as a torsion-free sheaf of rank 1 with trivial determinant. Choosing a relatively ample line bundle on $X/C$ , we may replace the ‘torsion-free’ condition by ‘Gieseker stable’. The substack $\operatorname{{Hilb}} (X/{\mathbb A}^1)$ then consists of perfect complexes whose Tor-amplitudes concentrate at degree 0, i.e. they are actually coherent sheaves, as well as being properly supported, Gieseker stable and stable (in the sense of Definition 2.17). However, these are all open conditions.Footnote ⁷ Hence we obtain an open embedding.

There is a derived enhancement $\mathbf{Hilb} (X/{\mathbb A}^1)$ of $\operatorname{{Hilb}} (X/{\mathbb A}^1)$ making the diagram

(3.2)

a homotopy pullback diagram [Reference Schürg, Toën and VezzosiSTV15, Proposition 2.1].

Similarly, fixing K-theory classes $P_t\in K_{c, \leqslant n -2}^{\mathrm{num}} (X_t)$ for all $t\in \mathbb{A}^1$ , we also have the open substack $\mathbf{Hilb}^{P_t}(X/{\mathbb A}^1)$ of $\mathbf{Hilb}(X/{\mathbb A}^1)$ which gives a derived enhancement of $\operatorname{{Hilb}}^{P_t} (X/{\mathbb A}^1)$ (Definition 2.18).

This follows from a family version of the construction of Preygel, [Reference PreygelPre],Footnote ⁸ which adapts the construction of [Reference Pantev, Toën, Vaquie and VezzosiPTV⁺13, Theorem 2.5] to a moduli of perfect complexes $(\mathcal{O}\to F)$ on a quasi-projective Calabi–Yau variety, with fixed determinant and the support of F being proper. We sketch the argument and adapt it to our case.

Step 1: Pullback. Let $\mathcal{E}$ be the universal family over $\textbf{RPerf}$ . By [Reference Pantev, Toën, Vaquie and VezzosiPTV⁺13, Theorem 2.12], the 2-shifted symplectic form on $\textbf{RPerf}$ is

\begin{align*}\Omega_{\textbf{RPerf}}=\operatorname{ch}_2(\mathcal{E}): \mathbb{C}\to NC(\textbf{RPerf})(2).\end{align*}

Here NC denotes the weighted negative cyclic complex, i.e. $NC^{w}$ used in [Reference Pantev, Toën, Vaquie and VezzosiPTV⁺13, Theorem 2.12], and $\operatorname{ch}_2$ denotes the second Chern character (see [Reference Pantev, Toën, Vaquie and VezzosiPTV⁺13, p. 317]).

Consider the evaluation map

\begin{align*}\mathop{\rm ev}\nolimits: {\mathcal X}\times_{{\mathcal C}}\mathbf{Hilb}(X/{\mathbb A}^1) \to \textbf{RPerf}\times {\mathcal C}, \end{align*}

and the pullback form

\begin{align*}\mathop{\rm ev}\nolimits^*\Omega_{\textbf{RPerf}}=\mathop{\rm ev}\nolimits^*\operatorname{ch}_2(\mathcal{E}):\mathbb{C} \to NC({\mathcal X}\times_{{\mathcal C}}\mathbf{Hilb}(X/{\mathbb A}^1) )(2). \end{align*}

By an abuse of notation, we denote its image in the negatively cyclic complex (relative to ${\mathcal C}$ ) by

\begin{align*}\mathop{\rm ev}\nolimits^*\Omega_{\textbf{RPerf}}=\mathop{\rm ev}\nolimits^*\operatorname{ch}_2(\mathcal{E}):\mathbb{C} \to NC(({\mathcal X}\times_{{\mathcal C}}\mathbf{Hilb}(X/{\mathbb A}^1))/{\mathcal C})(2). \end{align*}

Step 2: Lift to a compactly supported form. Let $\mathcal{F}:=\mathop{\rm ev}\nolimits^*\mathcal{E}$ and $\mathcal{Z}\subset {\mathcal X}\times_{{\mathcal C}}\mathbf{Hilb}(X/{\mathbb A}^1)$ be the universal substack. This is the support of the cone of the trace map

\begin{align*}\mathbf{R} \mathcal{H}om_{{\mathcal X}\times_{{\mathcal C}}\mathbf{Hilb}(X/{\mathbb A}^1)}(\mathcal{F},\mathcal{F}) \stackrel{\operatorname{tr}}{\to} \mathcal{O}_{{\mathcal X}\times_{{\mathcal C}}\mathbf{Hilb}(X/{\mathbb A}^1) }, \end{align*}

which is proper over $\mathbf{Hilb}(X/{\mathbb A}^1)$ . In other words, outside the support $\mathcal{Z}$ , the trace map is an isomorphism:

\begin{align*}\operatorname{tr}: \mathbf{R} \mathcal{H}om_{{\mathcal X}\times_{{\mathcal C}}\mathbf{Hilb}(X/{\mathbb A}^1)}(\mathcal{F},\mathcal{F})|_{({\mathcal X}\times_{{\mathcal C}}\mathbf{Hilb}(X/{\mathbb A}^1))\setminus \mathcal{Z}}\cong \mathcal{O}. \end{align*}

We now construct a lift of $\mathop{\rm ev}\nolimits^*\Omega_{\textbf{RPerf}}$ to a form which is supported on $\mathcal{Z}$ .

Recall that, for a map $M_X\to M$ of derived stacks over some base derived stack B and a closed derived substack $K\subset M_X$ with open immersion $i: M_X\setminus K\to M_X$ , we define

(3.3) \begin{equation}NC_{K}(M_X/B)(p):=\mathrm{fib}\{i^*: NC_{}(M_X/B)(p)\to NC_{}((M_X\setminus K)/B)(p)\}. \end{equation}

A section of $NC_{K}(M_X/B)(p)$ is called a closed p-form on $M_X/B$ supported on K.

Let $M_X={\mathcal X}\times_{{\mathcal C}}\mathbf{Hilb}(X/{\mathbb A}^1)$ , $M=\mathbf{Hilb}(X/{\mathbb A}^1)$ and $B={\mathcal C}$ in the above. There is a canonical null-homotopy $\operatorname{ch}_2(\mathcal{O}_{\text{pt}})\sim 0$ in $NC(\text{pt})(2)$ . Pulling it back along the natural projection $p: ({\mathcal X}\times_{{\mathcal C}}\mathbf{Hilb}(X/{\mathbb A}^1))\setminus \mathcal{Z}\to \text{pt}$ gives a null-homotopy in $NC(({\mathcal X}\times_{{\mathcal C}}\mathbf{Hilb}(X/{\mathbb A}^1))\setminus \mathcal{Z})(2)$ :

$\operatorname{ch}_2(\mathcal{F}|_{({\mathcal X}\times_{{\mathcal C}}\mathbf{Hilb}(X/{\mathbb A}^1))\setminus \mathcal{Z})}=\operatorname{ch}_2(\mathcal{O})=p^*\operatorname{ch}_2(\mathcal{O}_{\text{pt}})\sim 0.$

The fiber sequence

\[ NC_\mathcal{Z}({\mathcal X}\times_{{\mathcal C}}\mathbf{Hilb}(X/{\mathbb A}^1))(2)\to NC({\mathcal X}\times_{{\mathcal C}}\mathbf{Hilb}(X/{\mathbb A}^1))(2)\to NC(({\mathcal X}\times_{{\mathcal C}}\mathbf{Hilb}(X/{\mathbb A}^1))\setminus \mathcal{Z})(2) \]

then turns the null-homotopy above into a canonical lift

\[ \widetilde{\operatorname{ch}_2}({\mathcal F}):\mathbb{C}\to NC_\mathcal{Z}({\mathcal X}\times_{{\mathcal C}}\mathbf{Hilb}(X/{\mathbb A}^1))(2). \]

By an abuse of notation, we write this lift in the relative negatively cyclic homology as

\begin{align*}\mathop{\rm ev}\nolimits^*\Omega_{\textbf{RPerf}}:=\widetilde{\operatorname{ch}_2}({\mathcal F})\in HN_{{\mathcal Z}}^0(({\mathcal X}\times_{{\mathcal C}}\mathbf{Hilb}(X/{\mathbb A}^1))/{\mathcal C})(2). \end{align*}

Step 3: Integration. For any quasi-coherent complex F on $\mathbf{Hilb}(X/{\mathbb A}^1)$ , there is a trace map:Footnote ⁹

(3.4) \begin{equation}\operatorname{tr}_{{\mathcal X}/{\mathcal C}}: \pi_{M*}\circ \underline{\mathbf{R}\Gamma_{\mathcal{Z}}}(\omega_{{\mathcal X}/{\mathcal C}}[n]\boxtimes F)\to F, \end{equation}

where $\pi_M: {\mathcal X}\times_{{\mathcal C}}\mathbf{Hilb}(X/{\mathbb A}^1)\to \mathbf{Hilb}(X/{\mathbb A}^1)$ is the projection and

\begin{align*}\underline{\mathbf{R}\Gamma_{\mathcal{Z}}}: \mathop{\rm QCoh}\nolimits({\mathcal X}\times_{{\mathcal C}}\mathbf{Hilb}(X/{\mathbb A}^1))\to \mathop{\rm QCoh}\nolimits({\mathcal X}\times_{{\mathcal C}}\mathbf{Hilb}(X/{\mathbb A}^1)) \end{align*}

is the functorFootnote ¹⁰ of the section with support $\mathcal{Z}$ . By the argument in [Reference PreygelPre, Proof of Corollary A. 0.12], the complex F may be taken to be the de Rham complex. The same construction as in [Reference PreygelPre, Theorem 3.0.6] shows that the map $\operatorname{tr}_{{\mathcal X}/{\mathcal C}}$ and a trivialization

\begin{align*}\mathrm{vol}_{{\mathcal X}/{\mathcal C}}: \mathcal{O}_{{\mathcal X}}\stackrel{\cong}{\to} \omega_{{\mathcal X}/{\mathcal C}}, \end{align*}

defines an integration map

(3.5) \begin{equation}\int_{{\mathcal X}/{\mathcal C}}\mathrm{vol}_{{\mathcal X}/{\mathcal C}}\wedge(-):NC_{{\mathcal Z}}(({\mathcal X}\times_{{\mathcal C}}\mathbf{Hilb}(X/{\mathbb A}^1))/{\mathcal C})(2)\to NC_{}(\mathbf{Hilb}(X/{\mathbb A}^1)/{\mathcal C})[-n](2). \end{equation}

We define

\begin{align*}\Omega_{\mathbf{Hilb}(X/{\mathbb A}^1)/{\mathcal C}}:=\int_{{\mathcal X}/{\mathcal C}}\mathrm{vol}_{{\mathcal X}/{\mathcal C}}\wedge(\mathop{\rm ev}\nolimits^*\Omega_{\textbf{RPerf}})\in HN_{}^{-n}(\mathbf{Hilb}(X/{\mathbb A}^1)/{\mathcal C})(2). \end{align*}

The verification of non-degeneracy is a routine check using Grothendieck–Serre duality.

3.2 Lagrangian structure on $(\mathbf{Hilb} (Y, D)\to \mathbf{Hilb} ({\mathcal D}))/{\mathcal A}$

Let (Y,D) be a smooth log Calabi–Yau pair (Definition 2.4) so that D is a Calabi–Yau variety. Consider the induced map

\begin{align*}\pi : {\mathcal Y} \to {\mathcal A}\end{align*}

from the universal expanded pair ${\mathcal Y}$ to the stack ${\mathcal A}$ of expanded pairs, with universal relative divisor

\begin{align*} i_{{\mathcal D}\to {\mathcal Y}}: {\mathcal D}\cong D\times {\mathcal A} \hookrightarrow {\mathcal Y}\end{align*}

as in Definition 2.3.

Consider the restriction map between derived mapping stacks:

\begin{align*}r_{{\mathcal D}\to {\mathcal Y}}: \textbf{RPerf}({\mathcal Y}/{\mathcal A}) \to\textbf{RPerf}({\mathcal D}/{\mathcal A}).\end{align*}

This map is compatible with determinant maps, i.e. the diagram

is commutative. Therefore we have an induced restriction map

\begin{align*}r_{{\mathcal D}\to {\mathcal Y}}: \textbf{RPerf}({\mathcal Y}/{\mathcal A})_0 \to\textbf{RPerf}({\mathcal D}/{\mathcal A})_0, \end{align*}

of derived moduli stack of perfect complexes with trivial determinant.

As in Lemma 3.1, $\mathbf{Hilb}(Y,D)$ is an open substack in $\textbf{RPerf}({\mathcal D}/{\mathcal A})_{0, \mathop{\rm rk}\nolimits=1}$ . Further restriction to the open substack $\mathbf{Hilb}(Y,D)$ gives

(3.6) \begin{equation}r_{{\mathcal D}\to {\mathcal Y}}:\mathbf{Hilb}(Y,D)\to \mathbf{Hilb} ({\mathcal D})=\mathbf{Hilb} (D)\times {\mathcal A}. \end{equation}

For a fixed K-theory class $P\in K_{c, \leqslant \dim_{\mathbb{C}}Y-2}^{\mathrm{num}} (Y)$ , we also have the open substack $\mathbf{Hilb}^P(Y,D)$ of $\mathbf{Hilb}(Y,D)$ which gives a derived enhancement of $\operatorname{{Hilb}}^P (Y,D)$ (Definition 2.18). There is a similar restriction map

(3.7) \begin{equation}r_{{\mathcal D}\to {\mathcal Y}}:\mathbf{Hilb}^P(Y,D)\to \mathbf{Hilb}^{P|_D}(D)\times {\mathcal A}^P. \end{equation}

Fixing a trivialization $\mathrm{vol}_D: \mathcal{O}_D \stackrel{\cong}{\to} \omega_{D}$ , by a base change to ${\mathcal D}\cong D\times{\mathcal A}$ , we obtain a trivialization

(3.8) \begin{equation}\mathrm{vol}_{{\mathcal D}/{\mathcal A}}:\mathcal{O}_{{\mathcal D}}\cong \omega_{{\mathcal D}/{\mathcal A}}.\end{equation}

As in Theorem 3.2, $\mathrm{vol}_D$ induces a $(2-\dim_{\mathbb{C}}D)$ -shifted symplectic structure on $\mathbf{Hilb}(D)$ . Pulling back to $(\mathbf{Hilb}(D)\times {\mathcal A})/{\mathcal A}$ , we obtain a $(2-\dim_{\mathbb{C}}D)$ -shifted symplectic structure on $(\mathbf{Hilb}(D)\times {\mathcal A})/{\mathcal A}$ .

We have a commutative diagram

(3.9)

where $\mathop{\rm ev}\nolimits_{Y,D} and \mathop{\rm ev}\nolimits_{D}$ are evaluation maps. By Proposition 2.5, ${\mathcal D}\in|\omega_{{\mathcal Y}/{\mathcal A}}^{-1}|$ is an anticanonical divisor of ${\mathcal Y}/{\mathcal A}$ , and therefore we have the following fiber sequence:

(3.10) \begin{equation} \omega_{{\mathcal Y}/{\mathcal A}}\to \mathcal{O}_{{\mathcal Y}}\to i_{{\mathcal D}\to {\mathcal Y}*}\mathcal{O}_{{\mathcal D}}.\end{equation}

This follows from adapting the argument of [Reference CalaqueCal15, §§ 2.2.3 and 3.2.1]Footnote ¹¹ to the family case.

Proof. We prove the claim for (3.6). The other case follows in a similar way. Let

\begin{align*}\mathcal{Z}_{Y,D}\subset {\mathcal Y}\times_{{\mathcal A}}\mathbf{Hilb}(Y,D)\end{align*}

be the universal substack. It is the support of the cone of the trace map ( $\mathcal{F}_{Y,D}:=\mathop{\rm ev}\nolimits_{Y,D}^*\mathcal{E}$ ):

\begin{align*}\mathbf{R} \mathcal{H}om_{{\mathcal Y}\times_{{\mathcal A}}\mathbf{Hilb}(Y,D)}(\mathcal{F}_{Y,D},\mathcal{F}_{Y,D}) \stackrel{\operatorname{tr}}{\to} \mathcal{O}_{{\mathcal Y}\times_{{\mathcal A}}\mathbf{Hilb}(Y,D)}, \end{align*}

which is proper over $\mathbf{Hilb}(Y,D)$ . We have the following commutative diagrams (with the three boxes being homotopy pullback diagrams).

(3.11)

The vertical maps in the lower half of the diagram are natural projections, $\mathcal{Z}_{D}\subset {\mathcal D}\times_{{\mathcal A}} (\mathbf{Hilb}(D)\times {\mathcal A})$ is the universal substack of $\mathbf{Hilb}(D)\times {\mathcal A}$ and ${\mathcal Z}$ is given by the pullback diagram.

A base change implies that for any $F\in\mathop{\rm QCoh}\nolimits(\mathbf{Hilb}(Y,D))$ , $G\in \mathop{\rm QCoh}\nolimits(\mathbf{Hilb}(D)\times{\mathcal A})$ , we have

(3.12) \begin{align}\pi^{{\mathcal Y}}_{M_{Y,D}*}\underline{\mathbf{R}\Gamma_{\mathcal{Z}_{Y,D}}}(i_{{\mathcal D}\to {\mathcal Y}*}\mathcal{O}_{{\mathcal D}}\boxtimes F)&\cong\pi^{{\mathcal Y}}_{M_{Y,D}*}i_{{\mathcal D}\to {\mathcal Y}*}\underline{\mathbf{R}\Gamma_{\mathcal{Z}_{}}}(\mathcal{O}_{{\mathcal D}}\boxtimes F)\nonumber \\&\cong \pi^{{\mathcal D}}_{M_{Y,D}*}\underline{\mathbf{R}\Gamma_{\mathcal{Z}_{}}}(\mathcal{O}_{{\mathcal D}}\boxtimes F), \nonumber\\\pi^{{\mathcal D}}_{M_{Y,D}*}\underline{\mathbf{R}\Gamma_{\mathcal{Z}_{}}}(\mathcal{O}_{{\mathcal D}}\boxtimes r_{{\mathcal D}\to {\mathcal Y}}^*G) &\cong\pi^{{\mathcal D}}_{M_{Y,D}*}\underline{\mathbf{R}\Gamma_{\mathcal{Z}_{}}}\,r_{{\mathcal D}\to {\mathcal Y}}^*(\mathcal{O}_{{\mathcal D}}\boxtimes G) \nonumber\\&\cong \pi^{{\mathcal D}}_{M_{Y,D}*}\,r_{{\mathcal D}\to {\mathcal Y}}^*\,\underline{\mathbf{R}\Gamma_{\mathcal{Z}_{D}}}(\mathcal{O}_{{\mathcal D}}\boxtimes G) \nonumber\\&\cong r_{{\mathcal D}\to {\mathcal Y}}^*\,\pi_{M_{D}*}\,\underline{\mathbf{R}\Gamma_{\mathcal{Z}_{D}}}(\mathcal{O}_{{\mathcal D}}\boxtimes G)\end{align}

(3.13)

Applying $\pi^{{\mathcal Y}}_{M_{Y,D}*}\underline{\mathbf{R}\Gamma_{\mathcal{Z}_{Y,D}}}$ to (3.10), we obtain, for any $G\in \mathop{\rm QCoh}\nolimits(\mathbf{Hilb}(D)\times{\mathcal A})$ a commutative diagram

(3.14)

where $\operatorname{tr}_{(-)}$ is the trace map (3.4). The top and bottom row of the diagram are both fiber sequences.

Let $\mathcal{E}$ be the universal family over $\textbf{RPerf}$ . Then

\begin{align*}\Omega_{\textbf{RPerf}}=\operatorname{ch}_2(\mathcal{E})\in HN^0(\textbf{RPerf})(2)\cong HN^0((\textbf{RPerf}\times {\mathcal A})/ {\mathcal A})(2). \end{align*}

The pullback of the shifted symplectic form of $\mathbf{Hilb}(D)\times {\mathcal A}$ to $\mathbf{Hilb}(Y,D)$ is

(3.15) \begin{align}r_{{\mathcal D}\to {\mathcal Y}}^*\int_{{\mathcal D}/{\mathcal A}}\mathrm{vol}_{{\mathcal D}/{\mathcal A}}\wedge(\mathop{\rm ev}\nolimits_D^*\Omega_{\textbf{RPerf}})&=\int_{{\mathcal D}/{\mathcal A}}(\operatorname{id}\times r_{{\mathcal D}\to {\mathcal Y}})^*(\mathrm{vol}_{{\mathcal D}/{\mathcal A}}\wedge(\mathop{\rm ev}\nolimits_D^*\Omega_{\textbf{RPerf}})) \nonumber\\&=\int_{{\mathcal D}/{\mathcal A}}\mathrm{vol}_{{\mathcal D}/{\mathcal A}}\wedge i_{{\mathcal D}\to {\mathcal Y}}^*(\mathop{\rm ev}\nolimits_{Y,D}^*\Omega_{\textbf{RPerf}})\end{align}

where the second equality uses the commutativity of diagram (3.9), and the upper boxes in diagram (3.11) (which ensures the compactly supported lifts also coincide).

To construct a null-homotopy of (3.15), it suffices to construct a null-homotopy of the map (in the equation below, $d=\dim_{\mathbb C} D$ ):

(3.16) \begin{equation}\int_{{\mathcal D}/{\mathcal A}}\mathrm{vol}_{{\mathcal D}/{\mathcal A}}\wedge i_{{\mathcal D}\to {\mathcal Y}}^*(-):NC_{{\mathcal Z}_{Y,D}}(({\mathcal Y}\times_{{\mathcal A}}\mathbf{Hilb}(Y,D))/{\mathcal A})(2)\to NC_{}(\mathbf{Hilb}(Y,D)/{\mathcal A})[-d](2). \end{equation}

This is provided by $\operatorname{tr}_{{\mathcal Y}/{\mathcal A}}$ in diagram (3.14) following the argument in [Reference CalaqueCal15, §§ 3.2.1 and 2.2.3].

3.3 Composition with a Lagrangian fibration

The restriction map $r_{{\mathcal D}\to {\mathcal Y}}$ has a Lagrangian structure (Theorem 3.3). Fix a numerical K-theory class $P_D\in K_{c,\leqslant \dim_{\mathbb{C}}D-2}^\mathrm{{num}}(D)$ and consider the open substack $\mathbf{Hilb}^{P_D}(D)$ of $\mathbf{Hilb} (D)$ . We are interested in the case when $\mathbf{Hilb}^{P_D}(D)$ has a Lagrangian fibration structure in the following sense.

Definition 3.4 [Reference CalaqueCal19, Definition 1.9]. Let $\pi:M\to N$ be a map of derived Artin stacks locally of finite presentations. The structure of an n-shifted Lagrangian fibration on $\pi$ is an n-shifted symplectic structure $\Omega_{M}$ on M together with a null-homotopy of its image under the natural map $\mathcal{A}^{2,\textit{cl}}(M,n)\to \mathcal{A}^{2,\textit{cl}}(M/N,n)$ , such that the induced sequence

(3.17) \begin{equation}{\mathbb T}_{M/N}\to {\mathbb T}_M \stackrel{\Omega_{M}}{\cong} {\mathbb L}_M[n] \to {\mathbb L}_{M/N}[n] \end{equation}

is a fiber sequence.

Here is an example of Lagrangian fibration (in the $n=-1$ case).

Example 3.6 Let W be a smooth complex algebraic variety and $\phi$ a regular function. We define its derived critical locus $\textbf{Crit}(\phi)$ by the following homotopy pullback diagram.

$\textbf{Crit}(\phi)$ has a canonical $({-}1)$ -shifted symplectic structure by the Lagrangian intersection theorem of [Reference Pantev, Toën, Vaquie and VezzosiPTV⁺13, Theorem 0.5]. The natural embedding

\begin{align*}\textbf{Crit}(\phi)\to W\end{align*}

is a $({-}1)$ -shifted Lagrangian fibration [Reference GrataloupGra22, Remark 3.12].

Proof. This is a relative version of [Reference SafronovSaf23, Proposition 1.10]. We have the following homotopy pullback diagram.

Note that $L\times_A B \to M\times_A B $ and $M\to M\times_A B$ have n-shifted Lagrangian structures relative to B. As a Lagrangian intersection, we know that $L\to B$ has an induced $(n-1)$ -shifted symplectic structure.

As a consequence, we have the following proposition.

Proposition 3.8. Fix $P\in K_{c, \leqslant \dim_{\mathbb{C}}Y-2}^{\mathrm{num}} (Y)$ and $P_D:=P|_D\in K_c^{\mathrm{num}} (D)$ . Assume that the shifted symplectic stack $\mathbf{Hilb}^{P_D}(D)$ admits a Lagrangian fibration structure $\mathbf{Hilb}^{P_D} (D)\to W$ . Then its composition with $r_{{\mathcal D}\to {\mathcal Y}}$ (3.7),

\begin{align*}\mathbf{Hilb}^P(Y,D)\to (W \times {\mathcal A}^P),\end{align*}

has an induced $(1-\dim_{\mathbb{C}}D)$ -shifted symplectic structure.

Example 3.10. When $D=\mathbb{C}^3$ , it is believed that one has an equivalence of the $({-}1)$ -shifted symplectic derived scheme

(3.18) \begin{equation}\mathbf{Hilb}^n(D)\cong \textbf{Crit}(\phi: W\to \mathbb{C}). \end{equation}

Here W is the non-commutative Hilbert scheme

(3.19) \begin{equation}W=(\operatorname{{Hom}}(\mathbb{C}^n,\mathbb{C}^n)^{\times 3}\times \mathbb{C}^n)/\!\!/{\operatorname{GL}}(n), \end{equation}

and $\phi$ is the potential function:

\begin{align*}\phi(b_1,b_2,b_3,v)=\operatorname{tr} (b_1[b_2,b_3]). \end{align*}

By Example 3.6, $\mathbf{Hilb}^n(D)\to W$ then has a Lagrangian fibration structure.

4.1 Orientation on $(\operatorname{{Hilb}}(X/{\mathbb A}^1)\to {\mathcal C})$

Recall that a Calabi–Yau simple degeneration (Definitions 2.6 and 2.10) of quasi-projective n-folds is a family

\begin{align*}\pi : X \to {\mathbb A}^1, \end{align*}

where the generic fibers of $\pi$ are smooth quasi-projective Calabi–Yau n-folds. Associated to this data, using § 2.2, we have the stack ${\mathcal C}$ of expanded degeneration with universal family

\begin{align*}\pi: {\mathcal X} \to {\mathcal C}. \end{align*}

By Theorem 3.2, $\mathbf{Hilb}(X/{\mathbb A}^1)/{\mathcal C}$ has a $(2-n)$ -shifted symplectic structure, which induces an isomorphism

(4.1) \begin{equation}{\mathbb T}_{\mathbf{Hilb}(X/{\mathbb A}^1)/{\mathcal C}}\cong {\mathbb L}_{\mathbf{Hilb}(X/{\mathbb A}^1)/{\mathcal C}}[2-n]. \end{equation}

When n is even, taking the determinant induces an isomorphism

\begin{equation*}\det({\mathbb T}_{\mathbf{Hilb}(X/{\mathbb A}^1)/{\mathcal C}})^{\otimes 2}\cong \mathcal{O}_{\mathbf{Hilb}(X/{\mathbb A}^1)}. \end{equation*}

Here $\det({\mathbb T}_{\mathbf{Hilb}(X/{\mathbb A}^1)/{\mathcal C}})$ is the determinant line bundle. Restricting to the classical truncation $\operatorname{{Hilb}}(X/{\mathbb A}^1)=t_0(\mathbf{Hilb}(X/{\mathbb A}^1))$ , we obtain

(4.2) \begin{equation}\det({\mathbb T}_{\mathbf{Hilb}(X/{\mathbb A}^1)/{\mathcal C}}|_{\operatorname{{Hilb}}(X/{\mathbb A}^1)})^{\otimes 2}\cong \mathcal{O}_{\operatorname{{Hilb}}(X/{\mathbb A}^1)}. \end{equation}

By the construction of the shifted symplectic structure in Theorem 3.2, this isomorphism agrees with the one provided by the Grothendieck–Serre duality.

As in [Reference Borisov and JoyceBJ17, § 2.4], [Reference Cao, Gross and JoyceCGJ20, § 1.4], and [Reference Cao and LeungCL14, Reference Cao and LeungCL17a], we define the following.

Remark 4.2. In this paper, we focus on the case of $n=4$ , and a generic fiber of

\begin{align*}\operatorname{{Hilb}}(X/{\mathbb A}^1)\to {\mathcal C}\end{align*}

is the Hilbert scheme on a smooth quasi-projective Calabi–Yau 4-fold. The restriction of the isomorphism (4.2) to such a fiber has a square root under some assumptions [Reference Cao, Gross and JoyceCGJ20, Reference Cao, Gross and JoyceCGJ25, Reference Joyce and UpmeierJU25].Footnote ¹² The argument in the papers cited in the previous sentence is differential geometric in nature and uses gauge theoretic moduli spaces. Although it might be possible to adapt the argument to a family, here we provide a canonical orientation that is useful for calculations on (1.11). This is motivated by [Reference CaoCao20] and [Reference Cao, Kool and MonavariCKM22b, § 4.2].

Theorem 4.3. Let $U\to {\mathbb A}^1$ be a simple degeneration of 3-folds, and

\begin{align*}X=\operatorname{Tot}(\omega_{U/{\mathbb A}^1})\to {\mathbb A}^1\end{align*}

the associated simple degeneration of 4-folds. Then $\operatorname{{Hilb}}(X/{\mathbb A}^1)/{\mathcal C}$ has an orientation. Similarly, $\operatorname{{Hilb}}^{P_t}(X/{\mathbb A}^1)/{\mathcal C}^{P_t}$ has an orientation.

Proof. Let $M_{\mathrm{cpt}}({\mathcal X}/{\mathcal C})$ be the moduli stack of sheaves on ${\mathcal X}$ with support proper over $ {\mathcal C}$ . Consider the forgetful map

(4.3) \begin{equation}f: \operatorname{{Hilb}}(X/{\mathbb A}^1)\to M_{\mathrm{cpt}}({\mathcal X}/{\mathcal C}), \quad (\mathcal{O}\to F) \mapsto F.\end{equation}

Let $\bar{f}:=f\times_{\operatorname{id}_{{\mathcal C}}} \operatorname{id}_{{\mathcal X}}$ , and let

\begin{align*}\pi_{M_X}:M_{\mathrm{cpt}}({\mathcal X}/{\mathcal C})\times_{{\mathcal C}} {\mathcal X}\to M_{\mathrm{cpt}}({\mathcal X}/{\mathcal C}), \quad \pi_{H_X}: \operatorname{{Hilb}}(X/{\mathbb A}^1)\times_{{\mathcal C}} {\mathcal X}\to \operatorname{{Hilb}}(X/{\mathbb A}^1)\end{align*}

be the natural projections.

Let ${\mathbb F}$ be the universal sheaf on $M_{\mathrm{cpt}}({\mathcal X}/{\mathcal C})\times_{{\mathcal C}} {\mathcal X}$ and ${\mathbb I}=(\mathcal{O}_{\operatorname{{Hilb}}(X/{\mathbb A}^1)\times_{{\mathcal C}} {\mathcal X}}\to \bar{f}^*{\mathbb F})$ be the universal pair. Then we have the natural isomorphism

(4.4) \begin{align}\det({\mathbb T}_{\mathbf{Hilb}(X/{\mathbb A}^1)/{\mathcal C}}|_{\operatorname{{Hilb}}(X/{\mathbb A}^1)})&\cong \det(\pi_{H_X*}\mathbf{R}{\mathcal H} om({\mathbb I},{\mathbb I})_0[1]) \nonumber\\&\cong \det(\pi_{H_X*}\mathbf{R}{\mathcal H} om(\bar{f}^*{\mathbb F},\bar{f}^*{\mathbb F})[1]) \nonumber\\&\cong f^*\det(\pi_{M_X*}\mathbf{R}{\mathcal H} om({\mathbb F},{\mathbb F})[1]),\end{align}

where the second isomorphism uses the distinguished triangles (e.g. [Reference Cao, Maulik and TodaCMT22, § 3.2])

\begin{align*}\mathbf{R}{\mathcal H} om({\mathbb I},\bar{f}^*{\mathbb F}) &\to \mathbf{R}{\mathcal H} om({\mathbb I},{\mathbb I})_0[1]\to \mathbf{R}{\mathcal H} om(\bar{f}^*{\mathbb F}, \mathcal{O})[2], \\ \mathbf{R}{\mathcal H} om(\mathcal{O},\bar{f}^*{\mathbb F})& \to \mathbf{R}{\mathcal H} om({\mathbb I}, \bar{f}^*{\mathbb F})\to \mathbf{R}{\mathcal H} om(\bar{f}^*{\mathbb F},\bar{f}^*{\mathbb F})[1], \end{align*}

and Grothendieck–Serre duality

\begin{align*}\pi_{H_X*}\mathbf{R}{\mathcal H} om(\mathcal{O},\bar{f}^*{\mathbb F})\cong (\pi_{H_X*}\mathbf{R}{\mathcal H} om(\bar{f}^*{\mathbb F}, \mathcal{O}))^\vee[-4]. \end{align*}

As in (4.2), we also have

(4.5) \begin{equation}\det(\pi_{M_X*}\mathbf{R}{\mathcal H} om({\mathbb F},{\mathbb F})[1])^{\otimes 2}\cong \mathcal{O}, \end{equation}

given by Grothendieck–Serre duality on $\pi_{M_X*}\mathbf{R}{\mathcal H} om({\mathbb F},{\mathbb F})[1]$ . Notice that a choice of square root of (4.5) gives a choice of square root of (4.2) under the pullback map (4.4) (e.g. [Reference Cao, Maulik and TodaCMT22, Theorem 6.3]).

We provide two (equivalent) constructions of a square root of (4.5). As in Theorem 3.2 (see also [Reference Brav and DyckerhoffBD21]), by adapting [Reference Pantev, Toën, Vaquie and VezzosiPTV⁺13, Theorem 2.5] to (relatively) compactly supported sheaves, the natural derived enhancement $\textbf{M}_{\mathrm{cpt}}({\mathcal X}/{\mathcal C})$ of $M_{\mathrm{cpt}}({\mathcal X}/{\mathcal C})$ has a shifted symplectic structure relative to ${\mathcal C}$ . Since ${\mathcal X}=\operatorname{Tot}(\omega_{{\mathcal U}/{\mathcal C}})$ , $\textbf{M}_{\mathrm{cpt}}({\mathcal X}/{\mathcal C})$ is equivalent to the $(-2)$ -shifted symplectic cotangent bundle (relative to ${\mathcal C}$ ) of the derived moduli stack $\textbf{M}_{\mathrm{cpt}}({\mathcal U}/{\mathcal C})$ of compactly supported sheaves on ${\mathcal U}/{\mathcal C}$ [Reference Bozec, Calaque and ScherotzkeBCS24, Theorem 6.17]. Let $\textbf{p}$ be the projection

\begin{align*}\textbf{p}:\textbf{M}_{\mathrm{cpt}}({\mathcal X}/{\mathcal C})\to \textbf{M}_{\mathrm{cpt}}({\mathcal U}/{\mathcal C}). \end{align*}

Then there is a fiber sequence

(4.6) \begin{equation}\textbf{p}^*{\mathbb L}_{\textbf{M}_{\mathrm{cpt}}({\mathcal U}/{\mathcal C})/{\mathcal C}}[-2]\cong{\mathbb T}_{\textbf{p}}\to {\mathbb T}_{\textbf{M}_{\mathrm{cpt}}({\mathcal X}/{\mathcal C})/{\mathcal C}}\to \textbf{p}^*{\mathbb T}_{\textbf{M}_{\mathrm{cpt}}({\mathcal U}/{\mathcal C})/{\mathcal C}}. \end{equation}

Restricting to the classical truncation and taking the determinant, we obtain

(4.7) \begin{equation}\det({\mathbb T}_{\textbf{M}_{\mathrm{cpt}}({\mathcal X}/{\mathcal C})/{\mathcal C}}|_{M_{\mathrm{cpt}}({\mathcal X}/{\mathcal C})})\cong \mathcal{O}, \end{equation}

by Grothendieck–Serre duality.

Alternatively, let $\pi: {\mathcal X}=\operatorname{Tot}(\omega_{{\mathcal U}/{\mathcal C}}) \to {\mathcal U}$ be the projection, and consider the following diagram.

Spectral construction gives a distinguished triangle

(4.8) \begin{equation}\pi_{M_U*}\mathbf{R}{\mathcal H} om(\bar{\pi}_*{\mathbb F},\bar{\pi}_*{\mathbb F}\boxtimes \omega_{{\mathcal U}/{\mathcal C}})[-1]\to \pi_{M_X*}\mathbf{R}{\mathcal H} om({\mathbb F},{\mathbb F})\to \pi_{M_U*}\mathbf{R}{\mathcal H} om(\bar{\pi}_*{\mathbb F},\bar{\pi}_*{\mathbb F}). \end{equation}

Taking the determinant gives an isomorphism

(4.9) \begin{align}\det(\pi_{M_X*}\mathbf{R}{\mathcal H} om({\mathbb F},{\mathbb F})[1])& \cong \frac{\det(\pi_{M_U*}\mathbf{R}{\mathcal H} om(\bar{\pi}_*{\mathbb F},\bar{\pi}_*{\mathbb F}\boxtimes \omega_{{\mathcal U}/{\mathcal C}}))}{\det(\pi_{M_U*}\mathbf{R}{\mathcal H} om(\bar{\pi}_*{\mathbb F},\bar{\pi}_*{\mathbb F}) )}\cong \mathcal{O}\end{align}

where the second isomorphism is given by Grothendieck–Serre duality. It is easy to see that (4.9) and (4.7) are the same. Up to a constant determined in (4.10) below, the square of this isomorphism gives (4.5). The case of $\operatorname{{Hilb}}^{P_t}(X/{\mathbb A}^1)/{\mathcal C}^{P_t}$ is similar, by replacing ${\mathcal X}^{}/{\mathcal C}$ in the above by ${\mathcal X}^{P_t}/{\mathcal C}^{P_t}$ .

Remark 4.4. The orientation on $\operatorname{{Hilb}}(X/{\mathbb A}^1)/{\mathcal C}$ is constructed using the pullback of an orientation on the moduli stack $M_{\mathrm{cpt}}({\mathcal X}/{\mathcal C})$ of compactly supported sheaves on ${\mathcal X}/{\mathcal C}$ via the forgetful map (4.3). By spectral construction (4.8), the symmetric obstruction complex on $M_{\mathrm{cpt}}({\mathcal X}/{\mathcal C})$ is of the form

\begin{align*}{\mathbb E}=(\mathbb{V}\xrightarrow{\varphi} \mathbb{V}^\vee), \end{align*}

where $\mathbb{V}$ is a perfect complex and $\varphi$ is self-dual under the symmetric pairing on ${\mathbb E}$ (defined by Grothendieck–Serre duality). Therefore there is a canonical isomorphism $\det({\mathbb E})\cong {\mathcal O}$ and orientations of ${\mathbb E}$ are given by

(4.10) \begin{equation}{\mathcal O}\xrightarrow{\pm (-\sqrt{-1})^{\mathop{\rm rk}\nolimits(\mathbb{V})}} {\mathcal O} \end{equation}

on each connected component of ${\mathcal X}$ (as [Reference Oh and ThomasOT23, Equations (59) and (63)]). We choose the plus sign in the above as the canonical choice of orientation. Similarly, $\operatorname{{Hilb}}^{P_t}(X/{\mathbb A}^1)/{\mathcal C}^{P_t}$ has a canonical choice of orientation.

4.2 Orientation on $(\operatorname{{Hilb}}(Y,D)\to\operatorname{{Hilb}}({\mathcal D}))/{\mathcal A}$

Given a smooth log Calabi–Yau pair (Y,D) (Definition 2.4) so that D is a Calabi–Yau variety, we have a stack ${\mathcal A}$ of expanded pairs, a universal expanded pair ${\mathcal Y}$ , and a universal divisor ${\mathcal D}$ (Definition 2.3):

\begin{align*}\pi : {\mathcal Y} \to {\mathcal A}, \quad {\mathcal D}\cong D \times {\mathcal A}. \end{align*}

By Theorem 3.3, the restriction map

\begin{align*}r_{{\mathcal D}\to {\mathcal Y}}: \mathbf{Hilb}(Y,D)\to \mathbf{Hilb} ({\mathcal D})=\mathbf{Hilb} (D)\times {\mathcal A}\end{align*}

has a Lagrangian structure relative to ${\mathcal A}$ . Hence there is an isomorphism [Reference Pantev, Toën, Vaquie and VezzosiPTV⁺13, Definition 2.8]:

(4.11) \begin{equation}{\mathbb T}_{r_{{\mathcal D}\to {\mathcal Y}}}\cong {\mathbb L}_{\mathbf{Hilb}(Y,D)/{\mathcal A}}[1-\dim_{\mathbb{C}}D]. \end{equation}

Combining with the fiber sequence

\begin{align*}{\mathbb T}_{r_{{\mathcal D}\to {\mathcal Y}}} \to {\mathbb T}_{\mathbf{Hilb}(Y,D)/{\mathcal A}}\to r_{{\mathcal D}\to {\mathcal Y}}^*{\mathbb T}_{(\mathbf{Hilb}(D)\times {\mathcal A})/{\mathcal A}}, \end{align*}

we obtain an isomorphism of determinant line bundles:

(4.12) \begin{equation}(\det({\mathbb T}_{\mathbf{Hilb}(Y,D)/{\mathcal A}}))^{\otimes 2}\cong r_{{\mathcal D}\to {\mathcal Y}}^*\det({\mathbb T}_{(\mathbf{Hilb}(D)\times {\mathcal A})/{\mathcal A}} )\quad \mathrm{if }\dim_{\mathbb{C}}D\, \mathrm{is}\,\mathrm{odd}, \end{equation}

and $r_{{\mathcal D}\to {\mathcal Y}}^*\det({\mathbb T}_{(\mathbf{Hilb}(D)\times {\mathcal A})/{\mathcal A)}}\cong \mathcal{O}$ if $\dim_{\mathbb{C}}D$ is even, which reduces to Definition 4.1.

Restricting (4.12) to the classical truncation, we obtain

(4.13) \begin{equation}(\det({\mathbb T}_{\mathbf{Hilb}(Y,D)/{\mathcal A}}|_{\operatorname{{Hilb}}(Y,D)}))^{\otimes 2}\cong r_{{\mathcal D}\to {\mathcal Y}}^*\det({\mathbb T}_{(\mathbf{Hilb}(D)\times {\mathcal A})/{\mathcal A}}|_{\operatorname{{Hilb}}(D)\times {\mathcal A}} ), \end{equation}

where $r_{{\mathcal D}\to {\mathcal Y}}$ denotes its classical truncation by an abuse of notation.

As in [Reference Cao and LeungCL17a, Definition 4.4] and [Reference Cao and LeungCL17b, Definition 6.1], we use the following definition.

Definition 4.5. Assume $\dim_{\mathbb{C}}D$ is odd. A relative orientation on the map

\begin{align*}r_{{\mathcal D}\to {\mathcal Y}}: \operatorname{{Hilb}}(Y,D)\to \operatorname{{Hilb}}(D)\times {\mathcal A}\end{align*}

is the following data: a line bundle $K^{-1/2}_{\operatorname{{Hilb}}(D)}$ , an isomorphism

\[ (K^{-1/2}_{\operatorname{{Hilb}}(D)})^{\otimes 2}\cong \det({\mathbb T}_{(\mathbf{Hilb}(D)\times {\mathcal A})/{\mathcal A}}|_{\operatorname{{Hilb}}(D)\times {\mathcal A}} ), \]

and an isomorphism

\begin{align*}\det({\mathbb T}_{\mathbf{Hilb}(Y,D)/{\mathcal A}}|_{\operatorname{{Hilb}}(Y,D)})\cong r_{{\mathcal D}\to {\mathcal Y}}^*K^{-1/2}_{\operatorname{{Hilb}}(D)}, \end{align*}

whose square is the isomorphism (4.13).

As in Theorem 4.3, we have the following.

Theorem 4.6. Let (U,S) be a smooth pair (Definition 2.1) with $\dim_{\mathbb{C}} U=3$ and let

\begin{align*}Y=\operatorname{Tot}(\omega_{U}(S)), \quad D=\operatorname{Tot}(\omega_S)\end{align*}

be the (associated) log Calabi–Yau pair (Definition 2.4). Then

\begin{align*}r_{{\mathcal D}\to {\mathcal Y}}: \operatorname{{Hilb}}(Y,D)\to \operatorname{{Hilb}}(D)\times {\mathcal A}\end{align*}

has a relative orientation. Similarly, $\operatorname{{Hilb}}^P(Y,D)\to \operatorname{{Hilb}}^{P|_D}(D)\times {\mathcal A}^P$ has a relative orientation.

The proof is similar to the proof of Theorem 4.3.

Proof. Let $M_{\mathrm{cpt}}({\mathcal Y}/{\mathcal A})$ be the moduli stack of compactly supported sheaves on ${\mathcal Y}/{\mathcal A}$ which are normal to ${\mathcal D}$ in the sense of Definition 2.16. Let $M_{\mathrm{cpt}}(D)$ be the moduli stack of compactly supported sheaves on D. We have a commutative diagram

(4.14)

where r is the restriction map, and both $f_Y$ and $f_D$ are forgetful maps sending $(\mathcal{O}\to F)$ to F. Let $\bar{f}_Y:=f_Y\times_{\operatorname{id}_{{\mathcal A}}} \operatorname{id}_{{\mathcal Y}}$ , and

\begin{align*}\pi_{M_Y}:M_{\mathrm{cpt}}({\mathcal Y}/{\mathcal A})\times_{{\mathcal A}} {\mathcal Y}\to M_{\mathrm{cpt}}({\mathcal Y}/{\mathcal A}), \quad \pi_{H_Y}: \operatorname{{Hilb}}(Y,D)\times_{{\mathcal A}} {\mathcal Y}\to \operatorname{{Hilb}}(Y,D)\end{align*}

be the projections. The embedding $i:{\mathcal D}\hookrightarrow {\mathcal Y}$ induces $\bar{i}: M_{\mathrm{cpt}}({\mathcal Y}/{\mathcal A})\times_{{\mathcal A}} {\mathcal D}\hookrightarrow M_{\mathrm{cpt}}({\mathcal Y}/{\mathcal A})\times_{{\mathcal A}} {\mathcal Y}$ . Let ${\mathbb F}$ be the universal sheaf on $M_{\mathrm{cpt}}({\mathcal Y}/{\mathcal A})\times_{{\mathcal A}} {\mathcal Y}$ and ${\mathbb I}=(\mathcal{O}_{\operatorname{{Hilb}}(Y,D)\times_{{\mathcal A}} {\mathcal Y}}\to \bar{f}_Y^*{\mathbb F})$ be the universal pair. As in (4.4), we have

(4.15) \begin{align}& \det({\mathbb T}_{\mathbf{Hilb}(Y,D)/{\mathcal A}}|_{\operatorname{{Hilb}}(Y,D)})\cong \det(\pi_{H_Y*}\mathbf{R}{\mathcal H} om({\mathbb I},{\mathbb I})_0[1]) \nonumber\\&\quad\cong f_Y^*(\det(\pi_{M_Y*}\mathbf{R}{\mathcal H} om({\mathbb F},{\mathbb F})[1])\otimes \det(\pi_{M_Y*} \bar{i}_*\bar{i}^*{\mathbb F}) ),\end{align}

Consider the natural projection and the inclusion of a divisor:

\begin{align*}\pi: {\mathcal Y}=\operatorname{Tot}(\omega_{{\mathcal U}/{\mathcal A}}({\mathcal S})) \to {\mathcal U}, \quad j: {\mathcal S}:=S\times {\mathcal A}\hookrightarrow {\mathcal U}. \end{align*}

We have the commutative diagram

(4.16)

where $\pi_{\dagger}$ denotes the pushforward map from ${\mathcal Y}$ to ${\mathcal U}$ by $\pi$ and $j_{\dagger}$ denotes the pullback map from ${\mathcal U}$ to ${\mathcal S}$ by j. Note that we have

(4.17) \begin{equation}\bar{\pi}_*{\mathbb F}=\pi_{\dagger}^*{\mathbb F}_U, \quad \bar{j}^*{\mathbb F}_U=j_{\dagger}^*{\mathbb F}_S, \end{equation}

where ${\mathbb F}_U\to M_{\mathrm{cpt}}({\mathcal U}/{\mathcal A})\times_{{\mathcal A}} {\mathcal U}$ is the universal object of the moduli stack $M_{\mathrm{cpt}}({\mathcal U}/{\mathcal A})$ of compactly supported sheaves on ${\mathcal U}/{\mathcal A}$ (which are normal to ${\mathcal S}$ ) and ${\mathbb F}_S\to M_{\mathrm{cpt}}({\mathcal S}/{\mathcal A})\times_{{\mathcal A}} {\mathcal S}$ is the universal object of the moduli stack $M_{\mathrm{cpt}}({\mathcal S}/{\mathcal A})=M_{\mathrm{cpt}}(S)\times {\mathcal A}$ of compactly supported sheaves on $(S\times{\mathcal A})/{\mathcal A}$ .

As with (4.9), by using Serre duality and diagrams (4.16) and (4.17), we obtain

(4.18) \begin{align}&\quad \,\,\det(\pi_{M_Y*}\mathbf{R}{\mathcal H} om({\mathbb F},{\mathbb F})[1])\cong \pi_{\dagger}^*j_{\dagger}^*(\det(\pi_{M_S*}\mathbf{R}{\mathcal H} om({\mathbb F}_S,{\mathbb F}_S) ))^{-1}.\end{align}

Combining (4.15) and (4.18), we obtain

(4.19) \begin{equation}\det({\mathbb T}_{\mathbf{Hilb}(Y,D)/{\mathcal A}}|_{\operatorname{{Hilb}}(Y,D)})\cong \frac{f_Y^*\det(\pi_{M_Y*} \bar{i}_*\bar{i}^*{\mathbb F})}{f_Y^*\pi_{\dagger}^*j_{\dagger}^*\det(\pi_{M_S*}\mathbf{R}{\mathcal H} om({\mathbb F}_S,{\mathbb F}_S))}. \end{equation}

Consider the projection and the embedding:

\begin{align*}p: {\mathcal D}=\operatorname{Tot}(\omega_{{\mathcal S}/{\mathcal A}}) \to {\mathcal S}, \quad i:{\mathcal D}\hookrightarrow {\mathcal Y}. \end{align*}

We have a commutative diagram

(4.20)

where $p_{\dagger}$ is the pushforward of sheaves via p and r is the restriction map via i. As in (4.17), we have

(4.21) \begin{equation}\bar{i}^*{\mathbb F}=\bar{r}_{}^*{\mathbb F}_D , \quad \bar{p}_*{\mathbb F}_D=p_{\dagger}^*{\mathbb F}_S, \end{equation}

where ${\mathbb F}_D$ (respectively ${\mathbb F}_S$ ) is the universal sheaf on $M_{\mathrm{cpt}}({\mathcal D}/{\mathcal A})\times_{{\mathcal A}}{\mathcal D}$ (respectively on $M_{\mathrm{cpt}}({\mathcal S}/{\mathcal A})\times_{{\mathcal A}}{\mathcal S}$ ).

As in (4.15) and (4.18), with $f_D$ defined in diagram (4.14), we have

(4.22)

where the second isomorphism is derived as in (4.8). We define

(4.23) \begin{equation}K^{-1/2}_{\operatorname{{Hilb}}(D)}:=\frac{f_D^*\det(\pi_{M_D*}{\mathbb F}_D)}{f_D^*p_{\dagger}^*\det(\pi_{M_S*}\mathbf{R}{\mathcal H} om({\mathbb F}_S,{\mathbb F}_S))}.\end{equation}

Then, (4.22) provides an isomorphism

\[ (K^{-1/2}_{\operatorname{{Hilb}}(D)})^{\otimes 2}\cong \det({\mathbb T}_{(\mathbf{Hilb}(D)\times {\mathcal A})/{\mathcal A}}|_{\operatorname{{Hilb}}(D)\times {\mathcal A}} ). \]

Finally, we compute $r_{{\mathcal D}\to {\mathcal Y}}^*K^{-1/2}_{\operatorname{{Hilb}}(D)}$ :

where the last equality is by base change along the following Cartesian diagram.

Comparing with (4.19), we conclude that

(4.24) \begin{equation}\det({\mathbb T}_{\mathbf{Hilb}(Y,D)/{\mathcal A}}|_{\operatorname{{Hilb}}(Y,D)})\cong r_{{\mathcal D}\to {\mathcal Y}}^*K^{-1/2}_{\operatorname{{Hilb}}(D)}. \end{equation}

It is straightforward to show that this provides a relative orientation.

The case of $\operatorname{{Hilb}}^P(Y,D)\to \operatorname{{Hilb}}^{P|_D}(D)\times {\mathcal A}^P$ is similar, by replacing ${\mathcal Y}^{}/{\mathcal A}$ in the above by ${\mathcal Y}^{P}/{\mathcal A}^{P}$ .

Remark 4.7. We have isomorphisms

(4.25)

which is written using data purely on S.

4.3 Orientation on $(\operatorname{{Hilb}}(Y,D)\to W\times {\mathcal A}) $

Definition 4.8. Let S be a quasi-projective surface and $D=\operatorname{Tot}(\omega_S)$ , and fix $P_D\in K_{c,\leqslant 2}^\mathrm{{num}}(D)$ . A Lagrangian fibration

\begin{align*}p: \mathbf{Hilb}^{P_D}(D)\to W\end{align*}

is S-compatible if the line bundle (4.25) satisfies

(4.26) \begin{equation} K^{-1/2}_{\operatorname{{Hilb}}^{P_D}(D)}\cong p^*\det({\mathbb T}_W|_{t_0(W)}), \end{equation}

where p also denotes its classical truncation by an abuse of notation.

Recall from Proposition 3.8 that the composition map

\begin{align*}p\circ r_{{\mathcal D}\to {\mathcal Y}}: \mathbf{Hilb}^{P}(Y,D)\to W\times {\mathcal A}^{P}\end{align*}

has a (relative) $(-2)$ -shifted symplectic structure. By an abuse of notation, we denote its classical truncation by

\begin{align*}p\circ r_{{\mathcal D}\to {\mathcal Y}}:\operatorname{{Hilb}}^{P}(Y,D)\to W\times {\mathcal A}^{P}.\end{align*}

Proposition 4.9. Let (U,S) be a smooth pair (Definition 2.1) with $\dim_{\mathbb{C}} U=3$ and

(4.27) \begin{equation}Y=\operatorname{Tot}(\omega_{U}(S)), \quad D=\operatorname{Tot}(\omega_S). \end{equation}

If p is the S-compatible Lagrangian fibration given in Definition 4.8, then the map $p\circ r_{{\mathcal D}\to {\mathcal Y}}$ has an orientation in the sense of Definition 4.1.

Proof. The diagram

induces a fiber sequence

\begin{align*}r_{{\mathcal D}\to {\mathcal Y}}^*{\mathbb L}_{p} \to {\mathbb L}_{p\circ r_{{\mathcal D}\to {\mathcal Y}}} \to {\mathbb L}_{r_{{\mathcal D}\to {\mathcal Y}}}. \end{align*}

Therefore

\begin{align*}\det{\mathbb L}_{p\circ r_{{\mathcal D}\to {\mathcal Y}}}&\cong r_{{\mathcal D}\to {\mathcal Y}}^*\det {\mathbb L}_{p}\otimes \det{\mathbb L}_{r_{{\mathcal D}\to {\mathcal Y}}} \\&\cong r_{{\mathcal D}\to {\mathcal Y}}^*\bigg(\frac{\det {\mathbb L}_{\mathbf{Hilb}^{P|_D}(D)\times {\mathcal A}^{P}}}{p^*\det {\mathbb L}_{W\times {\mathcal A}^{P}}}\bigg)\otimes\bigg(\frac{\det {\mathbb L}_{\mathbf{Hilb}^{P}(Y,D)}}{r_{{\mathcal D}\to {\mathcal Y}}^*\det({\mathbb L}_{\mathbf{Hilb}^{P|_D}(D)\times {\mathcal A}^{P}})}\bigg) \\&\cong \frac{\det {\mathbb L}_{\mathbf{Hilb}^{P}(Y,D)}}{r_{{\mathcal D}\to {\mathcal Y}}^*p^*\det {\mathbb L}_{W\times {\mathcal A}^{P}}} \cong \frac{\det {\mathbb L}_{\mathbf{Hilb}^{P}(Y,D)/{\mathcal A}^{P}}}{r_{{\mathcal D}\to {\mathcal Y}}^*p^*\det {\mathbb L}_{(W\times {\mathcal A}^{P})/{\mathcal A}^{P}}}.\end{align*}

By (4.24) in Theorem 4.6 and Definition 4.8, we know that

\begin{align*}\det({\mathbb L}_{p\circ r_{{\mathcal D}\to {\mathcal Y}}}|_{\operatorname{{Hilb}}^{P}(Y,D)})\cong \mathcal{O}_{\operatorname{{Hilb}}^{P}(Y,D)}, \end{align*}

which can be checked to be an orientation.

Let (Y,D) be the log Calabi–Yau pair given by (4.27) and

\begin{align*}{\mathcal Y}=\operatorname{Tot}(\omega_{{\mathcal U}/{\mathcal A}}({\mathcal S})), \quad {\mathcal D}=\operatorname{Tot}(\omega_{{\mathcal S}/{\mathcal A}})\cong \operatorname{Tot}(\omega_{S})\times {\mathcal A}\end{align*}

be the universal expanded pair over the stack ${\mathcal A}$ of expanded pairs. Denote by $\textbf{M}_{\mathrm{cpt}}(-)$ the derived moduli stack of compactly supported sheaves on $(-)={\mathcal Y}/{\mathcal A}$ , ${\mathcal U}/{\mathcal A}$ , ${\mathcal D}/{\mathcal A}$ , ${\mathcal S}/{\mathcal A}$ . Then we have a pushforward map

\begin{align*}\pi_\dagger: \textbf{M}_{\mathrm{cpt}}({\mathcal Y}/{\mathcal A}) \to \textbf{M}_{\mathrm{cpt}}({\mathcal U}/{\mathcal A}), \end{align*}

and a restriction map

\begin{align*}r: \textbf{M}_{\mathrm{cpt}}({\mathcal Y}/{\mathcal A})\to \textbf{M}_{\mathrm{cpt}}({\mathcal D}/{\mathcal A}). \end{align*}

Note that $\textbf{M}_{\mathrm{cpt}}({\mathcal D}/{\mathcal A})$ is the $({-}1)$ -shifted cotangent bundle of $\textbf{M}_{\mathrm{cpt}}({\mathcal S}/{\mathcal A})$ over ${\mathcal A}$ :

\begin{align*}\textbf{M}_{\mathrm{cpt}}({\mathcal D}/{\mathcal A})={\mathbb T}_{\textbf{M}_{\mathrm{cpt}}({\mathcal S}/{\mathcal A})/{\mathcal A}}^*[-1], \end{align*}

with projection

\begin{align*}p_\dagger: \textbf{M}_{\mathrm{cpt}}({\mathcal D}/{\mathcal A})\to \textbf{M}_{\mathrm{cpt}}({\mathcal S}/{\mathcal A}). \end{align*}

As in Theorem 3.3, the map r has a Lagrangian structure (over ${\mathcal A}$ ). By [Reference CalaqueCal19, Theorem 2.2], $p_\dagger$ has a Lagrangian fibration structure (over ${\mathcal A}$ ). Therefore the composed map

\begin{align*}\textbf{M}_{\mathrm{cpt}}({\mathcal Y}/{\mathcal A}) \to \textbf{M}_{\mathrm{cpt}}({\mathcal S}/{\mathcal A})\end{align*}

has a $(-2)$ -shifted symplectic structure (Lemma 3.7). In particular, there is an isomorphism

(4.28) \begin{equation} {\mathbb T}_{\textbf{M}_{\mathrm{cpt}}({\mathcal Y}/{\mathcal A})/\textbf{M}_{\mathrm{cpt}}({\mathcal S}/{\mathcal A})}\cong {\mathbb L}_{\textbf{M}_{\mathrm{cpt}}({\mathcal Y}/{\mathcal A})/\textbf{M}_{\mathrm{cpt}}({\mathcal S}/{\mathcal A})}[-2]. \end{equation}

As in (4.6), there is a fiber sequence

(4.29) \begin{equation}\pi_\dagger^*{\mathbb L}_{\textbf{M}_{\mathrm{cpt}}({\mathcal U}/{\mathcal A})/\textbf{M}_{\mathrm{cpt}}({\mathcal S}/{\mathcal A})}[-2]\to{\mathbb T}_{\textbf{M}_{\mathrm{cpt}}({\mathcal Y}/{\mathcal A})/\textbf{M}_{\mathrm{cpt}}({\mathcal S}/{\mathcal A})} \to \pi_\dagger^*{\mathbb T}_{\textbf{M}_{\mathrm{cpt}}({\mathcal U}/{\mathcal A})/\textbf{M}_{\mathrm{cpt}}({\mathcal S}/{\mathcal A})}, \end{equation}

which is self-dual under the pairing (4.28). Its restriction to the classical truncation $M_{\mathrm{cpt}}({\mathcal Y}/{\mathcal A})$ provides a canonical isomorphism:

\begin{align*}{\mathbb E}:={\mathbb L}_{\textbf{M}_{\mathrm{cpt}}({\mathcal Y}/{\mathcal A})/\textbf{M}_{\mathrm{cpt}}({\mathcal S}/{\mathcal A})}|_{M_{\mathrm{cpt}}({\mathcal Y}/{\mathcal A})}\cong \mathcal{O}. \end{align*}

Define $\mathbb{V}:=\pi_\dagger^*{\mathbb T}_{\textbf{M}_{\mathrm{cpt}}({\mathcal U}/{\mathcal A})/\textbf{M}_{\mathrm{cpt}}({\mathcal S}/{\mathcal A})}|_{M_{\mathrm{cpt}}({\mathcal Y}/{\mathcal A})}$ . As in Remark 4.4, we choose

(4.30) \begin{equation}{\mathcal O}\xrightarrow{+(-\sqrt{-1})^{\mathop{\rm rk}\nolimits(\mathbb{V})}} {\mathcal O} \end{equation}

as the canonical choice of orientation on $M_{\mathrm{cpt}}({\mathcal Y}/{\mathcal A}) \to M_{\mathrm{cpt}}({\mathcal S}/{\mathcal A})$ .

There is a forgetful map

\begin{align*}f_Y: \operatorname{{Hilb}}^{P}(Y,D) \to M_{\mathrm{cpt}}({\mathcal Y}/{\mathcal A}). \end{align*}

By comparing (4.15), (4.18), (4.22), (4.23) and using (4.26), the above canonical orientation on

\begin{align*}M_{\mathrm{cpt}}({\mathcal Y}/{\mathcal A}) \to M_{\mathrm{cpt}}({\mathcal S}/{\mathcal A})\end{align*}

induces a canonical orientation on

\begin{align*}\operatorname{{Hilb}}^{P}(Y,D)\to W \times {\mathcal A}^{P}.\end{align*}

We give examples in which (4.26) in Definition 4.8 is satisfied.

Example 4.11. Consider the smooth pair

\begin{align*}U=\operatorname{Tot}(\mathcal{O}_{\mathbb{P}^1}(a_{})\oplus \mathcal{O}_{\mathbb{P}^1}(b_{})), \quad S=\mathbb{C}^2, \end{align*}

where the associated log Calabi–Yau pair is

\begin{align*}Y=\operatorname{Tot}(\omega_{U}(S))=\operatorname{Tot}(\mathcal{O}_{\mathbb{P}^1}(a)\oplus \mathcal{O}_{\mathbb{P}^1}(b)\oplus \mathcal{O}_{\mathbb{P}^1}(-1-a-b)), \quad D=\operatorname{Tot}(\omega_{S})=\mathbb{C}^3.\end{align*}

By Example 3.10, we have an inclusion into the non-commutative Hilbert scheme:

\begin{align*}p: \operatorname{{Hilb}}^n(D)\to W. \end{align*}

Then (4.26) holds, i.e. $p^*\det({\mathbb T}_W)\cong K^{-1/2}_{\operatorname{{Hilb}}^n(D)}$ for the square root $K^{-1/2}_{\operatorname{{Hilb}}^n(D)}$ in (4.23). Therefore

\begin{align*}p\circ r_{{\mathcal D}\to {\mathcal Y}}: \operatorname{{Hilb}}^P(Y,D)\to W\times {\mathcal A}^{P} \end{align*}

has an orientation by Proposition 4.9.

Proof. As $S=\mathbb{C}^2$ is symplectic, we have that

(4.31) \begin{align}K^{-1/2}_{\operatorname{{Hilb}}^n(D)}&\cong f_D^*\det(\pi_{M_D*}{\mathbb F}_D)\end{align}

is the determinant of the tautological bundle of $\operatorname{{Hilb}}^n(D)$ . By (3.19), we have

\begin{align*}{\mathbb T}_W\cong (\mathfrak {gl}_n\otimes \mathcal{O}\to (\mathfrak {gl}_n^{\oplus 3}\oplus \mathbb{C}^n)\otimes \mathcal{O}), \end{align*}

where the right-hand side (RHS) is written as a $\operatorname{GL}(n)$ -equivariant complex. Then

(4.32) \begin{equation}\det({\mathbb T}_W)\cong \det(\mathbb{C}^n\otimes \mathcal{O}), \end{equation}

where the RHS is the determinant of the tautological bundle. After restricting to $\operatorname{{Hilb}}^n(D)$ , we know that (4.31) and (4.32) are isomorphic.

5.1 Virtual pullback via symmetric obstruction theory

1. (1) a perfect complex ${\mathbb E}$ of tor-amplitude $[-2,0]$ on ${\mathcal X}$ ;

2. (2) a non-degenerate symmetric form $\theta$ on ${\mathbb E}$ , i.e. a morphism

   \begin{align*}\theta : {\mathcal O}_{{\mathcal X}} \to ({\mathbb E} \otimes{\mathbb E})[-2]\end{align*}
   in the derived category of ${\mathcal X}$ , invariant under the transposition $\sigma : {\mathbb E} \otimes {\mathbb E} \to {\mathbb E} \otimes {\mathbb E}$ , with the induced map $\iota_{\theta} : {\mathbb E}^\vee\to{\mathbb E}[-2]$ being an isomorphism;

3. (3) an orientation o of ${\mathbb E}$ , i.e. an isomorphism $o:{\mathcal O}_{\mathcal X} \to \det({\mathbb E})$ of line bundles such that $\det(\iota_\theta)=o\circ o^\vee$ .

For a symmetric complex ${\mathbb E}$ , there is a quadratic function [Reference ParkPar21, § A.2] from the virtual normal cone $\frak{C}_{{\mathbb E}}$ of ${\mathbb E}$ ,

(5.1) \begin{equation}\frak{q}_{\mathbb E}:\frak{C}_{{\mathbb E}}\to {\mathbb A}^1_{{\mathcal X}}, \end{equation}

characterized by some naturality conditions.

1. (1) ${\mathbb E}$ is a symmetric complex; and

2. (2) $\phi$ is an obstruction theory in the sense of Behrend and Fantechi [Reference Behrend and FantechiBF97], i.e. $h^0(\phi)$ is an isomorphism and $h^{-1}(\phi)$ is surjective, where ${\mathbf L}_f := \tau^{\geqslant -1}{\mathbb L}_f$ is the truncated cotangent complex.

The obstruction theory $\phi$ induces a closed embedding of the intrinsic normal cone

\begin{align*}\frak{C}_f\hookrightarrow \frak{C}_{{\mathbb E}}. \end{align*}

Isotropic symmetric obstruction theory implies the existence of square root virtual pullback which we now briefly recall. For a symmetric complex ${\mathbb E}$ on an algebraic stack ${\mathcal X}$ , let $\frak{Q}({\mathbb E})$ be the zero locus of the quadratic function $\frak{q}_{{\mathbb E}}:\frak{C}_{\mathbb E}\to {\mathbb A}^1_{\mathcal X}$ . There is a square root Gysin pullback [Reference ParkPar21, Definition A.2]

\begin{align*}\sqrt{0^!_{\frak{Q}({\mathbb E})}}:A_*(\frak{Q}({\mathbb E})) \to A_*({\mathcal X}),\end{align*}

if ${\mathcal X}$ is a quotient of a separated Deligne–Mumford stack by an algebraic group.

Definition 5.4. Assume that $f:{\mathcal X}\to{\mathcal Y}$ is a Deligne–Mumford morphism between algebraic stacks with an isotropic symmetric obstruction theory $\phi:{\mathbb E}\to{\mathbf L}_f$ . It induces a closed embedding $a:\frak{C}_f\to \frak{Q}({\mathbb E})$ . The square root virtual pullback is the composition

(5.2) \begin{equation} \sqrt{f^!}:A_*({\mathcal Y})\xrightarrow{sp_f}A_*(\frak{C}_f )\xrightarrow{a_*} A^{*} (\frak{Q}({\mathbb E}))\xrightarrow{\sqrt{0^!_{\frak{Q}({\mathbb E})}}} A_{*}({\mathcal X} ),\end{equation}

where $sp_f:A_*({\mathcal Y})\to A_*(\frak{C}_f)$ is the specialization map [Reference ManolacheMan12, Construction 3.6].

The map $\sqrt{f^!}$ commutes with projective pushforwards, smooth pullbacks, and Gysin pullbacks for regular immersions and can be used to define the square root virtual pullback for any base change of f (e.g. [Reference ParkPar21, Equation (1.14)]). Moreover, it has a functoriality with respect to morphisms compatible with symmetric obstruction theories [Reference ParkPar21, Theorem A.4] as explained below.

Let $f:{\mathcal X}\to{\mathcal Y}$ be a Deligne–Mumford (DM) morphism of algebraic stacks having reductive stabilizer groups and affine diagonals,Footnote ¹³ which are satisfied if ${\mathcal X}$ and ${\mathcal Y}$ are quotient stacks of separated DM stacks by algebraic tori. Let $g:{\mathcal Y}\to{\mathcal Z}$ be a DM morphism of algebraic stacks. Assume $\phi_g:{\mathbb E}_g\to {\mathbf L}_g$ , $\phi_{g\circ f}:{\mathbb E}_{g\circ f}\to {\mathbf L}_{g\circ f}$ are isotropic symmetric obstruction theories, $\phi_f:{\mathbb E}_f\to {\mathbf L}_f$ is a perfect obstruction theory [Reference Behrend and FantechiBF97] and they are compatible, i.e. there exists a perfect complex ${\mathbb D}$ and morphisms $\alpha:{\mathbb E}_{g\circ f}\to {\mathbb D}$ and $\beta:f^*{\mathbb E}_g\to {\mathbb D}$ fitting into the diagram of exact triangles

(5.3)

where the orientation of ${\mathbb E}_{g\circ f}$ is compatible with the orientation of ${\mathbb E}_{g}$ [Reference ParkPar21, Definition 2.1]. Here $\phi_{g\circ f}=\phi_{g\circ f}'\circ\alpha$ , $\phi_f=r\circ \phi'_f$ with ${\mathbf L}'_f$ is the cone of $\tau^{\geqslant -1}f^*{\mathbf L}_g\to{\mathbf L}_{g\circ f}$ and $r:{\mathbf L}'_f\to {\mathbf L}_f$ the truncation. Then we have

(5.4) \begin{equation}\sqrt{(g\circ f)^!}=f^!\circ\sqrt{g^!},\end{equation}

where $f^!$ is the virtual pullback of Manolache [Reference ManolacheMan12]. In the presence of a torus action, the above extends to the torus equivariant setting.

5.2 Relative invariants

Let (Y,D) be a smooth log Calabi–Yau pair (Definition 2.4) so that D is a Calabi–Yau 3-fold. Fixing a K-theory class $P\in K_{c, \leqslant 2}^{\mathrm{num}, } (Y)$ , we have a restriction map (3.7):

(5.5) \begin{equation}r_{{\mathcal D}\to {\mathcal Y}}:\mathbf{Hilb}^P(Y,D)\to \mathbf{Hilb}^{P|_D}(D)\times {\mathcal A}^P, \end{equation}

which has a Lagrangian structure (Theorem 3.3).

In this section, we work under the following assumption.

Assumption 5.5. We assume that there is a quasi-projective smooth scheme W, and a regular functionFootnote ¹⁴ $\phi$ on W, such that there is an equivalence

(5.6) \begin{equation}\mathbf{Hilb}^{P|_D}(D)\cong \textbf{Crit}(\phi:W\to \mathbb{C}) \end{equation}

of the $({-}1)$ -shifted symplectic derived scheme.

Under the above assumption, the composition (where p denotes the inclusion deduced by (5.6))

(5.7) \begin{equation}p\circ r_{{\mathcal D}\to {\mathcal Y}}: \mathbf{Hilb}^{P}(Y,D) \stackrel{r_{{\mathcal D}\to {\mathcal Y}}}{\to} \mathbf{Hilb}^{P|_D}(D)\times {\mathcal A}^P\stackrel{p}{\to} W\times {\mathcal A}^P \end{equation}

has a $(-2)$ -shifted symplectic structure (Proposition 3.8).

Restricting the cotangent complex of $p\circ r_{{\mathcal D}\to {\mathcal Y}}$ to the classical truncation, which, by an abuse of notation, we still write as

\begin{align*}p\circ r_{{\mathcal D}\to {\mathcal Y}}: \operatorname{{Hilb}}^{P}(Y,D) \to W\times {\mathcal A}^P, \end{align*}

we get a canonical symmetric obstruction theory in the sense of Definition 5.2:

(5.8) \begin{equation}{\mathbb E}_{\operatorname{{Hilb}}^{P}(Y,D)/(W\times {\mathcal A}^P)} \to {\mathbb L}_{\operatorname{{Hilb}}^{P}(Y,D)/(W\times {\mathcal A}^P)}. \end{equation}

By the footnote to Assumption 5.5, we have $\operatorname{Crit}(\phi)\subseteq Z(\phi)$ . By the construction, $p\circ r_{{\mathcal D}\to {\mathcal Y}}$ factors through $\operatorname{Crit}(\phi)$ . Therefore we obtain the following Cartesian diagram of classical stacks.

Hence we can pull back (5.8) to the upper line and get a canonical symmetric obstruction theory

(5.9) \begin{equation}{\mathbb E}_{\operatorname{{Hilb}}^{P}(Y,D)/(Z(\phi)\times {\mathcal A}^P)} \to {\mathbb L}_{\operatorname{{Hilb}}^{P}(Y,D)/(Z(\phi)\times {\mathcal A}^P)}. \end{equation}

The following isotropic result will be proven in Appendix C.

Therefore, we have the following theorem.

Theorem 5.7. Let $P\in K_{c,\leqslant 1}^{\mathrm{num}} (Y)$ . Assume that Assumption 5.5 holds and the classical truncation of (5.7) has an orientation. Then there is a square root virtual pullback

(5.10) \begin{equation}\sqrt{(p\circ r_{{\mathcal D}\to {\mathcal Y}})^!} : A_*(Z(\phi)\times {\mathcal A}^P)\to A_*(\operatorname{{Hilb}}^{P}(Y,D)) \end{equation}

which satisfies the usual bivariance properties and functoriality spelled out in § 5.1.

Given a smooth pair (Definition 2.1) (U,S) with $\dim_{\mathbb{C}} U=3$ , let

\begin{align*}Y=\operatorname{Tot}(\omega_{U}(S)), \quad D=\operatorname{Tot}(\omega_S)\end{align*}

be the (associated) log Calabi–Yau 4-fold pair. Recall the line bundle $K^{-1/2}_{\operatorname{{Hilb}}^{P|_D}(D)}$ (4.25).

Assumption 5.8. Given $\mathbf{Hilb}^{P|_D}(D)\cong \textbf{Crit}(\phi:W\to \mathbb{C})$ as in Assumption 5.5, we assume that

\begin{align*}K^{-1/2}_{\operatorname{{Hilb}}^{P|_D}(D)}\cong \det({\mathbb T}_W|_{\operatorname{{Hilb}}^{P|_D}(D)}).\end{align*}

By Proposition 4.9, the classical truncation of (5.7) then has an orientation in the sense of Definition 4.1 and we choose the canonical orientation as specified in Remark 4.10.

Definition 5.9. Let $Y=\operatorname{Tot}(\omega_{U}(S))$ , $D=\operatorname{Tot}(\omega_S)$ be as above. Fix a K-theory class $P \in K_{c,\leqslant 1}^\mathrm{{num}} (Y)$ . Under Assumption 5.8, we define relative invariants:

(5.11) \begin{equation} \Phi^{P}_{Y,D}:=\sqrt{(p\circ r_{{\mathcal D}\to {\mathcal Y}})^!}\circ \boxtimes : A_*(Z(\phi))\otimes A_{*}({\mathcal A}^{P}) \to A_*(\operatorname{{Hilb}}^P(Y,D)), \end{equation}

where $\boxtimes$ is the exterior product

\begin{align*}\boxtimes: A_*(Z(\phi))\otimes A_{*}({\mathcal A}^{P}) \to A_*(Z(\phi)\times {\mathcal A}^{P}), \quad (\alpha,\beta)\mapsto (\alpha \times \beta). \end{align*}

5.3 Family invariants

Let $\pi : X \to {\mathbb A}^1$ be a simple degeneration of Calabi–Yau 4-folds. Fixing K-theory classes $P_t\in K_{c, \leqslant 2}^{\mathrm{num}}(X_t)$ for all $t\in \mathbb{A}^1$ , the map

(5.12) \begin{equation}\mathbf{Hilb}^{P_t}(X/{\mathbb A}^1)\to {\mathcal C}^{P_t} \end{equation}

has a (relative) $(-2)$ -shifted symplectic structure (Theorem 3.2). By restricting the cotangent complex of $\mathbf{Hilb}^{P_t}(X/{\mathbb A}^1)/{\mathcal C}^{P_t}$ to the classical truncation, we obtain a canonical symmetric obstruction theory

(5.13) \begin{equation}\mathbb{E}_{\operatorname{{Hilb}}^{P_t}(X/{\mathbb A}^1)/{\mathcal C}^{P_t}}\to \mathbb{L}_{\operatorname{{Hilb}}^{P_t}(X/{\mathbb A}^1)/{\mathcal C}^{P_t}}. \end{equation}

The following isotropic result will be proven in Appendix C.

Therefore, we have the following theorem.

Theorem 5.13. Fix K-theory classes $P_t \in K_{c,\leqslant 1}^\mathrm{{num}} (X_t)$ for all $t\in \mathbb{A}^1$ and assume that the classical truncation of (5.12) has an orientation. Then there is a square root virtual pullback

(5.14) \begin{equation} \sqrt{\pi^!} : A_*({\mathcal C}^{P_t})\to A_*(\operatorname{{Hilb}}^{P_t}(X/{\mathbb A}^1)), \end{equation}

which satisfies the usual bivariance properties and functoriality spelled out in § 5.1.

Let $U\to {\mathbb A}^1$ be a simple degeneration of 3-folds and

\begin{align*}X=\operatorname{Tot}(\omega_{U/{\mathbb A}^1})\to {\mathbb A}^1\end{align*}

be the associated simple degeneration of Calabi–Yau 4-folds. By Theorem 4.3, the classical truncation of (5.12) has an orientation in the sense of Definition 4.1. We choose the canonical orientation as specified in Remark 4.4.

Definition 5.14. Let $X=\operatorname{Tot}(\omega_{U/{\mathbb A}^1})\to {\mathbb A}^1$ be as above. Given K-theory classes $P_t \in K_{c,\leqslant 1}^\mathrm{{num}} (X_t)$ for all $t\in {\mathbb A}^1$ , we define family invariants:

(5.15) \begin{equation}\Phi^{P_t}_{X/{\mathbb A}^1}:=\sqrt{\pi^!} : A_*({\mathcal C}^{P_t}) \to A_*(\operatorname{{Hilb}}^{P_t}(X/{\mathbb A}^1)). \end{equation}

5.4 Degeneration formulae

In this section, we will relate relative invariants (Definition 5.9) of $(Y_\pm,D)$ and family invariants (Definition 5.14) of $X/ {\mathbb A}^1$ in a degeneration formula. We work in the following setting.

Setting 5.16. Let $U\to {\mathbb A}^1$ be a simple degeneration of 3-folds (Definition 2.6) with central fiber $U_0=U_+\cup_S U_-$ . Consider

\begin{align*}X=\operatorname{Tot}(\omega_{U/{\mathbb A}^1})\to {\mathbb A}^1.\end{align*}

This is a CY simple degeneration (Definition 2.10) whose fiber satisfies:

• – $X_c=\operatorname{Tot}(\omega_{U_c})$ if $c\neq 0$ ; and

• – $X_0=Y_+\cup_D Y_-$ , where $Y_{\pm}=\operatorname{Tot}(\omega_{U_\pm}(S))$ and $D=\operatorname{Tot}(\omega_S)$ .

Data 5.17. In Setting 5.16 with Assumption 5.8, assume there is an automorphism

(5.16) \begin{equation}\bar{\sigma}: X\to X, \end{equation}

which preserves fibers of $X\to {\mathbb A}^1$ and also $Y_{\pm}$ , D in $X_0$ , and there is an automorphism

(5.17) \begin{equation}\sigma:W\to W \end{equation}

on the W in (5.6) such that:

• – $\sigma^{*}\phi=-\phi$ ;

• – the induced automorphism $(\bar{\sigma}|_{D})_*: \operatorname{{Hilb}}^{P_D}(D)\to \operatorname{{Hilb}}^{P_D}(D)$ coincides with $\sigma|_{\operatorname{{Hilb}}^{P_D}(D)}$ .

Example 5.18. As $X=\operatorname{Tot}(\omega_{U/{\mathbb A}^1})$ , we can choose the order two element in the $\mathbb{C}^*$ -action on fibers of $X\to U$ as a candidate for $\bar{\sigma}$ (5.16). We refer to [Reference Cao and ZhaoCZ26, Remark 4.18] for the choice of automorphism (5.17) in examples given by quivers with potentials. In particular, in Example 3.10, we can take

\begin{align*}\sigma(b_1,b_2,b_3,v)=(b_1,b_2,-b_3,v), \end{align*}

where the third coordinate corresponds to the fiber of $\mathbb{C}^3=D\to S=\mathbb{C}^2$ . Then $\sigma$ is homotopic to the identity map and compatible with $\bar{\sigma}$ as required in Data 5.17.

We define the sum function

\begin{align*}\boxplus^n \phi: W^n\to \mathbb{C}, \quad (x_1,x_2,\ldots,x_n)\mapsto \sum_{i=1}^n\phi(x_i), \end{align*}

and denote by $Z(\boxplus^n \phi)$ its zero locus.

Consider the diagonal embedding

\begin{align*}\Delta: W\to W\times W\end{align*}

and the anti-diagonal embedding

(5.18) \begin{equation}\bar{\Delta}: W\to W\times W, \quad x\mapsto (\sigma x,x). \end{equation}

We have the commutativity of the diagrams

(5.19)

where the automorphism

(5.20) \begin{equation}\bar{\sigma}: \operatorname{{Hilb}}^{P_-}(Y_-,D)\to \operatorname{{Hilb}}^{P_-}(Y_-,D) \end{equation}

is induced by the automorphism (5.17). The bottom square commutes because automorphisms $\sigma$ and $\bar{\sigma}$ are compatible in Data 5.17. where the map $r_{{\mathcal D}\to {\mathcal Y}_-}\times r_{{\mathcal D}\to {\mathcal Y}_+}$ goes to $\operatorname{{Hilb}}^{P_D}(D) \times \operatorname{{Hilb}}^{P_D}(D)$ (where $P_D=P_-|_D=P_+|_D$ ) and we omit the natural inclusion