2.1 The base locus and the incidence variety

As a warm-up, we first treat a simple version of HP duality where the derived category results are clear from the geometry. Let $X$ be a smooth, projective variety with a morphism $f:X\rightarrow \mathbb{P}(V)$ for some vector space $V$, with $f$ not factoring through any linear subspace of $\mathbb{P}(V)$, and let ${\mathcal{L}}=f^{\ast }({\mathcal{O}}(1))$. Choose a linear subspace $L\subset V^{\vee }$, which gives a linear system $\mathbb{P}(L)$ of divisors of class ${\mathcal{L}}$.

There are two natural schemes we can construct from this linear system. Firstly, we can intersect the divisors in the linear system to get the base locus $X_{L^{\bot }}\subset X$. Secondly, we can construct the incidence variety ${\mathcal{H}}_{L}\subset X\,\times \,\mathbb{P}(L)$, which consists of pairs $(x,H)$ such that $x\in H$.

Let us assume that $X_{L^{\bot }}$ has the expected dimension. The first step towards HP duality is the observation that $D^{b}(X_{L^{\bot }})$ then includes as a full subcategory of $D^{b}({\mathcal{H}}_{L})$.

Consider first the case where $\mathbb{P}(L)=\mathbb{P}^{1}$. Then ${\mathcal{H}}_{L}$ is the blow-up of $X$ in $X_{L^{\bot }}$, and by [Reference Bondal and OrlovBO95, Proposition 3.4] we get a semiorthogonal decompositionFootnote ²

$$\begin{eqnarray}D^{b}({\mathcal{H}}_{L})=\langle D^{b}(X_{L^{\bot }}),D^{b}(X)\rangle .\end{eqnarray}$$

More generally, if $\mathbb{P}(L)=\mathbb{P}^{l}$, $l\geqslant 1$, then the projection ${\mathcal{H}}_{L}\rightarrow X$ has fibres $\mathbb{P}^{l-1}$ over $X\setminus X_{L^{\bot }}$, which jump to $\mathbb{P}^{l}$ over $X_{L^{\bot }}$. This gives a semiorthogonal decomposition of $D^{b}({\mathcal{H}}_{L})$ with 1 piece isomorphic to $D^{b}(X_{L^{\bot }})$ and $l$ pieces isomorphic to $D^{b}(X)$. In general, the inclusion functor $D^{b}(X_{L^{\bot }})\rightarrow D^{b}({\mathcal{H}}_{L})$ is given by $i_{\ast }p^{\ast }$ with $p$ and $i$ as in the diagram

2.2 Lefschetz decompositions

Kuznetsov’s remarkable discovery is that this relation between the base locus $X_{L^{\bot }}$ and the universal hyperplane ${\mathcal{H}}_{L}$ can be turned into something more interesting if we can put a certain extra structure on $D^{b}(X)$. Namely, assume that the derived category $D^{b}(X)$ admits a semiorthogonal decomposition

$$\begin{eqnarray}D^{b}(X)=\langle {\mathcal{A}}_{0},{\mathcal{A}}_{1}(1),\ldots ,{\mathcal{A}}_{k}(k)\rangle ,\end{eqnarray}$$

where the ${\mathcal{A}}_{i}$ are full subcategories of $D^{b}(X)$ satisfying ${\mathcal{A}}_{i}\subseteq {\mathcal{A}}_{i-1}$ for all $i\geqslant 1$, and where ${\mathcal{A}}_{i}(i)$ denotes the full subcategory whose objects are ${\mathcal{E}}\otimes {\mathcal{L}}^{\otimes i}$, ${\mathcal{E}}\in {\mathcal{A}}_{i}$. Such a decomposition is called a Lefschetz decomposition.

For any hyperplane $H\subset \mathbb{P}(V)$ inducing a divisor $X_{H}:=f^{-1}(H)$, the functor

$$\begin{eqnarray}{\mathcal{A}}_{i}(i)\rightarrow D^{b}(X)\stackrel{-|_{X_{H}}}{\rightarrow }D^{b}(X_{H})\end{eqnarray}$$

is full and faithful for $1\leqslant i\leqslant k$. Furthermore, the image subcategories ${\mathcal{A}}_{i}(i)\subset D^{b}(X_{H})$ are semiorthogonal. Both of these facts are easy to show using our assumptions on ${\mathcal{A}}_{i}$ and the exact triangle

$$\begin{eqnarray}{\mathcal{E}}\otimes {\mathcal{L}}^{-1}\rightarrow {\mathcal{E}}\rightarrow {\mathcal{E}}|_{X_{H}}\quad {\mathcal{E}}\in D^{b}(X).\end{eqnarray}$$

We therefore have a full subcategory $\langle {\mathcal{A}}_{1}(1),\ldots ,{\mathcal{A}}_{k}(k)\rangle \subset D^{b}(X_{H})$, and letting ${\mathcal{C}}_{H}=\langle {\mathcal{A}}_{1}(1),\ldots ,{\mathcal{A}}_{k}(k)\rangle ^{\bot }$, we get a semiorthogonal decomposition

$$\begin{eqnarray}D^{b}(X_{H})=\langle {\mathcal{C}}_{H},{\mathcal{A}}_{1}(1),\ldots ,{\mathcal{A}}_{k}(k)\rangle .\end{eqnarray}$$

We see that $D^{b}(X_{H})$ decomposes into the parts ${\mathcal{A}}_{i}(i)$ inherited from $D^{b}(X)$, and the one new part ${\mathcal{C}}_{H}$. This motivates the term ‘Lefschetz decomposition’ (cf. the Lefschetz hyperplane theorem).

More generally, let $L\subset V^{\vee }$ be a linear subspace, let $L^{\bot }=\{v\in V\,\mid \,(v,L)=0\}\subset V$, and let $X_{L^{\bot }}:=f^{-1}(\mathbb{P}(L^{\bot }))$, which is the base locus of the linear system $\mathbb{P}(L)$. We then get a semiorthogonal decomposition

$$\begin{eqnarray}D^{b}(X_{L^{\bot }})=\langle {\mathcal{C}}_{L^{\bot }},{\mathcal{A}}_{l}(l),\ldots ,{\mathcal{A}}_{k}(k)\rangle .\end{eqnarray}$$

One way of summarising HP duality is that if we know the category ${\mathcal{C}}_{H}$ for all hyperplanes $H$ in the system $\mathbb{P}(L)$, then we get a description of the category ${\mathcal{C}}_{L^{\bot }}$, in terms of the ‘homological projective dual’ variety, which we now describe.

2.3 The homological projective dual

Let $Y$ be a variety equipped with a map $g:Y\rightarrow \mathbb{P}(V^{\vee })$, and assume that for every point $H\in \mathbb{P}(V^{\vee })$ the fibre $Y_{H}$ satisfies $D^{b}(Y_{H})\cong {\mathcal{C}}_{H}\subset D^{b}(X_{H})$. If $Y$ satisfies a certain strengthening of this condition,Footnote ³ then we say that $Y$ is a homological projective dual for $X$. Note that the existence of such a $Y$ is not automatic.

Here $Y$ is analogous to the incidence variety ${\mathcal{H}}_{V^{\vee }}\subset X\,\times \,\mathbb{P}(V^{\vee })$, with the difference that the ‘categorical fibre’ $D^{b}(Y_{H})$ at each $H\in \mathbb{P}(V^{\vee })$ is now the interesting part ${\mathcal{C}}_{H}\subset D^{b}(X_{H})$ rather than the whole of $D^{b}(X_{H})$.

For any $L\subset V^{\vee }$, let $Y_{L}=g^{-1}(\mathbb{P}(L))$. Just as we saw above that $D^{b}(X_{L^{\bot }})$ includes into $D^{b}({\mathcal{H}}_{L})$, we can now include ${\mathcal{C}}_{L^{\bot }}$ into $D^{b}(Y_{L})$.

To state the precise result, we will need some notation. Let ${\mathcal{O}}_{Y}(1)=g^{\ast }{\mathcal{O}}_{\mathbb{P}(V^{\vee })}(1)$. Kuznetsov shows that $D^{b}(Y)$ admits a ‘dual’ Lefschetz decomposition

$$\begin{eqnarray}D^{b}(Y)=\langle {\mathcal{B}}_{m}(-m),{\mathcal{B}}_{m-1}(-m-1),\ldots ,{\mathcal{B}}_{0}(0)\rangle ,\end{eqnarray}$$

where ${\mathcal{B}}_{i}\subseteq {\mathcal{B}}_{i+1}$ for $i\geqslant 1$. Let $l$ and $c$ be the dimension and codimension of $L$, respectively.

Theorem 2.1 [Reference KuznetsovKuz07, Theorem 1.1].

If $X_{L^{\bot }}$ and $Y_{L}$ have the expected dimensions, then we have semiorthogonal decompositions

$$\begin{eqnarray}\displaystyle & \displaystyle D^{b}(X_{L^{\bot }})=\langle {\mathcal{C}}_{L^{\bot }},{\mathcal{A}}_{l}(l),\ldots ,{\mathcal{A}}_{k}(k)\rangle , & \displaystyle \nonumber\\ \displaystyle & \displaystyle D^{b}(Y_{L})=\langle {\mathcal{B}}_{-m}(-m),{\mathcal{B}}_{-m-1}(-m-1),\ldots ,{\mathcal{B}}_{-c}(-c),{\mathcal{C}}_{L}\rangle , & \displaystyle \nonumber\end{eqnarray}$$

and ${\mathcal{C}}_{L}\cong {\mathcal{C}}_{L^{\bot }}$.

The most striking consequence of this theorem is that $D^{b}(X_{L^{\bot }})$ and $D^{b}(Y_{L})$ have the semiorthogonal piece ${\mathcal{C}}_{L}\cong {\mathcal{C}}_{L^{\bot }}$ in common. The functor ${\mathcal{C}}_{L^{\bot }}\stackrel{\cong }{\rightarrow }{\mathcal{C}}_{L}{\hookrightarrow}D^{b}(Y_{L})$ is obtained by composing the functor $D^{b}(X_{L^{\bot }}){\hookrightarrow}D^{b}({\mathcal{H}}_{L})$ with a certain projection $D^{b}({\mathcal{H}}_{L})\rightarrow D^{b}(Y_{L})$.

If the dimension of $L$ is sufficiently low (respectively, high) we get a fully faithful inclusion $D^{b}(Y_{L}){\hookrightarrow}D^{b}(X_{L^{\bot }})$ (respectively, $D^{b}(X_{L^{\bot }}){\hookrightarrow}D^{b}(Y_{L})$).

As an aside, we note that the notion of HP dual makes sense more generally than in the setting described here. In particular, one can drop the restriction of considering derived categories of varieties, and instead consider more general triangulated categories, linear over $D^{b}(\mathbb{P}(V))$ and $D^{b}(\mathbb{P}(V^{\vee }))$. For some nice such categories the same results can be shown. The main results of this paper deal with HP duality in this extended sense; see also [Reference Ballard, Deliu, Favero, Isik and KatzarkovBDFIK17] and the next section.

2.4 HP duality for quadrics

We will now explain the results of HP duality for the case of quadric hypersurfaces, worked out by Kuznetsov in [Reference KuznetsovKuz08]. This is both an instructive example of HP duality in general and formally quite similar to the case we treat in this paper.

In the terminology used above, we take $X=\mathbb{P}(V)$, and let the map $f$ be the Veronese embedding $\mathbb{P}(V){\hookrightarrow}\mathbb{P}(\operatorname{Sym}^{2}V)$ with associated line bundle ${\mathcal{L}}={\mathcal{O}}_{\mathbb{P}(V)}(2)$. Let $n=\dim \mathbb{P}(V)$. The semiorthogonal decomposition

$$\begin{eqnarray}D^{b}(\mathbb{P}(V))=\langle {\mathcal{O}},{\mathcal{O}}(1),\ldots ,{\mathcal{O}}(n)\rangle\end{eqnarray}$$

gives rise to a Lefschetz decomposition with ${\mathcal{A}}_{0}=\cdots ={\mathcal{A}}_{(n-1)/2}=\langle {\mathcal{O}},{\mathcal{O}}(1)\rangle$ when $n$ is odd, and a Lefschetz decomposition ${\mathcal{A}}_{0}=\cdots ={\mathcal{A}}_{n/2-1}=\langle {\mathcal{O}},{\mathcal{O}}(1)\rangle$, ${\mathcal{A}}_{n/2}=\langle {\mathcal{O}}\rangle$ when $n$ is even.

Let us focus on the case where $n$ is odd; similar results hold for even $n$. Consider a hyperplane $H\subset \mathbb{P}(\operatorname{Sym}^{2}V)$, such that the associated quadric $Q=f^{-1}(H)\subset \mathbb{P}(V)$ is non-singular. One can then show that there is a semiorthogonal decomposition

(2.2)$$\begin{eqnarray}D^{b}(Q)=\langle S_{+},S_{-},{\mathcal{O}}(2),{\mathcal{O}}(3),\ldots ,{\mathcal{O}}(n)\rangle ,\end{eqnarray}$$

where $S_{+},S_{-}$ denotes (some twist of) the so-called spinor bundles.

The spinor bundles are natural bundles defined on all non-singular quadrics; there are two spinor bundles on even-dimensional quadrics and one on odd-dimensional quadrics (see, for example, [Reference AddingtonAdd09]). In low dimensions the spinor bundles are easily described: for a two-dimensional quadric $\mathbb{P}^{1}\,\times \,\mathbb{P}^{1}$, they are ${\mathcal{O}}(1,0)$ and ${\mathcal{O}}(0,1)$, and for the four-dimensional quadric $\operatorname{Gr}(2,4)$, they are the universal quotient bundle and the dual of the universal subbundle.

Let us write ${\mathcal{C}}_{Q}$ for the component of $D^{b}(Q)$ denoted by ${\mathcal{C}}_{H}$ in the previous section. The decomposition (2.2) implies that ${\mathcal{C}}_{Q}=\langle S_{+},S_{-}\rangle$. The spinor bundles satisfy $\operatorname{RHom}(S_{\pm },S_{\pm })=\mathbb{C}$, and $\operatorname{RHom}(S_{\pm },S_{\mp })=0$, so we have ${\mathcal{C}}_{Q}=\langle S_{+},S_{-}\rangle =D^{b}(\text{pt}\,\sqcup \,\text{pt})$. If we now deform the non-singular quadric to a singular quadric $Q$ of corank 1, the bundles $S_{+}$ and $S_{-}$ become isomorphic, and we can show that in this case ${\mathcal{C}}_{Q}=D^{b}(\operatorname{Spec}k[\unicode[STIX]{x1D700}])$.

Ignoring a technical issue which we will discuss shortly, this tells us exactly what the HP dual variety $Y$ is over the locus in $\mathbb{P}(V^{\vee })$ corresponding to non-singular and corank-1 quadrics. Namely, we see that $Y$ is a double cover of the locus of non-singular quadrics, ramified in the locus of corank-1 quadrics.

The simplest application of Theorem 2.1 is now to the case of a general pencil $\mathbb{P}^{1}=\mathbb{P}(L)\subset \mathbb{P}(\operatorname{Sym}^{2}V^{\vee })$ generated by two quadrics $Q_{1},Q_{2}$. In this case the base locus $X_{L^{\bot }}=Q_{1}\,\cap \,Q_{2}$, and $Y_{L}$ is a double cover of $\mathbb{P}(L)=\mathbb{P}^{1}$, ramified at the $n+1$ points corresponding to singular quadrics in the pencil. Theorem 2.1 then gives an old result of Bondal and Orlov [Reference Bondal and OrlovBO95]:

$$\begin{eqnarray}D^{b}(Q_{1}\,\cap \,Q_{2})=\langle D^{b}(Y_{L}),{\mathcal{O}}(4),\ldots ,{\mathcal{O}}(n)\rangle .\end{eqnarray}$$

The technical issue ignored above is the fact that our description of the HP dual category was only true pointwise and may fail in a global setting. To explain this complication, let us first give Kuznetsov’s general description of the HP dual in terms of Clifford algebras.

Let $V$ be a vector space with a quadratic form $q$. The Clifford algebra $C_{q}$ is defined to be $T(V)/I$, where $T(V)$ is the tensor algebra, and $I$ is the two-sided ideal generated by $v\otimes v-q(v)$. Taking $q=0$ gives the exterior algebra $\wedge ^{\ast }V$, and the Clifford algebras are in this sense deformations of $\wedge ^{\ast }V$. The natural grading on $T(V)$ descends to a $\mathbb{Z}_{2}$-grading on $C_{q}$, and, taking the degree-0 part, we obtain the ‘even Clifford algebra’ $C_{q}^{0}\subset C_{q}$.

Now letting $q\in \mathbb{P}(\operatorname{Sym}^{2}V^{\vee })$ vary, one can fit these even Clifford algebras into a global family, that is, there is a sheaf of algebras $C$ on $\mathbb{P}(\operatorname{Sym}^{2}V^{\vee })$ such that the restriction to each $q\in \mathbb{P}(\operatorname{Sym}^{2}V^{\vee })$ is isomorphic to $C_{q}^{0}$. Kuznetsov shows that the HP dual of $\mathbb{P}(V)$ is the category $D^{b}(\mathbb{P}(\operatorname{Sym}^{2}(V^{\vee })),C)$, that is, the derived category of coherent $C$-modules on $\mathbb{P}(\operatorname{Sym}^{2}(V^{\vee }))$. This means, in particular, that Theorem 2.1 holds when we interpret $D^{b}(Y_{L})$ as $D^{b}(\mathbb{P}(L),C|_{\mathbb{P}(L)})$.

Let us consider what this means for a single quadric. For any $q\in \mathbb{P}(\operatorname{Sym}^{2}V^{\vee })$, if $Q\subset \mathbb{P}(V)$ is the associated quadric, we find ${\mathcal{C}}_{Q}\cong D^{b}(q,C|_{q})=D^{b}(C_{q}^{0})$. If we assume that $q$ and hence $Q$ is non-singular, then it is a classical fact that $C_{q}^{0}\cong \operatorname{End}(\mathbb{C}^{N})\,\oplus \,\operatorname{End}(\mathbb{C}^{N})$ for some $N$. By Morita equivalence we then get $D^{b}(C_{q}^{0})=D^{b}(\operatorname{End}(\mathbb{C}^{N}))\,\oplus \,D^{b}(\operatorname{End}(\mathbb{C}^{N}))\cong D^{b}(\text{pt})\,\oplus \,D^{b}(\text{pt})$. Thus we recover the statement that the fibre of the HP dual at $q$ is two points.

We can now explain the complication in the global description of the HP dual. Keeping to the locus of non-singular $q$, the above discussion shows that the centre of the algebra $C$ is a commutative algebra on $\mathbb{P}(\operatorname{Sym}^{2}V^{\vee })$, whose spectrum is a double cover $Z$. The algebra $C$ is then equivalent to an Azumaya algebra $A$ on $Z$, that is, an algebra which is étale locally isomorphic to ${\mathcal{E}}\mathit{nd}({\mathcal{O}}_{Z}^{N})$. The above results can be rephrased as saying that the HP dual is given by $D^{b}(Z,A)$.

If there exists a locally free sheaf ${\mathcal{E}}$ on $Z$ such that $A\cong {\mathcal{E}}\mathit{nd}({\mathcal{E}})$, we can define an equivalence $D^{b}(Z)\cong D^{b}(Z,A)$ by the inverse functors $-\otimes _{{\mathcal{O}}_{Z}}{\mathcal{E}}$ and $\text{R}{\mathcal{H}}\mathit{om}_{A}({\mathcal{E}},-)$. This can always be done locally, but there is a global obstruction to the existence of such an ${\mathcal{E}}$, known as the Brauer class of $A$, which lives in $H_{\text{an}}^{2}(Z,{\mathcal{O}}_{Z}^{\ast })$. In this example, the Brauer class does not always vanish, and in fact $D^{b}(Z,A)$ is not in general equivalent to $D^{b}(Z)$.

2.5 HP duality for $\operatorname{Sym}^{2}\mathbb{P}(V)$

The motivating problem for this paper is to construct the HP dual of $\operatorname{Sym}^{2}\mathbb{P}(V)$, with respect to the natural map $f:\mathbb{P}(\operatorname{Sym}^{2}\mathbb{P}(V))\rightarrow \mathbb{P}(\operatorname{Sym}^{2}V)$ and a Lefschetz decomposition of $D^{b}(\operatorname{Sym}^{2}\mathbb{P}(V))$ which we describe as follows.

We think of sheaves on $\operatorname{Sym}^{2}\mathbb{P}(V)$ as $\mathbb{Z}_{2}$-equivariant sheaves on $\mathbb{P}(V)^{2}$. For any distinct $i,j\in \mathbb{Z}$, there is a unique $\mathbb{Z}_{2}$-equivariant sheaf whose underlying sheaf on $\mathbb{P}(V)^{2}$ is ${\mathcal{O}}(i,j)\,\oplus \,{\mathcal{O}}(j,i)$. For any $i$, there are two $\mathbb{Z}_{2}$-equivariant structures on ${\mathcal{O}}(i,i)$. We let ${\mathcal{O}}(i,i)_{+}$ be the $\mathbb{Z}_{2}$-structure such that the $\mathbb{Z}_{2}$-action is trivial along the diagonal in $\mathbb{P}(V)^{2}$, and let ${\mathcal{O}}(i,i)_{-}$ be the other one. Note that then ${\mathcal{L}}=f^{\ast }({\mathcal{O}}_{\mathbb{P}(\operatorname{Sym}^{2}V)}(1))={\mathcal{O}}_{\operatorname{Sym}^{2}\mathbb{P}(V)}(1,1)_{+}$.

We take the initial piece in our Lefschetz decomposition of $D^{b}(\operatorname{Sym}^{2}\mathbb{P}(V))$ to be

$$\begin{eqnarray}{\mathcal{A}}_{0}=\langle {\mathcal{O}}(0,0)_{+},{\mathcal{O}}(0,0)_{-},\{{\mathcal{O}}(i,j)\,\oplus \,{\mathcal{O}}(j,i)\}_{(i,j)\in S}\rangle ,\end{eqnarray}$$

where $S=\{(i,j)\,\mid \,i+j\in [0,1],i>j,i-j\leqslant \lfloor n/2\rfloor \}$.Footnote ⁴

If $n$ is odd, we take ${\mathcal{A}}_{i}={\mathcal{A}}_{0}$ for all $i\in [0,n-1]$.

If $n$ is even, we let ${\mathcal{A}}_{0}={\mathcal{A}}_{1}=\cdots ={\mathcal{A}}_{n/2-1}$. We remove 1 element from $S$ to get $S^{\prime }=\{(i,j)\,\mid \,i+j\in [0,1],i>j,i-j\leqslant n/2-1\}$. We let

$$\begin{eqnarray}{\mathcal{A}}^{\prime }=\langle {\mathcal{O}}(0,0)_{+},{\mathcal{O}}(0,0)_{-},\{{\mathcal{O}}(i,j)\,\oplus \,{\mathcal{O}}(j,i)\}_{(i,j)\in S^{\prime }}\rangle ,\end{eqnarray}$$

and then let ${\mathcal{A}}_{n/2}=\cdots ={\mathcal{A}}_{n-1}={\mathcal{A}}^{\prime }$.

By Proposition 5.17, this gives a Lefschetz decomposition

$$\begin{eqnarray}D^{b}(\operatorname{Sym}^{2}\mathbb{P}(V))=\langle {\mathcal{A}}_{0},{\mathcal{A}}_{1}(1),\ldots ,{\mathcal{A}}_{n-1}(n-1)\rangle\end{eqnarray}$$

in both the even and the odd case. Let $X_{L^{\bot }}=f^{-1}(\mathbb{P}(L^{\bot }))$, which is denoted by $X$ in Theorem 1.1. Theorem 1.1 and the computation of the orthogonal complements in Proposition 5.17 then show that we have

$$\begin{eqnarray}D^{b}(X_{L^{\bot }})=\langle {\mathcal{C}}_{L^{\bot }},{\mathcal{A}}_{l}(l),\ldots ,{\mathcal{A}}_{n-1}(n-1)\rangle ,\end{eqnarray}$$

with ${\mathcal{C}}_{L^{\bot }}={\mathcal{W}}_{+}\,\cap \,{\mathcal{W}}_{-,\text{res}}$, which is a fully faithful subcategory of $D^{b}({\mathcal{Y}}_{L},C)_{\text{res}}$. If $D^{b}({\mathcal{Y}},C)_{\text{res}}$ were the HP dual of $\operatorname{Sym}^{2}\mathbb{P}(V)$, this is in accordance with what Theorem 2.1 would give. In view of this and the similar results obtained in [Reference Ballard, Deliu, Favero, Isik and KatzarkovBDFIK17], our results are strong evidence that $D^{b}({\mathcal{Y}},C)_{\text{res}}$ is indeed the HP dual of $\operatorname{Sym}^{2}\mathbb{P}(V)$.

The obstacles to actually proving this are of a technical nature. In a forthcoming paper [Reference RennemoRen17], combining the techniques of [Reference KuznetsovKuz07, Reference Ballard, Deliu, Favero, Isik and KatzarkovBDFIK17], we give a slightly more general definition of HP duality which is more adapted to our VGIT techniques and show that the main theorem of [Reference KuznetsovKuz07] still holds with this definition. We then use the results of this paper to show that Theorem 2.1 indeed holds for the pair of $\operatorname{Sym}^{2}\mathbb{P}(V)$ and $D^{b}({\mathcal{Y}},C)_{\text{res}}$.

2.5.1 Geometric interpretation of the HP dual

Our Proposition 1.2 can be rephrased as saying that when $n=\dim V$ is odd, then away from the corank ${\geqslant}3$ locus the HP dual is a double cover of the corank-1 locus in $\mathbb{P}(\operatorname{Sym}^{2}V^{\vee })$, ramified in the corank-2 locus. Similarly, Proposition 1.3 says that when $n$ is even, the HP dual is a double cover ramified in the corank-1 locus.

Let us show concretely what this means in the case where $n$ is odd. First of all, if $H\subset \operatorname{Sym}^{2}\mathbb{P}(V)$ is such that $X_{H}$ is a non-singular bilinear divisor, then we have

$$\begin{eqnarray}D^{b}(X_{H})=\langle {\mathcal{A}}_{1}(1),\ldots ,{\mathcal{A}}_{n}(n)\rangle ,\end{eqnarray}$$

that is, the interesting part ${\mathcal{C}}_{H}$ is trivial.

Correspondingly, if $X_{H}$ is of corank 1, we would like to say that ${\mathcal{C}}_{H}$ corresponds to the derived category of two points. This is almost correct, but must be modified slightly because the fibre $Y_{H}$ of the double cover $Y\rightarrow \mathbb{P}(\operatorname{Sym}^{2}V^{\vee })$ has higher dimension than expected. The correct statement is that ${\mathcal{C}}_{H}$ is the derived category of the derived fibre product of $0{\hookrightarrow}\mathbb{A}^{1}$ and $(0\,\sqcup \,0)\rightarrow \mathbb{A}^{1}$.

A somewhat surprising aspect of our description is that our HP dual is globally a variety, and that there is no need for an Azumaya algebra or Brauer class as in the case of quadrics. One way of thinking about this is that in the quadric case the spinor bundles, which are pointwise generators for the category of the HP dual, do not extend to globally defined bundles, and this can be explained by the presence of a Brauer twist. In our case, it turns out that we can write down an explicit global object which locally generates the HP dual category; this is the object called $K$ in § 8.

3.1 Coherent and locally free factorisations

We say a factorisation is coherent if the underlying sheaf is. We define the category $D^{b}({\mathcal{X}},W)\subset D({\mathcal{X}},W)$ to be the full subcategory of objects isomorphic to coherent factorisations. The category $D^{b}({\mathcal{X}},W)$ is a generalisation of the usual bounded derived category, which is the special case where $W=0$.

Proposition 3.3 [Reference Ballard, Deliu, Favero, Isik and KatzarkovBDFIK17, Proposition 2.1.6].

If $\mathbb{C}_{R}^{\ast }$ acts trivially on $X$, then

$$\begin{eqnarray}D^{b}({\mathcal{X}},0)\cong D^{b}(X/G).\end{eqnarray}$$

We say that a factorisation is locally free if the underlying sheaf is.

We record the following lemma, which gives a useful criterion for checking that a complex is acyclic.

3.2 Functors

Suppose we are given a map of LG models $f:({\mathcal{X}},W_{{\mathcal{X}}})\rightarrow ({\mathcal{Y}},W_{{\mathcal{Y}}})$, that is, a morphism of stacks $f:{\mathcal{X}}\rightarrow {\mathcal{Y}}$ such that $f^{\ast }{\mathcal{O}}_{{\mathcal{Y}}}[1]\cong {\mathcal{O}}_{{\mathcal{X}}}[1]$, and such that $f^{\ast }W_{{\mathcal{Y}}}=W_{{\mathcal{X}}}$.

For any factorisation ${\mathcal{E}}$, the pushforward $f_{\ast }{\mathcal{E}}$ of its underlying sheaf inherits a differential map which squares to $W_{{\mathcal{Y}}}$, and so becomes a factorisation on $({\mathcal{Y}},W_{{\mathcal{Y}}})$. This defines a pushforward functor $f_{\ast }:K({\mathcal{X}},W_{{\mathcal{X}}})\rightarrow K({\mathcal{Y}},W_{{\mathcal{Y}}})$, which admits a right derived functor

$$\begin{eqnarray}Rf_{\ast }:D({\mathcal{X}},W_{{\mathcal{X}}})\rightarrow D({\mathcal{Y}},W_{{\mathcal{Y}}});\end{eqnarray}$$

see [Reference Ballard, Favero and KatzarkovBFK14, Definition 3.36]. We can compute the pushforward functor by replacing a factorisation ${\mathcal{E}}$ with a quasi-isomorphic injective factorisation ${\mathcal{I}}$.

If $f$ is a closed immersion, then since the pushforward functor of sheaves $f_{\ast }$ is exact, we do not have to choose injective representatives, that is, we always have $Rf_{\ast }({\mathcal{E}})=f_{\ast }({\mathcal{E}})$. Moreover, since the pushforward functor of sheaves then preserves coherent sheaves, it follows that $Rf_{\ast }({\mathcal{E}})$ is coherent if ${\mathcal{E}}$ is; in other words, $Rf_{\ast }$ sends $D^{b}({\mathcal{X}},W_{{\mathcal{X}}})$ to $D^{b}({\mathcal{Y}},W_{{\mathcal{Y}}})$.

We similarly get a functor $f^{\ast }$, which we may left derive by taking locally free replacements to get a functor

$$\begin{eqnarray}Lf^{\ast }:D({\mathcal{Y}},W_{{\mathcal{Y}}})\rightarrow D({\mathcal{X}},W_{{\mathcal{X}}}).\end{eqnarray}$$

This functor clearly sends $D^{b}({\mathcal{Y}},W_{{\mathcal{Y}}})$ to $D^{b}({\mathcal{X}},W_{{\mathcal{X}}})$.

We also have a tensor product. If ${\mathcal{E}}$ is a factorisation on $({\mathcal{X}},W_{1})$ and ${\mathcal{F}}$ a factorisation on $({\mathcal{X}},W_{2})$, we may equip the sheaf ${\mathcal{E}}\otimes {\mathcal{F}}$ with a differential which squares to $W_{1}+W_{2}$, and so becomes a factorisation on $({\mathcal{X}},W_{1}+W_{2})$.

To be precise, the differential is as follows. We have assumed that there exists a 2-torsion element $g\in G$ such that $(g,-1)\in G\,\times \,\mathbb{C}_{R}^{\ast }$ acts trivially on $X$. As a consequence, any sheaf ${\mathcal{E}}$ on ${\mathcal{X}}$ splits canonically into eigensheaves ${\mathcal{E}}_{+}\,\oplus \,{\mathcal{E}}_{-}$, where ${\mathcal{E}}_{\pm }$ is the subsheaf on which $(g,-1)\in G\,\times \,\mathbb{C}_{R}^{\ast }$ acts by $\pm 1$. We can then equip ${\mathcal{E}}\otimes {\mathcal{F}}$ with the differential which is $d_{{\mathcal{E}}}\otimes 1+1\otimes d_{{\mathcal{F}}}$ on ${\mathcal{E}}_{+}\otimes {\mathcal{F}}$ and which is $d_{{\mathcal{E}}}\otimes 1-1\otimes d_{{\mathcal{F}}}$ on ${\mathcal{E}}_{-}\otimes {\mathcal{F}}$. This is essentially the standard sign rule for the tensor product of dg objects.

Replacing either ${\mathcal{E}}$ or ${\mathcal{F}}$ by a locally free factorisation, we obtain a derived tensor product

$$\begin{eqnarray}-\otimes _{{\mathcal{X}}}^{L}\,-:D({\mathcal{X}},W_{1})\,\times \,D({\mathcal{X}},W_{2})\rightarrow D({\mathcal{X}},W_{1}+W_{2}).\end{eqnarray}$$

Given factorisations ${\mathcal{E}},{\mathcal{F}}$ on $({\mathcal{X}},W)$ with ${\mathcal{E}}$ coherent, we have a sheaf hom ${\mathcal{H}}\mathit{om}({\mathcal{E}},{\mathcal{F}})$. This is the usual sheaf hom with the induced differential, where we use the standard sign rule together with the splitting of ${\mathcal{E}}$ and ${\mathcal{F}}$ into even and odd graded parts. The differential on ${\mathcal{H}}\mathit{om}({\mathcal{E}},{\mathcal{F}})$ satisfies $d^{2}=0$. We may derive this to get $\text{R}{\mathcal{H}}\mathit{om}({\mathcal{E}},{\mathcal{F}})\in D({\mathcal{X}},0)$. The derived sheaf hom can be computed by taking either an injective replacement of ${\mathcal{F}}$ or a locally free replacement of ${\mathcal{E}}$. We have an isomorphism $\text{R}{\mathcal{H}}\mathit{om}({\mathcal{E}},{\mathcal{F}})\cong {\mathcal{F}}\otimes ^{L}{\mathcal{E}}^{\vee }$, where ${\mathcal{E}}=\text{R}{\mathcal{H}}\mathit{om}({\mathcal{E}},{\mathcal{O}}_{{\mathcal{X}}})$.

3.2.1 Twisted inverse image functor

The following basic version of Grothendieck duality will be useful. Let $({\mathcal{X}},W_{{\mathcal{X}}})$ and $({\mathcal{Y}},W_{{\mathcal{Y}}})$ be gauged LG models, and let $i:{\mathcal{X}}\rightarrow {\mathcal{Y}}$ be a closed immersion of ${\mathcal{X}}$ in ${\mathcal{Y}}$ of codimension $c$. Since both ${\mathcal{X}}$ and ${\mathcal{Y}}$ are smooth over $\mathbb{C}$, this is a regular embedding, and so the normal sheaf $N_{{\mathcal{X}}/{\mathcal{Y}}}$ on ${\mathcal{X}}$ is free of rank $c$. We define $\unicode[STIX]{x1D714}_{{\mathcal{X}}/{\mathcal{Y}}}=\wedge ^{c}N_{{\mathcal{X}}/{\mathcal{Y}}}$, and let $i^{!}:D^{b}({\mathcal{Y}},W_{{\mathcal{Y}}})\rightarrow D^{b}({\mathcal{X}},W_{{\mathcal{X}}})$ be defined by $i^{!}({\mathcal{E}})=i^{\ast }({\mathcal{E}})\otimes \unicode[STIX]{x1D714}_{{\mathcal{X}}/{\mathcal{Y}}}[-c]$.

Lemma 3.6. The functor $i^{!}$ is right adjoint to $i_{\ast }$.

Proof. The object ${\mathcal{H}}\mathit{om}_{{\mathcal{Y}}}(i_{\ast }{\mathcal{O}}_{{\mathcal{X}}},{\mathcal{O}}_{{\mathcal{Y}}})\in D^{b}({\mathcal{Y}},0)$ has an ${\mathcal{O}}_{{\mathcal{X}}}$-module structure and so is canonically the pushforward along $i$ of an object we denote by ${\mathcal{H}}\mathit{om}_{{\mathcal{Y}}}({\mathcal{O}}_{{\mathcal{X}}},{\mathcal{O}}_{{\mathcal{Y}}})\in D^{b}({\mathcal{X}},0)$. A standard computation (cf. [Sta19, Lemma 0BQZ]) gives the equivalence

$$\begin{eqnarray}{\mathcal{H}}\mathit{om}_{{\mathcal{Y}}}({\mathcal{O}}_{{\mathcal{X}}},{\mathcal{O}}_{{\mathcal{Y}}})\cong \unicode[STIX]{x1D714}_{{\mathcal{X}}/{\mathcal{Y}}}[-c].\end{eqnarray}$$

Moreover, for any ${\mathcal{E}}\in D^{b}({\mathcal{X}},0)$, we have $\operatorname{RHom}(i_{\ast }{\mathcal{E}},{\mathcal{O}}_{{\mathcal{Y}}})\cong \operatorname{RHom}({\mathcal{E}},{\mathcal{H}}\mathit{om}_{{\mathcal{Y}}}({\mathcal{O}}_{{\mathcal{X}}},{\mathcal{O}}_{{\mathcal{Y}}}))$; cf. [Sta19, Lemma 0E2I].

Now if ${\mathcal{E}}\in D^{b}({\mathcal{X}},W_{{\mathcal{X}}})$ and ${\mathcal{F}}\in D^{b}({\mathcal{Y}},W_{{\mathcal{Y}}})$, we have

$$\begin{eqnarray}\displaystyle \operatorname{RHom}(i_{\ast }{\mathcal{E}},{\mathcal{F}}) & \cong & \displaystyle \operatorname{RHom}(i_{\ast }({\mathcal{E}}\otimes i^{\ast }{\mathcal{F}}^{\vee }),{\mathcal{O}}_{{\mathcal{Y}}})\nonumber\\ \displaystyle & \cong & \displaystyle \operatorname{RHom}({\mathcal{E}}\otimes i^{\ast }({\mathcal{F}}^{\vee }),{\mathcal{H}}\mathit{om}_{{\mathcal{Y}}}(i_{\ast }{\mathcal{O}}_{{\mathcal{X}}},{\mathcal{O}}_{{\mathcal{Y}}}))\nonumber\\ \displaystyle & \cong & \displaystyle \operatorname{RHom}({\mathcal{E}}\otimes i^{\ast }({\mathcal{F}}^{\vee }),\unicode[STIX]{x1D714}_{{\mathcal{X}}/{\mathcal{Y}}}[-c])\nonumber\\ \displaystyle & \cong & \displaystyle \operatorname{RHom}({\mathcal{E}}\otimes ,i^{\ast }{\mathcal{F}}\otimes \unicode[STIX]{x1D714}_{{\mathcal{X}}/{\mathcal{Y}}}[-c])\cong \operatorname{RHom}({\mathcal{E}},i^{!}{\mathcal{F}}).\Box \nonumber\end{eqnarray}$$

3.3 Resolutions

A sheaf ${\mathcal{F}}$ on ${\mathcal{X}}$ supported on $\{W=0\}$, equipped with the trivial differential, is a factorisation on ${\mathcal{X}}$. This provides a useful supply of objects in $D^{b}({\mathcal{X}},W)$. Lemma 3.8 gives a way of constructing a locally free representative of such an ${\mathcal{F}}$.

As a matter of notation, we write

(3.7)$$\begin{eqnarray}{\mathcal{E}}_{n}\overset{d_{r}}{\underset{d_{l}}{\rightleftarrows }}\cdots \overset{d_{r}}{\underset{d_{l}}{\rightleftarrows }}{\mathcal{E}}_{1}\overset{d_{r}}{\underset{d_{l}}{\rightleftarrows }}{\mathcal{E}}_{0}\end{eqnarray}$$

to mean the factorisation $({\mathcal{E}},d)$, where ${\mathcal{E}}=\,\bigoplus \,{\mathcal{E}}_{i}$ and $d=d_{r}+d_{l}$. Note that the arrows $d_{l},d_{r}$ in (3.7) are then maps of degree $1$ with respect to the $\mathbb{C}_{R}^{\ast }$-action.

Lemma 3.8. Let

$$\begin{eqnarray}{\mathcal{E}}={\mathcal{E}}_{n}\overset{d_{r}}{\underset{d_{l}}{\rightleftarrows }}\cdots \overset{d_{r}}{\underset{d_{l}}{\rightleftarrows }}{\mathcal{E}}_{0}\end{eqnarray}$$

be a factorisation, and suppose that there is a map of sheaves ${\mathcal{E}}_{0}\stackrel{f}{\rightarrow }{\mathcal{F}}$, such that the sequence

$$\begin{eqnarray}0\rightarrow {\mathcal{E}}_{n}\stackrel{d_{r}}{\rightarrow }\cdots \stackrel{d_{r}}{\rightarrow }{\mathcal{E}}_{0}\stackrel{f}{\rightarrow }{\mathcal{F}}\rightarrow 0\end{eqnarray}$$

is exact. Then ${\mathcal{F}}$ is scheme-theoretically supported on $\{W=0\}$.

Thinking of ${\mathcal{F}}$ as a factorisation on $({\mathcal{X}},W)$ with trivial differential, the induced map of factorisations ${\mathcal{E}}\rightarrow {\mathcal{E}}_{0}\stackrel{f}{\rightarrow }{\mathcal{F}}$ is a quasi-isomorphism.

3.4 Change of $R$-grading

For the purpose of constructing $D({\mathcal{X}},W)$ and $D^{b}({\mathcal{X}},W)$, the definition of gauged LG model that we use contains some superfluous information, because the splitting of $\overline{G}=G\,\times \,\mathbb{C}_{R}^{\ast }$ can be replaced with the choice of a surjection $\overline{G}\rightarrow \mathbb{C}_{R}^{\ast }$. Indeed, the only information used in the definition of $D({\mathcal{X}},W)$ is the structure ${\mathcal{X}}=X/\overline{G}$ as a stack and the line bundle ${\mathcal{O}}_{{\mathcal{X}}}[1]$.

We draw the following consequence. Suppose that there is an automorphism $\unicode[STIX]{x1D70E}:\overline{G}\rightarrow \overline{G}$ commuting with the projection $G\,\times \,\mathbb{C}_{R}^{\ast }\rightarrow \mathbb{C}_{R}^{\ast }$. Then we may replace the action $\unicode[STIX]{x1D70C}:\overline{G}\rightarrow \operatorname{Aut}(X)$ by $\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70E}$, without modifying the category $D^{b}({\mathcal{X}},W)$. In particular, fixing the action of $G$ on $X$, different choices of $\mathbb{C}_{R}^{\ast }$-action may give the same derived category $D^{b}({\mathcal{X}},W)$.

Consider, for instance, the LG model where $X=\mathbb{C}^{n}\setminus 0$, $G=\mathbb{C}^{\ast }$, $W=0$, and both $G$ and $\mathbb{C}_{R}^{\ast }$ act by the usual multiplication. Then by the above remarks we may instead take the same model with trivial $\mathbb{C}_{R}^{\ast }$-action, without changing the category of factorisations. Hence by Proposition 3.3 we have $D^{b}({\mathcal{X}},0)\cong D^{b}(\mathbb{P}^{n-1})$.

We will use this flexibility to choose different $\mathbb{C}_{R}^{\ast }$-actions at various points throughout the proof. The choice of $\unicode[STIX]{x1D70E}\in \operatorname{Aut}(\overline{G})$ will in our case be unique, so that the categories corresponding to different choices are canonically equivalent.

3.5 Notational abuse

From the next section onwards we will drop the $\mathbb{C}_{R}^{\ast }$-action from the notation and denote the LG model simply by $(X/G,W)$, and the category of factorisations by either $D^{b}(X/G,W)$ or $D_{G}^{b}(X,W)$. As justification for this abuse, we offer Proposition 3.3, whose conclusion then has the natural form $D^{b}(X/G,0)=D^{b}(X/G)$.

4.1 The positive GIT quotient

Choosing $\unicode[STIX]{x1D712}$ as our linearisation, the unstable locus is

$$\begin{eqnarray}Z_{+}^{\text{us}}=(0\,\times \,V\,\times \,L)\cup (V\,\times \,0\,\times \,L).\end{eqnarray}$$

Recall that by ${\mathcal{O}}_{\operatorname{Sym}^{2}\mathbb{P}(V)}(-1,-1)_{+}$ we mean the $\mathbb{Z}_{2}$-equivariant line bundle ${\mathcal{O}}_{\mathbb{P}(V)^{2}}(-1,-1)$ on $\mathbb{P}(V)^{2}$, equipped with the unique lifting of the $\mathbb{Z}_{2}$-action which leaves the restriction of ${\mathcal{O}}_{\mathbb{P}(V)^{2}}(-1,-1)$ to the diagonal of $\mathbb{P}(V)^{2}$ fixed. The GIT quotient ${\mathcal{Z}}_{+}=(Z\setminus Z_{+}^{\text{us}})/G$ is then the stacky vector bundle

$$\begin{eqnarray}p:{\mathcal{O}}_{\operatorname{Sym}^{2}\mathbb{P}(V)}(-1,-1)_{+}^{\oplus l}\rightarrow \operatorname{Sym}^{2}\mathbb{P}(V).\end{eqnarray}$$

We get a superpotential $W$ on ${\mathcal{Z}}_{+}$ by restriction from ${\mathcal{Z}}$. In each fibre of $p$, the function $W$ is linear, and so dually gives a section

$$\begin{eqnarray}s\in \unicode[STIX]{x1D6E4}(\operatorname{Sym}^{2}\mathbb{P}(V),{\mathcal{O}}(1,1)_{+}^{\oplus l}).\end{eqnarray}$$

We let $X\subset \operatorname{Sym}^{2}\mathbb{P}(V)$ be defined by the vanishing of this $s$. Equivalently, $X$ is defined by the Cartesian diagram

and therefore corresponds to the $X$ in Theorem 1.1.

Let $i$ denote the inclusion $p^{-1}(X){\hookrightarrow}{\mathcal{Z}}_{+}$.

Proposition 4.1. If $X$ has the expected dimension, then the functor

$$\begin{eqnarray}i_{\ast }\,\circ \,p^{\ast }:D^{b}(X)\rightarrow D^{b}({\mathcal{Z}}_{+},W)\end{eqnarray}$$

is an equivalence.

4.1.1 An alternative Knörrer functor

Let $K$ denote the functor $i_{\ast }\,\circ \,p^{\ast }$ from Proposition 4.1. If ${\mathcal{E}}\in D^{b}(X)$ is restricted from $\operatorname{Sym}^{2}\mathbb{P}(V)$, then we may describe $K({\mathcal{E}})$ differently, a result which will be needed in § 5.3.

Let $j_{1}:X\rightarrow \operatorname{Sym}^{2}\mathbb{P}(V)$ be the inclusion, and let $j_{2}:\operatorname{Sym}^{2}\mathbb{P}(V)\rightarrow {\mathcal{Z}}_{+}$ be the inclusion along the 0-section of $p$. Let the functors $\unicode[STIX]{x1D6F7}_{1},\unicode[STIX]{x1D6F7}_{2}:D^{b}(\operatorname{Sym}^{2}\mathbb{P}(V))\rightarrow D^{b}({\mathcal{Z}}_{+},W)$ be given by

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{1}=K\,\circ \,j_{1}^{\ast }\end{eqnarray}$$

and

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{2}=(j_{2})_{\ast }(-\otimes {\mathcal{O}}(-1,-1)_{+}^{\otimes l}[l]).\end{eqnarray}$$

Lemma 4.2. Assume that $X$ has the expected codimension. Then $\unicode[STIX]{x1D6F7}_{1}$ and $\unicode[STIX]{x1D6F7}_{2}$ are equivalent.

Proof. We ignore the cohomological shifts for the sake of ease of notation. We have

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{1}({\mathcal{E}})=i_{\ast }p^{\ast }j_{1}^{\ast }({\mathcal{E}})=i_{\ast }i^{\ast }p^{\ast }({\mathcal{E}})=p^{\ast }({\mathcal{E}})\otimes i_{\ast }{\mathcal{O}}_{p^{-1}(X)},\end{eqnarray}$$

where $p^{\ast }({\mathcal{E}})\in D^{b}({\mathcal{Z}},0)$ and ${\mathcal{O}}_{p^{-1}(X)}\in D^{b}({\mathcal{Z}}_{+},W)$. Next we have

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{2}({\mathcal{E}})=(j_{2})_{\ast }({\mathcal{E}}\otimes {\mathcal{O}}(-1,-1)_{+}^{\otimes l})=p^{\ast }({\mathcal{E}})\otimes (j_{2})_{\ast }({\mathcal{O}}_{\operatorname{Sym}^{2}\mathbb{P}(V)}\otimes {\mathcal{O}}(-1,-1)_{+}^{\oplus l}),\end{eqnarray}$$

with $p^{\ast }({\mathcal{E}})\in D^{b}({\mathcal{Z}}_{+},0)$ and $(j_{2})_{\ast }({\mathcal{O}}_{\operatorname{Sym}^{2}\mathbb{P}(V)}\otimes {\mathcal{O}}(-1,-1)_{+}^{\otimes l})\in D^{b}({\mathcal{Z}}_{+},W)$.

So the functors $\unicode[STIX]{x1D6F7}_{i}$ for $i=1,2$ are equivalent to $(-\otimes {\mathcal{K}}_{i})\,\circ \,p^{\ast }$, where $p^{\ast }:D^{b}(\operatorname{Sym}^{2}\mathbb{P}(V))\rightarrow D^{b}({\mathcal{Z}}_{+},0)$ and ${\mathcal{K}}_{i}\in D^{b}({\mathcal{Z}}_{+},W)$ are given by

$$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{K}}_{1}=i_{\ast }{\mathcal{O}}_{p^{-1}(X)}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle {\mathcal{K}}_{2}=(j_{2})_{\ast }({\mathcal{O}}_{\operatorname{Sym}^{2}\mathbb{P}(V)}\otimes {\mathcal{O}}(-1,-1)_{+}^{\oplus l}). & \displaystyle \nonumber\end{eqnarray}$$

The claim thus follows once we show that $i_{\ast }{\mathcal{O}}_{p^{-1}(X)}\cong (j_{2})_{\ast }({\mathcal{O}}_{\operatorname{Sym}^{2}\mathbb{P}(V)}\otimes {\mathcal{O}}(-1,-1)_{+}^{\otimes l})$. Let us first work on ${\mathcal{Z}}$. Choose coordinates $p_{1},\ldots ,p_{l}$ on $L$. Then the potential $W$ decomposes as $\sum p_{i}f_{i}$, where the $f_{i}$ are symmetric bilinear forms on $V\,\times \,V$. Define an object ${\mathcal{K}}_{3}\in D^{b}({\mathcal{Z}},W)$ by the explicit factorisation

where the $d_{r}$ are contractions with $(p_{i})\in \unicode[STIX]{x1D6E4}(p^{\ast }({\mathcal{O}}(-1,-1)_{+})^{\oplus l})$ and the $d_{l}$ are exterior multiplications with $(f_{i})\in \unicode[STIX]{x1D6E4}(p^{\ast }({\mathcal{O}}(-1,-1)_{+})^{\oplus l})$ (see [Reference DyckerhoffDyc11, § 2.3]). Restricting ${\mathcal{K}}_{3}$ to ${\mathcal{Z}}_{+}$, we see that since $\operatorname{Sym}^{2}\mathbb{P}(V)$ is cut out by the $p_{i}$, since $p^{-1}(X)$ is cut out by the $f_{i}$, and since both of these have codimension $l$, it follows from Lemma 3.8 that

$$\begin{eqnarray}i_{\ast }{\mathcal{O}}_{p^{-1}(X)}\cong {\mathcal{K}}_{3}|_{{\mathcal{Z}}_{+}}\cong (j_{2})_{\ast }({\mathcal{O}}_{\operatorname{Sym}^{2}\mathbb{P}(V)}\otimes {\mathcal{O}}(-1,-1)_{+}^{\otimes l}).\Box\end{eqnarray}$$

4.2 The negative GIT quotient

If we take the character $\unicode[STIX]{x1D712}^{-1}$ as our linearisation, then the unstable locus is

$$\begin{eqnarray}Z_{-}^{\text{us}}=V\,\times \,V\,\times \,0.\end{eqnarray}$$

We let $Z_{-}^{\text{ss}}=Z\setminus Z_{-}^{\text{us}}$ and let ${\mathcal{Z}}_{-}=(Z_{-}^{\text{ss}})/G$.

When dealing with ${\mathcal{Z}}_{-}$ and its substacks, we will assume the $\mathbb{C}_{R}^{\ast }$-action on $Z$ is the one which scales the $V$-factors by $t_{R}$ and leaves $L$ fixed. The remarks in § 3.4 show that the category $D^{b}({\mathcal{Z}}_{-},W)$ is the same for this $\mathbb{C}_{R}^{\ast }$-action as for the one described before.

The projection $Z_{-}^{\text{ss}}\rightarrow L\setminus 0$ and the character $\unicode[STIX]{x1D712}:G\rightarrow \mathbb{C}^{\ast }$ together give a map of quotient stacks

$$\begin{eqnarray}f:{\mathcal{Z}}_{-}=Z_{-}^{\text{ss}}/G\rightarrow (L\setminus 0)/\mathbb{C}^{\ast }=\mathbb{P}(L).\end{eqnarray}$$

The kernel of $\unicode[STIX]{x1D712}$ is isomorphic to $O(2)$, and each fibre of $f$ is isomorphic to $V\,\times \,V/O(2)$. The action of $O(2)$ on $V\,\times \,V$ is the one given by identifying $V\,\times \,V$ with $V\otimes \mathbb{C}^{2}$, where $\mathbb{C}^{2}$ is the standard representation of $O(2)$. In particular, each fibre of $f$ contains a point with positive-dimensional stabiliser group, so that ${\mathcal{Z}}_{-}$ is not a Deligne–Mumford stack.

4.2.1 Grade-restricted objects

Because ${\mathcal{Z}}_{-}$ contains points with positive-dimensional isotropy groups, the category $D^{b}({\mathcal{Z}}_{-},W)$ is too big to be directly compared with $D^{b}({\mathcal{Z}}_{+},W)$ in the way we would like. For example, for each stacky point $p$ and each representation $\unicode[STIX]{x1D70C}$ of $O(2)$, we have an object ${\mathcal{O}}_{p}(\unicode[STIX]{x1D70C})\in D^{b}({\mathcal{Z}}_{-},W)$. It will therefore be useful to consider a full subcategory of $D^{b}({\mathcal{Z}}_{-},W)$, defined as follows.

Choose a point $p\in \mathbb{P}(L)$, and let $\mathbb{C}^{\ast }$ be the connected component of its isotropy group $O(2)$ in ${\mathcal{Z}}_{-}$. The category $D_{\mathbb{C}^{\ast }}^{b}(p)$ splits as $\bigoplus _{k}\langle {\mathcal{O}}_{p}(k)\rangle$. If ${\mathcal{E}}\in D_{\mathbb{C}^{\ast }}^{b}(p)$, then we let ${\mathcal{E}}_{k}$ denote the projection of ${\mathcal{E}}$ to $\langle {\mathcal{O}}(k)\rangle$.

The inclusion $i_{p}:p/\mathbb{C}^{\ast }\rightarrow {\mathcal{Z}}_{-}$ induces a restriction functor

$$\begin{eqnarray}i_{p}^{\ast }:D^{b}({\mathcal{Z}}_{-},W)\rightarrow D_{\mathbb{C}^{\ast }}^{b}(p).\end{eqnarray}$$

For ${\mathcal{E}}\in D^{b}({\mathcal{Z}}_{-},W)$, we let the ‘weights of ${\mathcal{E}}$ at $p$’ be the set $\{k\,\mid \,(i_{p}^{\ast }{\mathcal{E}})_{k}\not =0\}$. We say that ${\mathcal{E}}$ is grade restricted if at every $p$ the weights of ${\mathcal{E}}$ are contained in the interval $[-\lfloor n/2\rfloor ,\lfloor n/2\rfloor ]$, where $n=\dim V$. We denote the full subcategory of grade-restricted objects in $D^{b}({\mathcal{Z}}_{-},W)$ by $D^{b}({\mathcal{Z}}_{-},W)_{\text{res}}$.

5.1 Essential surjectivity

The goal of this section is to prove that $\unicode[STIX]{x1D6F7}_{+}:{\mathcal{W}}_{+}\rightarrow D^{b}({\mathcal{X}}_{+},W)$ is essentially surjective, which follows from Lemma 5.4 and Proposition 5.5.

Proof. Let $\overline{G}=G\,\times \,\mathbb{C}_{R}^{\ast }$, let $j$ be the inclusion $U/\overline{G}{\hookrightarrow}X/\overline{G}$, and let ${\mathcal{E}}$ be a coherent factorisation on $U/\overline{G}$. The (underived) pushforward $j_{\ast }({\mathcal{E}})$ is a quasi-coherent factorisation on $X/\overline{G}$; in particular, its underlying sheaf is a quasi-coherent sheaf on $X/\overline{G}$. By [Reference Laumon and Moret-BaillyLM00, Proposition 15.4], the underlying sheaf of $j_{\ast }({\mathcal{E}})$ is the union of its coherent subsheaves; that is, thought of as a $\overline{G}$-equivariant sheaf on $X$, it is the union of its $\overline{G}$-equivariant coherent subsheaves. Since $j^{\ast }j_{\ast }({\mathcal{E}})={\mathcal{E}}$, we may find a coherent subsheaf $\unicode[STIX]{x1D719}:{\mathcal{F}}{\hookrightarrow}j_{\ast }({\mathcal{E}})$ such that the composition $j^{\ast }({\mathcal{F}})\stackrel{j^{\ast }(\unicode[STIX]{x1D719})}{\rightarrow }j^{\ast }j_{\ast }{\mathcal{E}}\rightarrow {\mathcal{E}}$ is an isomorphism.

Let $\unicode[STIX]{x1D713}$ be the composition ${\mathcal{F}}[-1]\stackrel{\unicode[STIX]{x1D719}[-1]}{\rightarrow }j_{\ast }{\mathcal{E}}[-1]\stackrel{d}{\rightarrow }j_{\ast }{\mathcal{E}}$, and let $\overline{{\mathcal{F}}}=\unicode[STIX]{x1D719}({\mathcal{F}})+\unicode[STIX]{x1D713}({\mathcal{F}}[-1])\subset j_{\ast }{\mathcal{E}}$. We now claim that the differential $d:{\mathcal{E}}\rightarrow {\mathcal{E}}[1]$ restricts to induce a differential on $\overline{{\mathcal{F}}}$, that is, that $d(\overline{{\mathcal{F}}})\subset \overline{{\mathcal{F}}}[1]$. Indeed, we have

$$\begin{eqnarray}d(\unicode[STIX]{x1D719}({\mathcal{F}}))=f({\mathcal{F}}[-1])[1]\subset \overline{{\mathcal{F}}}[1]\end{eqnarray}$$

and

$$\begin{eqnarray}\displaystyle d(\unicode[STIX]{x1D713}({\mathcal{F}}[-1])) & = & \displaystyle d\,\circ \,d\,\circ \,(\unicode[STIX]{x1D719}[-1])({\mathcal{F}}[-1])\nonumber\\ \displaystyle & = & \displaystyle W\,\circ \,(\unicode[STIX]{x1D719}[-1])({\mathcal{F}}[-1])\subset (\unicode[STIX]{x1D719}[-1])({\mathcal{F}}[-1])[2]\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D719}({\mathcal{F}})[1]\subset \overline{{\mathcal{F}}}[1].\nonumber\end{eqnarray}$$

Thus the differential on $j_{\ast }{\mathcal{E}}$ restricts to a differential on $\overline{{\mathcal{F}}}$. We also have that $j^{\ast }{\mathcal{F}}\rightarrow j^{\ast }\overline{{\mathcal{F}}}$ is an isomorphism, and so the map $j^{\ast }\overline{{\mathcal{F}}}\rightarrow j^{\ast }j_{\ast }{\mathcal{E}}\rightarrow {\mathcal{E}}$ is a morphism of factorisations which induces an isomorphism on the underlying sheaves. Hence $j^{\ast }\overline{{\mathcal{F}}}\cong {\mathcal{E}}$ as objects of $D^{b}(X/G,W)$, and so ${\mathcal{E}}$ is in the image of the restriction functor $j^{\ast }$.◻

Recall that $j_{+}$ denotes the inclusion ${\mathcal{Z}}_{+}{\hookrightarrow}{\mathcal{Z}}$. We write $\ker j_{+}^{\ast }$ for the full subcategory of $D^{b}({\mathcal{Z}},W)$ consisting of objects ${\mathcal{E}}$ such that $j_{+}^{\ast }({\mathcal{E}})\cong 0$.

The proof of this proposition occupies the rest of § 5.1; the main idea is as follows. If${\mathcal{E}}\in D^{b}({\mathcal{Z}},W)$ and the weights of ${\mathcal{E}}$ are not contained in $S_{+}$, then we can construct a factorisation ${\mathcal{G}}$ supported on ${\mathcal{Z}}\setminus {\mathcal{Z}}_{+}$ (so that ${\mathcal{G}}\in \ker j_{+}^{\ast }$), which admits a map to or from ${\mathcal{E}}$ with cone ${\mathcal{F}}$. By setting this up correctly, we can ensure that the weights of ${\mathcal{F}}$ are closer to being contained in $S_{+}$ than those of ${\mathcal{E}}$, and by repeatedly replacing ${\mathcal{E}}$ with ${\mathcal{F}}$ we eventually arrive at an ${\mathcal{E}}$ whose weights are in $S_{+}$.

5.1.1 Affine spaces with torus actions

Let $X=\operatorname{Spec}\mathbb{C}[x_{1},\ldots ,x_{n}]$, and let $T$ be a $k$-dimensional torus acting linearly on $X$. We assume that the action is such that $\unicode[STIX]{x1D6E4}(X,{\mathcal{O}}_{X})^{T}=\mathbb{C}$. We will need some simple lemmas about $D_{T}^{b}(X)$.

Remark 5.6. The condition that $\unicode[STIX]{x1D6E4}(X,{\mathcal{O}}_{X})^{T}=\mathbb{C}$ can be reformulated as follows. Let $\unicode[STIX]{x1D70C}_{1},\ldots ,\unicode[STIX]{x1D70C}_{n}\in \unicode[STIX]{x1D712}(T)\cong \mathbb{Z}^{k}$ be the weights of the $T$-action on $X$. Then $\unicode[STIX]{x1D6E4}(X,{\mathcal{O}}_{X})^{T}=\mathbb{C}$ if and only if $0\in \unicode[STIX]{x1D712}(T)$ is not in the convex hull of $\{\unicode[STIX]{x1D70C}_{1},\ldots ,\unicode[STIX]{x1D70C}_{n}\}$.

In the setting of this paper, we have $T=(\mathbb{C}^{\ast })^{2}$ acting on $V\,\times \,V\,\times \,L$ with weights $(1,0)$, $(0,1)$, $(-1,-1)$. We see that the condition on $X$ fails for $X=V\,\times \,V\,\times \,L$, but holds for $X=V\,\times \,0\,\times \,L$, $X=0\,\times \,V\,\times \,L$ and $X=V\,\times \,V\,\times \,0$.

Let $\unicode[STIX]{x1D70C}_{i}\in \unicode[STIX]{x1D712}(T)$ be the character such that $x_{i}$ is a $T$-invariant section of ${\mathcal{O}}_{X}(\unicode[STIX]{x1D70C}_{i})$. Define a partial ordering on $\mathbb{Z}^{k}=\unicode[STIX]{x1D712}(T)$ by saying that $\unicode[STIX]{x1D70C}\leqslant \unicode[STIX]{x1D70C}^{\prime }$ if there exist $i_{1},\ldots ,i_{k}\geqslant 0$ such that $\unicode[STIX]{x1D70C}^{\prime }=\unicode[STIX]{x1D70C}+\sum i_{j}\unicode[STIX]{x1D70C}_{k}$. This is equivalent to saying that $\unicode[STIX]{x1D70C}\leqslant \unicode[STIX]{x1D70C}^{\prime }$ if and only if $\operatorname{Hom}_{X}({\mathcal{O}}(\unicode[STIX]{x1D70C}),{\mathcal{O}}(\unicode[STIX]{x1D70C}^{\prime }))^{T}\not =0$.

We say that a triangulated category ${\mathcal{C}}$ is generated by a set $S\subset Ob({\mathcal{C}})$ if ${\mathcal{C}}$ is the smallest full triangulated subcategory containing all objects in $S$.

Lemma 5.7. The category $D_{T}^{b}(X)$ is generated by $\{{\mathcal{O}}_{X}(\unicode[STIX]{x1D70C})\}_{\unicode[STIX]{x1D70C}\in \unicode[STIX]{x1D712}(T)}$, and for any $n\in \mathbb{Z}$ we have $\operatorname{Hom}({\mathcal{O}}_{X}(\unicode[STIX]{x1D70C}),{\mathcal{O}}_{X}(\unicode[STIX]{x1D70C}^{\prime })[n])=0$ unless $\unicode[STIX]{x1D70C}\leqslant \unicode[STIX]{x1D70C}^{\prime }$.

Proof. By [Reference Ballard, Favero and KatzarkovBFK19, Proposition 2.2.10], any $T$-equivariant complex of sheaves on $X$ is isomorphic to a complex of locally free $T$-equivariant sheaves. By [Reference Masuda, Moser-Jauslin and PetrieMMP96, Main Theorem], any $T$-equivariant locally free sheaf on $X$ decomposes as a direct sum of copies of ${\mathcal{O}}_{X}(\unicode[STIX]{x1D70C})$. This proves generation.

For the second statement, note that as $T$ is reductive, we have

$$\begin{eqnarray}\operatorname{Hom}_{X/T}({\mathcal{O}}(\unicode[STIX]{x1D70C}),{\mathcal{O}}(\unicode[STIX]{x1D70C}^{\prime })[n])=\operatorname{Ext}_{X}^{n}({\mathcal{O}}(\unicode[STIX]{x1D70C}),{\mathcal{O}}(\unicode[STIX]{x1D70C}^{\prime }))^{T}.\end{eqnarray}$$

As $X$ is affine and ${\mathcal{O}}(\unicode[STIX]{x1D70C})$ locally free, this vanishes if $n\not =0$, and the claim follows.◻

Let ${\mathcal{E}}\in D_{T}^{b}(X)$. We let the weights of ${\mathcal{E}}$, denoted by $\operatorname{wt}({\mathcal{E}})$, be the set of $\unicode[STIX]{x1D70C}\in \unicode[STIX]{x1D712}(T)$ such that $({\mathcal{E}}|_{0})_{\unicode[STIX]{x1D70C}}\not =0$, where the subscript $\unicode[STIX]{x1D70C}$ denotes the projection to the subcategory $\langle {\mathcal{O}}(\unicode[STIX]{x1D70C})\rangle \subset D_{T}^{b}(0)$.

We say that $S\subset \operatorname{wt}({\mathcal{E}})$ is a maximal set if no element of $S$ is smaller than an element of $\operatorname{wt}({\mathcal{E}})\setminus S$, and we say that $S\subset \operatorname{wt}({\mathcal{E}})$ is a minimal set if no element of $S$ is bigger than an element of $\operatorname{wt}({\mathcal{E}})\setminus S$.

For any $S\subset \unicode[STIX]{x1D712}(T)$, we let $D_{T}^{b}(X)_{S}\subset D_{T}^{b}(X)$ be the full subcategory of objects ${\mathcal{E}}$ with weights in $S$.

Lemma 5.8. If ${\mathcal{E}}\in D_{T}^{b}(X)$ is such that ${\mathcal{E}}|_{0}=0\in D_{T}^{b}(0)$, then ${\mathcal{E}}=0$.

Proof. We reuse the argument from the proof of Corollary 5.3. The support of ${\mathcal{E}}$ is a closed $T$-invariant subset of $Y$. Assume that ${\mathcal{E}}|_{0}=0$. Then the support of ${\mathcal{E}}$ does not intersect $0$. The assumption $\unicode[STIX]{x1D6E4}(X,{\mathcal{O}}_{X})^{T}=\mathbb{C}$ implies that the closure of every $T$-orbit intersects $0$, hence the support of ${\mathcal{E}}$ is empty and so ${\mathcal{E}}=0$.◻

For any $S\subset \unicode[STIX]{x1D712}(T)$, we let $\overline{S}$ be the set of all $\unicode[STIX]{x1D70C}$ such that $\unicode[STIX]{x1D70C}\geqslant s$ for some $s\in S$, and let $\text{}\underline{S}$ be the set of $\unicode[STIX]{x1D70C}$ such that $\unicode[STIX]{x1D70C}\leqslant s$ for some $s\in S$.

Lemma 5.9. For any $S\subset \unicode[STIX]{x1D712}(T)$, there are semiorthogonal decompositions

$$\begin{eqnarray}D_{T}^{b}(X)=\langle D_{T}^{b}(X)_{\unicode[STIX]{x1D712}(T)\setminus \overline{S}},D_{T}^{b}(X)_{\overline{S}}\rangle\end{eqnarray}$$

and

$$\begin{eqnarray}D_{T}^{b}(X)=\langle D_{T}^{b}(X)_{\text{}\underline{S}},D_{T}^{b}(X)_{\unicode[STIX]{x1D712}(T)\setminus \text{}\underline{S}}\rangle .\end{eqnarray}$$

Proof. We prove the first claim; the second is proved in the same way. We work in the homotopy category ${\mathcal{C}}$ of bounded locally free complexes on $X/T$, which is equivalent to $D_{T}^{b}(X)$. By [Reference Masuda, Moser-Jauslin and PetrieMMP96, Main Theorem], any locally free sheaf is a direct sum of sheaves of the form ${\mathcal{O}}(\unicode[STIX]{x1D70C})$.

For ${\mathcal{E}}\in {\mathcal{C}}$, let ${\mathcal{E}}_{\overline{S}}$ be the subcomplex consisting of those ${\mathcal{O}}(\unicode[STIX]{x1D70C})$ with $\unicode[STIX]{x1D70C}\in \overline{S}$. The fact that this is indeed a subcomplex (i.e. that $d({\mathcal{E}}_{\overline{S}})\subset {\mathcal{E}}_{\overline{S}}$) follows from the fact that any map

$$\begin{eqnarray}{\mathcal{O}}_{X}(\unicode[STIX]{x1D70C})\rightarrow {\mathcal{O}}_{X}(\unicode[STIX]{x1D70C}^{\prime })\end{eqnarray}$$

with $\unicode[STIX]{x1D70C}\in \overline{S}$ and $\unicode[STIX]{x1D70C}^{\prime }\not \in \overline{S}$ must be trivial. The operation ${\mathcal{E}}\mapsto {\mathcal{E}}_{\overline{S}}$ is functorial.

Now for any ${\mathcal{E}}\in {\mathcal{C}}$ we have a functorial short exact sequence

$$\begin{eqnarray}0\rightarrow {\mathcal{E}}_{\overline{S}}\rightarrow {\mathcal{E}}\rightarrow {\mathcal{E}}_{\unicode[STIX]{x1D712}(T)\setminus \overline{S}}\rightarrow 0,\end{eqnarray}$$

where ${\mathcal{E}}_{\overline{S}}\in D_{T}^{b}(X)_{\overline{S}}$ and ${\mathcal{E}}_{\unicode[STIX]{x1D712}(T)\setminus \overline{S}}\in D_{T}^{b}(X)_{\unicode[STIX]{x1D712}(T)\setminus \overline{S}}$.

If ${\mathcal{E}}\in D_{T}^{b}(X)_{\overline{S}}$, then ${\mathcal{E}}_{\unicode[STIX]{x1D712}(T)\setminus \overline{S}}|_{0}\cong 0$, and therefore ${\mathcal{E}}_{\unicode[STIX]{x1D712}(T)\setminus \overline{S}}\cong 0$ by Lemma 5.8. It follows that ${\mathcal{E}}\cong {\mathcal{E}}_{\overline{S}}$. Similarly, if ${\mathcal{F}}\in D_{T}^{b}(X)_{\unicode[STIX]{x1D712}(T)\setminus \overline{S}}$, then ${\mathcal{F}}\cong {\mathcal{F}}_{\unicode[STIX]{x1D712}(T)\setminus \overline{S}}$. It now follows from Lemma 5.7 that $\operatorname{Hom}({\mathcal{E}},{\mathcal{F}})=\operatorname{Hom}({\mathcal{E}}_{\overline{S}},{\mathcal{F}}_{\unicode[STIX]{x1D712}(T)\setminus \overline{S}})=0$, which is what we needed.◻

Lemma 5.10. Let ${\mathcal{E}}\in D_{T}^{b}(X)$. If $\unicode[STIX]{x1D70C}\in \operatorname{wt}({\mathcal{E}})$ is maximal, then

$$\begin{eqnarray}\operatorname{RHom}({\mathcal{O}}_{X}(\unicode[STIX]{x1D70C}),{\mathcal{E}})\not =0.\end{eqnarray}$$

Proof. By the semiorthogonal decomposition of Lemma 5.9, we have a map ${\mathcal{E}}_{\overline{\unicode[STIX]{x1D70C}}}\rightarrow {\mathcal{E}}$, where ${\mathcal{E}}_{\overline{\unicode[STIX]{x1D70C}}}$ is locally free and the underlying sheaf is a direct sum of ${\mathcal{O}}_{X}(\unicode[STIX]{x1D70C}^{\prime })$ with $\unicode[STIX]{x1D70C}^{\prime }\geqslant \unicode[STIX]{x1D70C}$. Next we claim that the map

$$\begin{eqnarray}{\mathcal{E}}_{\overline{\unicode[STIX]{x1D70C}}}\rightarrow ({\mathcal{E}}_{\overline{\unicode[STIX]{x1D70C}}})_{\text{}\underline{\unicode[STIX]{x1D70C}}}\end{eqnarray}$$

is an isomorphism. This holds because, by construction, the map is surjective and its kernel has weights in $(\overline{\unicode[STIX]{x1D70C}}\setminus \unicode[STIX]{x1D70C})\,\cap \,\operatorname{wt}({\mathcal{E}})=\varnothing$.

Now by construction the underlying sheaf of $({\mathcal{E}}_{\overline{\unicode[STIX]{x1D70C}}})_{\text{}\underline{\unicode[STIX]{x1D70C}}}$ is a direct sum of copies of ${\mathcal{O}}(\unicode[STIX]{x1D70C})$ (with cohomological shifts). Since we assume that $\unicode[STIX]{x1D6E4}(X,{\mathcal{O}}_{X})^{T}=\mathbb{C}$, every differential in the complex $({\mathcal{E}}_{\overline{\unicode[STIX]{x1D70C}}})_{\text{}\underline{\unicode[STIX]{x1D70C}}}$ must be constant on $X$. As the restriction of the complex to $0$ does not vanish, it follows that there is a non-trivial map ${\mathcal{O}}(\unicode[STIX]{x1D70C})\rightarrow ({\mathcal{E}}_{\overline{\unicode[STIX]{x1D70C}}})_{\text{}\underline{\unicode[STIX]{x1D70C}}}\cong {\mathcal{E}}_{\overline{\unicode[STIX]{x1D70C}}}$. By Lemma 5.9, we have the exact triangle

$$\begin{eqnarray}{\mathcal{E}}_{\overline{\unicode[STIX]{x1D70C}}}\rightarrow {\mathcal{E}}\rightarrow {\mathcal{E}}_{\unicode[STIX]{x1D712}(T)\setminus \overline{\unicode[STIX]{x1D70C}}}.\end{eqnarray}$$

Since ${\mathcal{O}}(\unicode[STIX]{x1D70C})\in D_{T}^{b}(X)_{\overline{\unicode[STIX]{x1D70C}}}$, we have $\operatorname{RHom}({\mathcal{O}}(\unicode[STIX]{x1D70C}),{\mathcal{E}}_{\unicode[STIX]{x1D712}(T)\setminus \overline{\unicode[STIX]{x1D70C}}})=0$ and so

$$\begin{eqnarray}\operatorname{RHom}({\mathcal{O}}_{X}(\unicode[STIX]{x1D70C}),{\mathcal{E}})=\operatorname{RHom}({\mathcal{O}}_{X}(\unicode[STIX]{x1D70C}),{\mathcal{E}}_{\overline{\unicode[STIX]{x1D70C}}})\not =0.\Box\end{eqnarray}$$

5.1.2 Constructing objects supported on $Z_{+}^{\text{us}}$

We now return to our example and apply the above results to $X=V\,\times \,0\,\times \,L$, equipped with the previously described action of $T=(\mathbb{C}^{\ast })^{2}\subset G$. We have $\unicode[STIX]{x1D712}(T)=\mathbb{Z}^{2}$, and the partial ordering is given by letting $(i,j)\leqslant (i^{\prime },j^{\prime })$ if there exist $l,m\geqslant 0$ such that $(i^{\prime },j^{\prime })=(i+l-m,j-m)$.

We let $i:X{\hookrightarrow}Z$ be the inclusion, let ${\mathcal{F}}\in D_{T}^{b}(X,0)$, and let $S=\operatorname{wt}({\mathcal{F}})$.

Lemma 5.11. The weights of $i_{\ast }{\mathcal{F}}$ are contained in

$$\begin{eqnarray}\mathop{\bigcup }_{i=0}^{n}S-(0,i).\end{eqnarray}$$

The weights of the cone over $i^{\ast }i_{\ast }{\mathcal{F}}\rightarrow {\mathcal{F}}$ are contained in

$$\begin{eqnarray}\mathop{\bigcup }_{i=1}^{n}S-(0,i).\end{eqnarray}$$

Proof. Let $p$ be the projection $Z\rightarrow X$. Then $p\,\circ \,i=\operatorname{id}_{X}$, and so ${\mathcal{F}}=i^{\ast }p^{\ast }({\mathcal{F}})$. It follows that the weights of $p^{\ast }({\mathcal{F}})$ are equal to those of ${\mathcal{F}}$, hence are also contained in $S$. We have $i_{\ast }({\mathcal{F}})=i_{\ast }i^{\ast }p^{\ast }({\mathcal{F}})=p^{\ast }({\mathcal{F}})\otimes i_{\ast }({\mathcal{O}}_{X})$, where $p^{\ast }({\mathcal{F}})\in D_{T}^{b}(Z,0)$ and $i_{\ast }({\mathcal{O}}_{X})\in D_{T}^{b}(Z,W)$. The weights of $i_{\ast }({\mathcal{O}}_{X})$ are easy to compute using a Koszul resolution like the one in the proof of Lemma 4.2; they are $(0,-i)$, $0\leqslant i\leqslant n$. The first claim follows.

For the second claim, let ${\mathcal{G}}=p^{\ast }{\mathcal{F}}\in D_{T}^{b}(Z,0)$, so that we have $i^{\ast }{\mathcal{G}}={\mathcal{F}}$. The cone is taken over the map $g:i^{\ast }i_{\ast }(i^{\ast }({\mathcal{G}}))\rightarrow i^{\ast }({\mathcal{G}})$ induced by the counit, so our claim regards the weights of $C(g)$. We also have the map $f:i^{\ast }({\mathcal{G}})\rightarrow i^{\ast }(i_{\ast }i^{\ast }{\mathcal{G}})$ induced by the unit, which satisfies $g\,\circ \,f=\operatorname{id}_{i^{\ast }({\mathcal{G}})}$, by [Reference Mac LaneMac98, p. 85]. Thus, by the octahedral axiom, we have the exact triangle

$$\begin{eqnarray}C(f)\rightarrow C(g\,\circ \,f)\rightarrow C(g),\end{eqnarray}$$

and, since $C(g\,\circ \,f)=0$, that $C(g)\cong C(f)[1]$. The map $f:i^{\ast }({\mathcal{G}})\rightarrow i^{\ast }i_{\ast }i^{\ast }({\mathcal{G}})=i^{\ast }({\mathcal{G}}\otimes {\mathcal{O}}_{X})$ has cone $C(f)\cong i^{\ast }({\mathcal{G}}\otimes {\mathcal{I}}_{X})[1]$. The weights of ${\mathcal{I}}_{X}$ are easily computed; they are $(0,-i)$, $1\leqslant i\leqslant n$. The claim about the weights of $C(g)$ follows.◻

Now let ${\mathcal{E}}\in D_{T}^{b}(Z,W)$, and let $S$ be a minimal subset of $\operatorname{wt}({\mathcal{E}})$.

Lemma 5.12. The weights of the cone over the natural map ${\mathcal{E}}\rightarrow i_{\ast }(({\mathcal{E}}|_{X})_{\text{}\underline{S}})$ are contained in

$$\begin{eqnarray}(\operatorname{wt}({\mathcal{E}})\setminus S)\cup \biggl(\mathop{\bigcup }_{i=1}^{n}S-(0,i)\biggr).\end{eqnarray}$$

Proof. Let ${\mathcal{F}}$ be the cone. Pulling back the triangle defining ${\mathcal{F}}$ along $i$, we get a triangle

$$\begin{eqnarray}i^{\ast }({\mathcal{E}})\stackrel{g}{\rightarrow }i^{\ast }i_{\ast }i^{\ast }({\mathcal{E}}_{\text{}\underline{S}})\rightarrow C(g)=i^{\ast }{\mathcal{F}}.\end{eqnarray}$$

The weights of ${\mathcal{F}}$ equal those of $C(g)$, so it is enough to prove the claim for the weights of $C(g)$.

We claim that the map

$$\begin{eqnarray}f:i^{\ast }({\mathcal{E}})\rightarrow (i^{\ast }({\mathcal{E}}))_{\text{}\underline{S}}\end{eqnarray}$$

induced by the semiorthogonal decomposition of Lemma 5.9 is equal to the composition

$$\begin{eqnarray}i^{\ast }({\mathcal{E}})\stackrel{g}{\rightarrow }i^{\ast }(i_{\ast }(i^{\ast }{\mathcal{E}})_{\text{}\underline{S}})\stackrel{h}{\rightarrow }(i^{\ast }({\mathcal{E}}))_{\text{}\underline{S}},\end{eqnarray}$$

where $h$ is induced by the counit $i^{\ast }i_{\ast }\rightarrow \operatorname{id}$.

To see this, let $P$ the functor $D_{T}^{b}(X,W)\rightarrow D_{T}^{b}(X,W)$ given by ${\mathcal{E}}\mapsto {\mathcal{E}}_{\text{}\underline{S}}$. The map $f$ is induced by the natural transformation $i^{\ast }\rightarrow Pi^{\ast }$, which equals the composition of unit and counit transformations $i^{\ast }\rightarrow i^{\ast }i_{\ast }i^{\ast }\rightarrow i^{\ast }\rightarrow Pi^{\ast }$, by the triangle equalities of [Reference Mac LaneMac98, p. 85]. The composition $h\,\circ \,g$ is induced by the natural transformation $i^{\ast }\rightarrow i^{\ast }i_{\ast }i^{\ast }\rightarrow i^{\ast }i_{\ast }Pi^{\ast }\rightarrow Pi^{\ast }$. We see that the two natural transformations are the same, hence $f=h\,\circ \,g$.

By the octahedral axiom, there is a distinguished triangle whose vertices are $C(f)$, $C(g)$ and $C(h)$. The weights of $C(f)$ are contained in $\operatorname{wt}({\mathcal{E}})\setminus S$, while the weights of $C(h)$ are contained in

$$\begin{eqnarray}S-(0,1),\ldots ,S-(0,n)\end{eqnarray}$$

by Lemma 5.11. The claim about the weights of $C(g)$ follows.◻

Now let ${\mathcal{E}}\in D_{G}^{b}(Z,W)$, and let ${\mathcal{E}}^{\prime }\in D_{T}^{b}(Z,W)$ be the underlying $T$-equivariant object. Let $\unicode[STIX]{x1D70E}\in G$ be the order-2 element which permutes the factors of $V$ in $Z$. The $G$-equivariant object induced from $({\mathcal{E}}^{\prime }|_{X})_{\text{}\underline{S}}$ is

$$\begin{eqnarray}i_{\ast }(({\mathcal{E}}^{\prime }|_{X})_{\text{}\underline{S}})\,\oplus \,\unicode[STIX]{x1D70E}_{\ast }i_{\ast }(({\mathcal{E}}^{\prime }|_{X})_{\text{}\underline{S}}),\end{eqnarray}$$

and there is a canonical $G$-equivariant map

$$\begin{eqnarray}\unicode[STIX]{x1D719}:{\mathcal{E}}\rightarrow i_{\ast }(({\mathcal{E}}^{\prime }|_{X})_{\text{}\underline{S}})\,\oplus \,\unicode[STIX]{x1D70E}_{\ast }i_{\ast }(({\mathcal{E}}^{\prime }|_{X})_{\text{}\underline{S}}).\end{eqnarray}$$

We also denote by $\unicode[STIX]{x1D70E}$ the involution of $\mathbb{Z}^{2}$ which permutes the $\mathbb{Z}$-factors. We say that $S\subset \mathbb{Z}^{2}$ is good if

$$\begin{eqnarray}\biggl(\mathop{\bigcup }_{i=0}^{n}\unicode[STIX]{x1D70E}S-(i,0)\biggr)\,\cap \,S=\varnothing .\end{eqnarray}$$

Lemma 5.13. If $S\subset \operatorname{wt}({\mathcal{E}})$ is minimal and good, then the weights of $C(\unicode[STIX]{x1D719})$ are contained in

$$\begin{eqnarray}\biggl(\mathop{\bigcup }_{i=1}^{n}(S-(0,i))\cup (\unicode[STIX]{x1D70E}S-(i,0))\biggr)\cup (\operatorname{wt}({\mathcal{E}})\setminus (S\cup \unicode[STIX]{x1D70E}S)).\end{eqnarray}$$

Proof. Let $\unicode[STIX]{x1D719}=(\unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{2})$ be the decomposition of $\unicode[STIX]{x1D719}$ corresponding to the splitting of its codomain

$$\begin{eqnarray}i_{\ast }(({\mathcal{E}}^{\prime }|_{X})_{\text{}\underline{S}})\,\oplus \,\unicode[STIX]{x1D70E}_{\ast }i_{\ast }(({\mathcal{E}}^{\prime }|_{X})_{\text{}\underline{S}}),\end{eqnarray}$$

and let $p_{1}$ and $p_{2}$ be the projections of this object onto each factor. Then the identity $\unicode[STIX]{x1D719}_{i}=p_{i}\unicode[STIX]{x1D719}$ and the octahedral axiom imply that the cones $C(\unicode[STIX]{x1D719}_{i})$, $C(\unicode[STIX]{x1D719})$ and $C(p_{i})$ fit into a distinguished triangle for $i=1,2$. We thus have

$$\begin{eqnarray}\operatorname{wt}(C(\unicode[STIX]{x1D719}))\subset \mathop{\bigcap }_{i=1,2}(\operatorname{wt}(C(\unicode[STIX]{x1D719}_{i}))\cup \operatorname{wt}(C(p_{i}))).\end{eqnarray}$$

We have $\unicode[STIX]{x1D70E}\operatorname{wt}(C(\unicode[STIX]{x1D719}_{1}))=\operatorname{wt}(C(\unicode[STIX]{x1D719}_{2}))$ and $\unicode[STIX]{x1D70E}\operatorname{wt}(C(p_{1}))=\operatorname{wt}(C(p_{2}))$. By Lemmas 5.11 and 5.12, we get

$$\begin{eqnarray}\operatorname{wt}(C(p_{1}))\subset \mathop{\bigcup }_{i=0}^{n}(\unicode[STIX]{x1D70E}S-(i,0))\end{eqnarray}$$

and

$$\begin{eqnarray}\operatorname{wt}(C(\unicode[STIX]{x1D719}_{1}))\subset (\operatorname{wt}({\mathcal{E}})\setminus S)\cup \biggl(\mathop{\bigcup }_{i=1}^{n}S-(0,i)\biggr).\end{eqnarray}$$

The claim now follows using the goodness of $S$.◻

Proposition 5.14. Let ${\mathcal{E}}\in D^{b}({\mathcal{Z}},W)$. If $S\subset \operatorname{wt}({\mathcal{E}})$ is minimal and good (respectively, maximal and $-S$ is good), then there is a distinguished triangle in $D^{b}({\mathcal{Z}},W)$

$$\begin{eqnarray}{\mathcal{F}}\rightarrow {\mathcal{E}}\rightarrow {\mathcal{G}}\quad (\text{respectively }{\mathcal{G}}\rightarrow {\mathcal{E}}\rightarrow {\mathcal{F}})\end{eqnarray}$$

such that ${\mathcal{G}}$ is supported on $Z\setminus Z_{+}=X\cup \unicode[STIX]{x1D70E}(X)$ and such that

$$\begin{eqnarray}\displaystyle & \displaystyle \operatorname{wt}({\mathcal{F}})\subset \biggl(\mathop{\bigcup }_{i=1}^{n}(S-(0,i))\cup \unicode[STIX]{x1D70E}(S-(i,0))\biggr)\cup (\operatorname{wt}({\mathcal{E}})\setminus (S\cup \unicode[STIX]{x1D70E}S)) & \displaystyle \nonumber\\ \displaystyle & \displaystyle \biggl(\text{respectively }\operatorname{wt}({\mathcal{F}})\subset \biggl(\mathop{\bigcup }_{i=1}^{n}(S+(0,i))\cup \unicode[STIX]{x1D70E}(S+(i,0))\biggr)\cup (\operatorname{wt}({\mathcal{E}})\setminus (S\cup \unicode[STIX]{x1D70E}S))\biggr). & \displaystyle \nonumber\end{eqnarray}$$

Proof. The statements for maximal $S$ follow from those of minimal $S$ by dualising. So assume that $S$ is minimal, in which case we may take

$$\begin{eqnarray}{\mathcal{G}}=i_{\ast }(({\mathcal{E}}^{\prime }|_{X})_{\text{}\underline{S}})\,\oplus \,\unicode[STIX]{x1D70E}_{\ast }i_{\ast }(({\mathcal{E}}^{\prime }|_{X})_{\text{}\underline{S}})\end{eqnarray}$$

by Lemma 5.13. ◻

We can now give the proof of Proposition 5.5.

Proposition 5.5. The subcategories ${\mathcal{W}}_{+}$ and $\ker j_{+}^{\ast }$ generate $D^{b}({\mathcal{Z}},W)$ as a triangulated category.

Proof. Let ${\mathcal{E}}\in D^{b}({\mathcal{Z}},W)$, and suppose that ${\mathcal{E}}^{\prime }$ is a cone over a map between ${\mathcal{E}}$ and an object in $\ker j_{+}^{\ast }$. If ${\mathcal{E}}^{\prime }\in \langle {\mathcal{W}}_{+},\ker j_{+}^{\ast }\rangle$, then ${\mathcal{E}}\in \langle {\mathcal{W}}_{+},\ker j_{+}^{\ast }\rangle$. This means that if we can always, after replacing ${\mathcal{E}}$ with such an ${\mathcal{E}}^{\prime }$ a finite number of times, get ${\mathcal{E}}\in {\mathcal{W}}_{+}$, then the claim of the proposition follows.

We first see that we can get $\operatorname{wt}({\mathcal{E}})\subset \{(i,j)\,\mid \,i+j\in [0,2n-1]\}$. This is an application of [Reference Ballard, Favero and KatzarkovBFK19, Proposition 3.3.2], where we apply the result to the unstable locus $0\,\times \,0\,\times \,L\subset Z$ and the one-parameter diagonal subgroup $\mathbb{C}^{\ast }\subset G$.

Define the width of ${\mathcal{E}}$ as the maximal value of $|i-j|$ for $(i,j)\in \operatorname{wt}({\mathcal{E}})$. Let $k$ be the width of ${\mathcal{E}}$, assume that $k>n/2$, and let $\operatorname{wt}({\mathcal{E}})_{k}$ be the set of $(i,j)\in \operatorname{wt}({\mathcal{E}})$ such that $|i-j|=k$. Then either $\max _{(i,j)\in \operatorname{wt}({\mathcal{E}})_{k}}(i+j)\geqslant n$ or $\min _{(i,j)\in \operatorname{wt}({\mathcal{E}})_{k}}(i+j)<n$.

In the first case let $S\subset \operatorname{wt}({\mathcal{E}})_{k}$ be the set of $(i,j)$ such that $i+j\geqslant n$ and $i<j$. The set $S$ is minimal and good, and we modify ${\mathcal{E}}$ by taking a cone over ${\mathcal{G}}$ as in Proposition 5.14. In the second case we let $S$ be the set of $(i,j)\in \operatorname{wt}({\mathcal{E}})_{k}$ with $i+j<n$ and such that $i>j$. This $S$ is maximal and $-S$ is good, and again we modify ${\mathcal{E}}$ by taking a cone over ${\mathcal{G}}$ as in Proposition 5.14.

In either case, we see that replacing ${\mathcal{E}}$ by this cone will either decrease the width of ${\mathcal{E}}$ or leave it unchanged. If the width of ${\mathcal{E}}$ is unchanged, the cardinality of $\operatorname{wt}({\mathcal{E}})_{k}$ will decrease. Furthermore, we still keep the property that $\operatorname{wt}({\mathcal{E}})\subset \{(i,j)\,\mid \,i\,+\,j\in [0,2n-1]\}$. Thus by repeating this procedure a finite number of times we obtain an ${\mathcal{E}}$ with width less than or equal to $n/2$. If $n$ is odd, we then have $\operatorname{wt}({\mathcal{E}})\subseteq S_{+}$, hence ${\mathcal{E}}\in {\mathcal{W}}_{+}$, and we are done.

If $n$ is even, let $S$ be the set of $(i,j)\in \operatorname{wt}({\mathcal{E}})_{n/2}$ with $i+j>n$ and such that $i<j$. This $S$ is minimal and good, and, by taking the cone over ${\mathcal{G}}$ as in Proposition 5.14, we get $\operatorname{wt}({\mathcal{E}})\subseteq S_{+}$.◻