3.1 Geometric genus bounds

First, we recall the bounds for the geometric genus from [Reference NémethiNém22, § 6.8.B]. These bounds are constructive and can be explicitly computed in our examples. For a more self-contained exposition, we review the relevant material from [Reference NémethiNém22, § 6.8.B].

Fix a good resolution $\tilde{X}$ of (X, 0), with dual-resolution graph $\Gamma$ whose vertices $v\in \Gamma$ correspond to exceptional curves $E_v$ , $v\in \Gamma$ . Let $g_v$ be the genus of $E_v$ .

The lattice $L = H_2(\tilde{X}, \mathbb{Z})$ is freely generated over $\mathbb{Z}$ by the classes of $E_v$ , $v \in \Gamma$ ; its intersection form is negative definite. Let $L' = H^2(\tilde{X}, \mathbb{Z}) = H_2(\tilde{X}, \partial \tilde{X}, \mathbb{Z})$ be the dual lattice; L’ can be identified with $\mathrm{Hom} (L, \mathbb{Z})$ . Extending the intersection form on L to $L \otimes \mathbb{Q}$ , we can think of $\mathrm{Hom} (L, \mathbb{Z})$ as the set of elements $l' \in L\otimes \mathbb{Q}$ such that $(l', l) \in \mathbb{Z}$ . Thus, we will identify L’ with a sublattice of $L \otimes \mathbb{Q}$ and write its elements as rational linear combinations of the generators $E_v$ .

Let $Z_K= -c_1 (\Omega_{\tilde{X}}^2) \in L'$ be the canonical cycle of the resolution. The cycle $Z_K$ satisfies the adjunction relations

(3.1) \begin{equation}Z_K E_v = {E_v}\cdot E_v +2 - 2 g_v, \quad v \in \Gamma,\end{equation}

and is in fact uniquely determined by these relations as a rational linear combination of classes $E_v$ , since the intersection form corresponding to $\Gamma$ is negative definite. It is known (see [Reference NémethiNém22, Example 6.3.4]) that all coefficients of $Z_K$ are positive if $\tilde{X}$ is a minimal good resolution and if (X, o) is not a rational double point. For rational double points, the minimal resolution is good, $Z_K=0$ , and $p_g=0$ . Let $\lfloor Z_k \rfloor \in L$ be the integral cycle given by the floor of $Z_K \in L'$ ; the floor of a rational cycle $Z=\sum r_v E_v$ is defined as $\lfloor Z \rfloor = \sum \lfloor r_v \rfloor E_v$ , where $\lfloor r_v \rfloor$ stands for the greatest integer less than or equal to $r_v \in \mathbb{Q}$ .

In what follows, we use short exact sequences of the form

(3.2) \begin{align} 0 & \longrightarrow \mathcal{O}_{\tilde{X}}(-Z) \longrightarrow \mathcal{O}_{\tilde{X}} \longrightarrow \mathcal{O}_{Z} \longrightarrow 0, \end{align}

(3.3) \begin{align} 0 & \longrightarrow \mathcal{O}_{F}(-E) \longrightarrow \mathcal{O}_{F+E} \longrightarrow \mathcal{O}_{E} \longrightarrow 0\end{align}

for effective non-zero divizors Z, E, and F on $\tilde{X}$ . Here, as usual, $\mathcal{O}_{F}(-E)$ stands for $\mathcal{O}_{\tilde{X}}(-E)|_F= \mathcal{O}_{\tilde{X}}(-E)/\mathcal{O}_{\tilde{X}}(-E-F)$ , and $\mathcal{O}_{\tilde{X}}(-Z)$ is the sheaf of holomorphic functions vanishing to order at least $n_i$ along each irreducible component $E_i$ of $Z=\sum_i n_i E_i$ . These exact sequences follow from the definition of the structure sheaves. We will also use the fact that

(3.4) \begin{equation}H^{\geqslant 2}(\mathcal{O}_{\tilde{X}}(-Z))=0, \quad H^{\geqslant 2}(\mathcal{O}_{F}(-E))=0,\end{equation}

because higher cohomologies vanish for sheaves of holomorphic sections of line bundles on complex curves, and, more generally, on one-dimensional complex spaces, and in the case of $\tilde{X}$ it follows from the Formal Function Theorem, see e.g. [Reference NémethiNém22, § 6].

Lemma 3.1. The geometric genus of (X, 0) can be expressed as

(3.5) \begin{equation}p_g=h^1(\tilde{X}, \mathcal{O}_{\lfloor Z_K \rfloor}).\end{equation}

Proof. Consider the short exact sequence of sheaves

\begin{align*}0 \longrightarrow \mathcal{O}_{\tilde{X}}(-\lfloor Z_K \rfloor) \longrightarrow \mathcal{O}_{\tilde{X}} \longrightarrow \mathcal{O}_{\lfloor Z_K \rfloor} \longrightarrow 0.\end{align*}

From the cohomological long exact sequence, we have

\begin{align*}\cdots \longrightarrow H^1 ( \mathcal{O}_{\tilde{X}}(-\lfloor Z_K \rfloor) ) \longrightarrow H^1( \mathcal{O}_{\tilde{X}}) \longrightarrow H^1 (\mathcal{O}_{\lfloor Z_K \rfloor}) \longrightarrow H^2 (\mathcal{O}_{\tilde{X}}(-\lfloor Z_K \rfloor)) \longrightarrow \cdots.\end{align*}

By (3.4), we get zero on the right; we also have $H^1 ( \mathcal{O}_{\tilde{X}}(-\lfloor Z_K \rfloor) )=0$ by the Generalized Grauert–Riemenschneider Vanishing Theorem, see [Reference NémethiNém22, § 6.4]. Therefore, $p_g=h^1(\mathcal{O}_{\tilde{X}})= h^1(\mathcal{O}_{\lfloor Z_K \rfloor})$ .

We can now estimate $h^1(\lfloor Z_K \rfloor)$ via a dévissage-type argument. Consider a path in the lattice L from 0 to $\lfloor Z_K \rfloor$ , that is, take a sequence of integral cycles $\{l_i\}\in L$ such that

\begin{align*}l_0=0, \quad \dots, \quad l_{i+1}=l_i+E_{v_i}, \quad \dots, \quad l_t= \lfloor Z_K \rfloor,\end{align*}

where $E_{v_i}$ is the basis element of L corresponding to some $v_i \in \Gamma$ . (We also denote by $l_i$ the corresponding effective divisor in $\tilde{X}$ .) Writing $\mathcal{O}_{l_i}= \mathcal{O}_{\tilde{X}}|_{l_i}$ , we have short exact sequences at each step, $i=0, \dots, t-1$ ,

\begin{align*}0 \longrightarrow \mathcal{O}_{E_{v_i}}(-l_i) \longrightarrow \mathcal{O}_{l_{i+1}} \longrightarrow \mathcal{O}_{l_i} \longrightarrow 0,\end{align*}

and the corresponding long exact sequences in cohomology,

\begin{align*}\cdots\longrightarrow H^0(\mathcal{O}_{l_i} ) \longrightarrow H^1(\mathcal{O}_{E_{v_i}}(-l_i) )\longrightarrow H^1(\mathcal{O}_{l_{i+1}} ) \longrightarrow H^1 (\mathcal{O}_{l_i} ) \longrightarrow H^2 (\mathcal{O}_{E_{v_i}}(-l_i) ) \longrightarrow \cdots.\end{align*}

Since $H^2 (\mathcal{O}_{E_{v_i}}(-l_i) )=0$ , we get

(3.6) \begin{equation}h^1(\mathcal{O}_{l_{i+1}})- h^1(\mathcal{O}_{l_i}) \leqslant h^1 (\mathcal{O}_{E_{v_i}}(-l_i)).\end{equation}

Here, $E_{v_i}$ is a smooth curve of genus $g_{v_i}$ , and $\mathcal{O}_{E_{v_i}}(-l_i)$ is the sheaf of sections of a line bundle of degree $d_i = - l_i \cdot E_{v_i}$ on $E_{v_i}$ . Recall that for a line bundle L of degree d on a smooth curve C of genus g, we have the following inequalities:

\begin{align*}h^1(L) \left\{\begin{array}{l@{\quad}l}=g-d-1 &\mbox{if}\ d\leqslant -1,\\ \leqslant g-\lceil d/2\rceil & \mbox{if} \ 0 \leqslant d\leqslant 2g-2,\\=0 & \mbox{if}\ 2g-1\leqslant d.\end{array}\right.\end{align*}

Indeed, by Riemann and Roch, $h^1(L)=h^0(L)+g-d-1$ . In particular, if $d\leqslant -1$ , then $h^0(L)=0$ and $h^1(L)=g-d-1$ . If $0\leqslant d\leqslant 2g-2$ , then $h^0(L)\leqslant 1+\lfloor d/2\rfloor$ by Clifford’s Theorem (see e.g. [Reference Arbarello, Cornalba, Griffiths and HarrisACG⁺85, p. 107]). On the other hand, for any d, $h^1(C, L)= h^0(C, L^{-1}\otimes K_C)$ by the Kodaira–Serre duality, where $K_C$ is the canonical bundle of C with degree $2g-2$ . Hence, if $d\geqslant 2g-1$ , then $\mathrm{deg}(L^{-1}\otimes K_C)\lt0$ and $h^0(L^{-1}\otimes K_C)=0$ .

In particular, along our path:

\begin{align*}h^1 (\mathcal{O}_{E_{v_i}}(-l_i)) \leqslant B_i := \left\{\begin{array}{l@{\quad}l}g_{v_i}-d_i-1 &\mbox{if} \ d_i\leqslant-1,\\g_{v_i}-\lceil d_{i}/2\rceil& \mbox{if} \ 0\leqslant d_i\leqslant 2g_{v_i}-2,\\0 & \mbox{if}\ 2g_{v_i}-1\leqslant d_i.\end{array}\right.\end{align*}

Now add the inequalities (3.6) for all $0\leqslant i\lt t$ to get

(3.7) \begin{equation}p_g= h^1(\lfloor Z_K \rfloor) \leqslant \sum_{i=0}^{t-1} B_i.\end{equation}

Note that the resulting bound (3.7) depends only on the resolution graph but not on the analytic structure. The bound from the right-hand side of (3.7) depends also on the choice of the path; with special, well-chosen paths we can realize sharper bounds.

The bound (3.7) is not very sharp in general; this weakness usually occurs when some of the genera $g_v$ are non-zero. However, if the link is a rational homology sphere, then in many cases the bound (3.7) is optimal, that is, for a well-chosen path, the topological right-hand side can be realized as the geometric genus of a (very special) analytic structure. See § 11.3.A or Corollary 11.6.11 of [Reference NémethiNém22].

Example 3.2. Consider the case where the minimal resolution has a unique exceptional curve E of genus g with self-intersection $-b$ , where $g, b\geqslant 1$ . This is the simplest case of Theorem 1.4, where for $b\leqslant 2g -4$ we are able to generate infinitely many Stein fillings. From (3.1), we have

\begin{align*}Z_K = \frac{2g-2+b}{b}\, E, \quad \lfloor Z_K \rfloor =\bigg(\bigg\lfloor \frac{2g-2}b \bigg\rfloor +1\!\bigg) E.\end{align*}

The upper bound as in (3.7) is then obtained from the path

\begin{align*}0, E, 2E, \dots, \bigg(\bigg\lfloor \frac{2g-2}b \bigg\rfloor +1\!\bigg) E.\end{align*}

Then we apply (3.7), where at the ith stage we consider the line bundle of degree $d_i=-(i E)\cdot E = bi $ on E, $0\leqslant i\leqslant \lfloor(2g-2)/b\rfloor$ . Note that $d_i=ib \leqslant 2g-2$ . Therefore,

(3.8) \begin{equation}p_g \leqslant\sum_{i=0}^{\lfloor(2g-2)/b\rfloor} \bigg(g-\bigg\lceil\frac{ib}{2}\bigg\rceil\bigg).\end{equation}

The reader is invited to compute the bound for some specific values of b and g. For example, if $b=1$ , then we obtain $p_g\leqslant g^2$ . If $b=2$ , then $p_g\leqslant (g^2+g)/2$ .

When $g\gt3$ and $b=2g-4$ (the maximum value of b where we will be able to produce an infinitude of fillings), the sum (3.7) comes from the path 0, E, 2E and has only two terms: $B_0=g$ and $B_1=2$ . Hence,

(3.9) \begin{equation}p_g \leqslant B_0 +B_1 = g+2.\end{equation}

If $g=3$ and $b=2g-4=2$ , then $p_g\leqslant B_0+B_1+B_2=3+2+1=6$ .

A straightforward computation shows that for any such $b,g\geqslant 1$ , the following inequality holds:

(3.10) \begin{equation}p_g \leqslant \sum_{i=0}^{t-1} B_i \leqslant g^2.\end{equation}

3.2 Computing topological invariants

To constrast these bounds with Proposition 2.3 and pinpoint specific Stein fillings that cannot come from Milnor fibers, we compute the topological invariants for a particular family of Stein fillings we will describe explicitly, through allowable Lefschetz fibrations they are equipped with. We focus on the examples where the link is an $S^1$ -bundle over a surface, with its Boothby–Wang structure. As noted earlier, in this case, we have a compatible open book with page $\Sigma_g^b$ , a compact genus-g surface with b boundary components, and the monodromy is the boundary multitwist $\Delta:=t_{\delta_1} \cdots t_{\delta_{b}}$ . We assume that $g \geqslant 4$ , $b \geqslant 1$ , and $b \leqslant 2g-4$ . We will describe the allowable Lefschetz fibrations on our Stein fillings via their monodromy factorizations coming from [Reference Baykur, Monden and Van Horn-MorrisBMV17] and compute the invariants of these fillings.

For a product P of positive Dehn twists in $\operatorname{Mod}(\Sigma_g^b)$ , let the length $|P|$ be the number of Dehn twist factors in P. Recall that for any Dehn twist $t_c$ and $ \phi \in \operatorname{Mod}(\Sigma_g^b)$ , the conjugate of the Dehn twist $t_c^\phi=\phi \, t_c \, \phi^{-1}=t_{\phi(c)}$ . In particular, for a product P of positive Dehn twists, $P^\phi$ is also a product of positive Dehn twists with $|P^\phi|=|P|$ .

We are going to describe factorizations of the multitwist $\Delta$ into arbitrarily large numbers of positive Dehn twists in $\operatorname{Mod}(\Sigma_g^b)$ . We begin with the maximal case $b=2g-4$ , where in our factorizations, each Dehn twist curve c will satisfy the following additional property: under an embedding of $\Sigma_g^b \hookrightarrow \Sigma_g^{b'}$ , the curve $c \subset \Sigma_g^b$ maps to a homologically essential curve in $\Sigma_g^{b'}$ , for any $b' \leqslant b$ . (Clearly, this then holds for any embedding.) One can view the embedding as capping off some of the boundary components of $\Sigma_g^b$ with discs to get $\Sigma_g^{b'}$ .

Let $c_i, d, e, x_j, \delta_j$ be the curves on $\Sigma:=\Sigma_g^{2g-4}$ as shown in Figures 2 and 3, and let $d_k, e_k, f_k$ be the curves

\begin{align*}& d_k=t_{c_{k-3}}^{-1}t_{c_{k-2}}^{-1}t_{c_{k-1}}^{-1}(c_k),\\& e_k=t_{c_{k-3}} t_{c_{k-2}} t_{c_{k-1}}(c_k), & \\& f_k=t_{c_{k-5}}t_{c_{k-4}}t_{c_{k-3}}t_{c_{k-2}}t_{c_{k-1}}(c_k),&\end{align*}

whenever the indices are well-defined. With these in mind, we consider the following products of positive Dehn twists:

\begin{align*}\psi&:= t_{c_4} t_{c_3} t_{c_2} t_{c_1} t_{c_5} t_{c_4} t_{c_3} t_{c_2} t_{c_6} t_{c_5} t_{c_4} t_{c_3} t_{c_7} t_{c_6} t_{c_5} t_{c_4}, \\A_m&:= (t_{c_4} t_{c_3} t_{c_2} t_{c_1} t_{c_1} t_{c_2} t_{c_3} t_{c_4} t_{c_4} t_d t_{c_3} t_{c_4})^{t_{c_3}^{-m}t_e^m}((t_{c_1} t_{c_2} t_{c_3})^2 t_{c_2} t_{c_1} t_{c_3} t_{c_2})^m,\text{ } m \in \mathbb{N},\\B&:=t_{(t_{c_4} t_d t_{c_3} t_{c_4})^{-1}(c_5)} t_{c_2} t_{(t_{c_5}t_{c_4})^{-1}(c_6)} t_d t_{t_{c_4}^{-1}(c_3)} t_{c_7} t_{(t_{c_5}t_{c_4})^{-1}(c_6)} t_d t_{t_{c_4}^{-1}(e)} \, t_{c_5} t_{c_3} t_{c_4} t_{c_2} t_{c_3}, \\C&:= (t_{d_{10}} t_{d_{11}} {\kern-1pt}\cdots t_{d_{2g+1}}) \, (t_{e_{2g+1}} {\kern-1pt}\cdots t_{e_9} t_{e_8}), \\D&:=D_0^2 \, (t_{x_1} t_{x_2} {\kern-1pt}\cdots t_{x_{g-2}}) \, (t_{x_{g-1}} t_{x_g} {\kern-1pt}\cdots t_{x_{2g-4}}) \, (t_{f_9} t_{f_8} t_{f_7} t_{f_6} ),\end{align*}

where in the last product, $D_0:=t_{c_1} t_{c_2} t_{c_3}$ for even g and $D_0:=t_{c_3} t_{c_2} t_{c_1}$ for odd g.

Figure 2.

The curves $c_i, d, e$ on $\Sigma_g^{2g-4}$ with boundary components $\delta_j$ .

Figure 3.

The curves $x_j$ drawn on the two subsurfaces of $\Sigma_g^{2g-4}$ containing all the boundary components.

It was shown in [Reference Baykur, Monden and Van Horn-MorrisBMV17, Proof of Theorem 9] that, for $\Sigma=\Sigma_g^{2g-4}$ , the following hold:

(3.11) \begin{equation}P_m:=A_m B \,C \,D^\psi = \Delta \quad \text{in}\, \operatorname{Mod}(\Sigma) \quad \text{for all } m \in \mathbb{N} .\end{equation}

Counting the Dehn twists in the factorization, we see that $|P_m|=6g+10m+18$ . Observe that as we cap off $\partial \Sigma$ with discs, each one of the Dehn twist curves of the positive factorization $P_m$ of $\Delta$ maps to a non-separating curve in $\Sigma_g$ ; so its image remains homologically essential under any embedding $\Sigma=\Sigma_g^{2g-4} \hookrightarrow \Sigma_g^{b'}$ for $1 \leqslant b' \leqslant 2g-4$ , satisfying the additional property we asked for. In particular, the Lefschetz fibration $X_m$ over the disc with monodromy factorization (3.11) is allowable, and yields a Stein filling of the contact boundary $Y:=\partial X_m$ , taken with the contact structure compatible with the fixed-boundary open book.

Next, we calculate the homology groups of the Stein fillings $X_m$ given by (3.11).

3.2.1 First homology

We will show that $H_1(X_m)=0$ for each $m \in \mathbb{N}$ . We have $G:=H_1(X_m) = H_1(\Sigma) / N$ , where $N \leqslant H_1(\Sigma)$ is generated by the homology classes of the vanishing cycles of the monodromy factorization of $X_m$ . Note that the homology classes of the curves $\{ c_i, \delta_j\}$ , taken with auxiliary orientations, generate $H_1(\Sigma)$ . So, to prove our claim, it suffices to show that there is a collection of monodromy curves $ \{v_k\}$ whose homology classes generate those of all $c_i$ , $\delta_j$ in $H_1(\Sigma)$ , or equivalently, by arguing that the images of $[c_i], [\delta_j]$ in the quotient $H_1(\Sigma) / \langle \{[v_k]\} \rangle$ will all be trivial. By a slight abuse of notation, whenever $x \in H_1(\Sigma)$ can be seen to be generated by $ \{[v_k]\}$ , we are going to say that $x=0$ in G.

Since the order of the Dehn twists in the monodromy factorization here is irrelevant, we can carry out our calculation for both odd- and even-genus g at the same time, as the two cases only differ by the order of the Dehn twists in the subproduct $D_0$ . Also, without explicitly mentioning it after the first few instances, we are going to repeatedly appeal to the Picard–Lefschetz formula whenever a monodromy curve is expressed as the image of a curve under a mapping class that is expressed as a product of (possibly negative) Dehn twists.

From the monodromy curves $c_1, c_2, c_3$ of $A_m$ and $c_4, c_5, c_7, d$ of B, we immediately have $[c_i]=0$ for $i=1, 2,3,4,5$ and 7, and $[d]=0$ in G. (When $m=0$ , the curves $c_1, c_2, c_3$ are in the now trivially conjugated factor of $A_m$ .) Moreover, by Picard and Lefschetz, the homology class of the monodromy curve $(t_{c_5}t_{c_4})^{-1}(c_6)$ of B is equal to $[c_6] + x$ in $H_1(\Sigma)$ for some $x \in \langle [c_4], [c_5]\rangle$ , and since $x=0$ in G, we conclude that $[c_6]=0$ in G too. Similarly, the monodromy curve $t_{c_4}^{-1}(e)$ of B yields $[e]=0$ in G. Next, applying Picard–Lefschetz to each monodromy curve $e_i=t_{c_{i-3}} t_{c_{i-2}} t_{c_{i-1}}(c_i)$ in C, we see that $[e_i]=[c_i]+x$ for some $x \in \langle [c_{i-3}], [c_{i-2}], [c_{i-1}] \rangle$ . Since we already have $[c_i]=0$ for all $i=1, \ldots, 7$ , using the monodromy curves $e_8, \ldots , e_{2g+1}$ , we inductively see that $[c_i]=0$ also for $i=8, \ldots, 2g+1$ .

Therefore, we have $[c_i]=0$ for all $i=1, \ldots, 2g+1$ and $[d]=[e]=0$ in G. Now, the homology classes of the monodromy curves $x_j$ in $D^\psi$ are all of the form $[x_j] +x$ for $x \in \langle \{[c_i]\} \rangle$ , since the conjugating mapping class $\psi \in \langle \{t_{c_i}\} \rangle$ . So we also get $[x_j]=0$ for all $j=1, \ldots, 2g-4$ . Orienting $\{x_j\}$ and $\{\delta_j\}$ suitably, we have

\[[x_j]=\begin{cases} [d]+[\delta_j]& \text{if } j=1, \ldots , g-2, \\ [e]+[\delta_j]& \text{if } j=g-1, \ldots, 2g-4.\end{cases} \]

Since $[x_j]=[d]=[e]=0$ , it follows that $[\delta_j]=0$ in G for all $j=1 ,\ldots, 2g-4$ .

Hence, all the generators $[c_i]=[\delta_j]=0$ in G, so $H_1(X_m)=0$ .Footnote ¹

3.2.2 Second homology

First of all, we have $H_1(X_m, Y)=0$ , since $H_1(X_m)=0$ and Y is connected. By the universal coefficients theorem and Poincaré–Lefschetz duality, we conclude that $H_2(X_m)$ has no torsion.

Since $|P_m|=6g+10m+18$ is the number of nodes $\ell$ in $X_m$ , we calculate the Euler characteristic of $X_m$ as

\[ \chi(X_m)=\chi(D^2)\, \chi (\Sigma_g^{2g-4}) + \ell=6-4g+(6g+10m+18)=2g +10m+24 . \]

Since $X_m$ admits a 2-handlebody, we also have

\[\chi(X_m)= b_0(X_m) - b_1(X_m)+b_2(X_m) = 1+ b_2(X_m) . \]

It follows that $H_2(X_m) \cong \mathbb{Z}^{2g +10m+23}$ .

By (2.2), we have $b_2^0(X_m)=b_1(Y)-b_1(X_m)= b_1(Y)$ . Here, $-Y$ is diffeomorphic to an $S^1$ -bundle $\Sigma_g$ with Euler number $b=2g-4 \neq 0$ . (Note that $-Y$ is the boundary of the tubular neighborhood of a regular fiber $\Sigma_g$ transversally intersecting $2g-4$ disjoint $(-1)$ -spheres; the claim follows from blowing down the $(-1)$ -spheres in the usual Kirby diagram for the boundary of this plumbed 4-manifold.) So, $H_1(Y) \cong \mathbb{Z}^{2g} \oplus \mathbb{Z}_b$ . Thus, $b_2^0(X_m)=2g$ , so the rank of the maximal non-degenerate subspace of $H_2(X_m)$ with respect to the intersection form is $b_2(X_m)-b_2^0(X_m)=10m+23$ .

In order to calculate the signature of $X_m$ , let us first calculate the signature of the associated Lefschetz pencil $\widehat{X}_m$ prescribed by the (same) monodromy factorization (3.11). We calculate the latter following the approach in [Reference Endo and NagamiEN05]: take the infinite presentation of $\operatorname{Mod}(\Sigma_g^b)$ with generators as all Dehn twists along homotopically essential curves, and then $\sigma(\widehat{X}_m)$ is the algebraic sum of the signatures of the relators involved in the derivation of (3.11) from the trivial word. As we examine the derivation of the relation (3.11) in [Reference Baykur, Monden and Van Horn-MorrisBMV17], we see that the only relators with non-trivial signature contributions employed here are odd-chain relators (of different lengths and some are inverses) and a pair of daisy relators, each of which are derived from $(g-3)$ lantern relators. Denoting a length-k chain relator by $C_k$ and the lantern relator by L, we have

\begin{eqnarray*}\sigma(\widehat{X}_m)&=&\sigma(C_{2g+1})+\sigma(C_{2g-3}^{-1})+(g-2)\sigma(C_{3}^{-1}) +m\sigma(C_3) + (2g-6) \sigma(L)\\&=& -2g(g + 2)+ 2(g-2)g+(g-2-m)\,6+(2g-6)\\&=& -6m-18 .\end{eqnarray*}

As $X_m$ is the complement of a regular fiber of self-intersection $b=2g-4\gt0$ in the pencil $\widehat{X}_m$ , by Novikov additivity, we have

\[\sigma(X_m)=\sigma(\widehat{X}_m)-1=-6m-19. \]

Combining $b_2^+(X_m)+b_2^-(X_m)=10m+23$ and $b_2^+(X_m)-b_2^-(X_m)=-6m-19$ , we then get $b_2^+(X_m)=2m+2$ and $b_2^-(X_m)=8m+21$ .

Moreover, we can easily see that $X_m$ has odd intersection form. This follows from the observation that the monodromy factorization (3.11) is Hurwitz equivalent to another factorization of the form

\begin{equation*} t_{c_1}t_{c_3} t_d P'_m = \Delta \quad \text{in}\, \operatorname{Mod}(\Sigma) \quad \text{for all } m \in \mathbb{N},\end{equation*}

which is simply obtained by conjugating away all Dehn twists other than $t_{c_1}, t_{c_3}, t_d$ that appear in the original factorization. Clustering these three vanishing cycles on the same fiber, we get another Lefschetz fibration whose total space is still $X_m$ , but now it has a reducible fiber component $\Sigma' \cong S^2$ , which is obtained by pinching these three vanishing cycles. As the fiber framing of each Lefschetz 2-handle is $-1$ , the surface $\Sigma'$ has self-intersection $-3$ .

Hence, $Q_{X_m}$ is indefinite and odd, and over $\mathbb{R}$ ,

\[Q_{X_m}{\kern-1.2pt}\otimes\mathbb{R} \cong (2m+2)\langle 1 \rangle \oplus (8m+21) \langle -1 \rangle \oplus (2g) \langle 0 \rangle . \]

More generally, let $X_{g,b, m}$ denote the allowable Lefschetz fibration of genus $g \geqslant 4$ with monodromy factorization

\begin{equation*}P_{g, b,m}:=A_m B \,C \,D^\psi = \Delta \quad \text{in}\,\operatorname{Mod}(\Sigma_g^{b}) \quad \text{for all } m, b \in \mathbb{N}\text{ with } 1 \leqslant b \leqslant 2g-4 ,\end{equation*}

where for simplicity we use the same letters for the images of the products of Dehn twists in (3.11) under the homomorphism induced by the embedding $\Sigma_g^{2g-4} \hookrightarrow \Sigma_g^b$ and $\Delta$ is the boundary multitwist on $ \Sigma_g^b$ . These are Stein fillings of the contact 3-manifold $Y_{g,b}$ taken with the contact structure compatible with the boundary open book with monodromy $\Delta \in \operatorname{Mod}(\Sigma_g^b)$ . From our calculations above for the maximal case $X_m=X_{g, 2g-4, m}$ , we easily conclude that $H_1(X_{g, b, m})=0$ as well and

(3.12) \begin{equation} Q_{X_{g,b,m}}{\kern-1.2pt}\otimes\mathbb{R} \cong (2m+2)\langle 1 \rangle \oplus (2g-b+8m+17) \langle -1 \rangle \oplus (2g) \langle 0 \rangle .\end{equation}

Example 3.3 For the fillings $X_{g,b,m}$ of $Y_{g,b}$ discussed above, we can now contrast the invariants we calculated with the bounds for geometric genus and the formulas of Proposition 2.3.

We have $b_2^0(X_{g,b,m})=2g$ and $b_2^+(X_{g,b,m})=2m+2$ , thus, if $X_{g,b,m}$ is a Milnor fiber, we must have $2p_g=2g+2m+2$ . Since, by (3.10), we know that $p_g \leqslant g^2$ , $X_{g,b,m}$ cannot be a Milnor filling for $m\gt g^2-g-1$ . If $g\gt3$ and $b=2g -4$ , then $p_g \leqslant g+2$ by (3.9), so taking $m\gt1$ will ensure that $X_{g,2g-4,m}$ is not a Milnor fiber.

It might be worth noting that, alternatively, we can avoid the signature calculation by combining (2.5), (2.7) and (3.10) to show that

\begin{align*}\chi(W) \leqslant 12 g^2 - 2g + 2b -1\end{align*}

for all Milnor fillings, and therefore $X_{g,b,m}$ cannot be a Milnor fiber when $10m\gt12g^2 -6g+3b -21$ . Similarly, for the case $b=2g -4$ this argument gives a bound $\chi \leqslant 12 g^2+2g-9 $ , and it follows that $X_{g,2g-4,m}$ is an unexpected filling for $10m\gt12 g^2-33$ . This simpler calculation gives a larger bound for m compared to the bound obtained in the previous paragraph, but it has the advantage of generalizing easily to handle other examples covered by Theorem 1.4.

The minimal resolution for a surface singularity with link $Y_{g, b}$ has a unique exceptional curve E of genus g with self-intersection ${-}b$ . By [Reference Gay and MarkGM13], a compatible Lefschetz fibration for the corresponding Stein filling has the fiber $\Sigma^g_b$ , with the monodromy factorization corresponding to the boundary Dehn twists in the multitwist $\Delta=t_{\delta_1} \cdots t_{\delta_{b}}$ , so that $b_2=1$ . It is then clear that the fillings that correspond to non-trivial factorizations we constructed above cannot be diffeomorphic to the minimal resolution, and thus are not in the set of Milnor fillings.

4.1 An infinite family of minimally elliptic links

Consider the family of minimally elliptic singularities prescribed by the dual-resolution graphs in Figure 4, with $n=2k+1$ vertices in the $(-2)$ -chain. These are of the form $A_{n, ***}$ in Laufer’s notation [Reference LauferLau77a].

The fundamental cycle here is $Z_{\rm min}= Z_K=(1, 1, 2, 2, \dots, 2, 2, 1, 1)$ , where $Z_{\rm min}$ has coefficient 1 on each end vertex of the graph and coefficient 2 on each vertex in the chain of the n vertices with weights $-2$ .

To get an open-book decomposition of the link, we use Bhupal’s algorithm in [Reference BhupalBhu09, Reference BhupalBhu21]. The link Y is obtained by plumbing circle bundles over spheres, as dictated by the resolution graph. Bhupal’s procedure gives an open book whose binding is the union of a certain number of the fibers of these circle bundles, according to the numerical data from the intersection form of the resolution and the fundamental cycle. By [Reference Neumann and PichonNP07, Reference PichonPic01], this open book arises from a Milnor-type fibration associated to a holomorphic function vanishing at the isolated singular point of a germ of an appropriate analytic surface (X,0), such that Y is the link of (X, 0). Uniqueness results of [Reference Caubel, Némethi and Popescu-PampuCNP06, Reference Caubel and Popescu-PampuCP04] then imply that the open book is compatible with the canonical contact structure on Y.

In Bhupal’s algorithm, the page of the open book is built out of subsurfaces that correspond to the individual exceptional curves. (In our situation, all exceptional curves have $g=0$ , although sometimes the algorithm works more generally.) When the good vertex condition (1.1) is satisfied, each subsurface has the same genus as the corresponding exceptional curve, with the number of holes determined by the Euler number and the valency of the vertex in the graph, and these subsurfaces form the page as dictated by the graph, see also [Reference Gay and MarkGM13, Reference Etgü and OzbagciEO06]. When we have a bad vertex, that is a vertex violating the condition (1.1), the subsurfaces are coverings of the exceptional spheres with punctures; the number of sheets is determined by the corresponding coefficient of $Z_{\rm min}$ . In this construction, the monodromy of the resulting open book is presented as a product that may contain fractional multitwists. Under favorable circumstances, the latter can be expressed as a product of positive Dehn twists. For further details of the algorithm, we refer the reader to [Reference BhupalBhu09, Reference BhupalBhu21].

Applying this algorithm, we get a compatible open book on the link $(Y_k, \xi_k)$ with monodromyFootnote ³

(4.2) \begin{equation}\phi= t_{\delta_1} t_{\delta_2} t_{\delta_3} t_{a_1}^k t_{a_3}^kt_{a_1} t_{a_3} t_{b} t_{a_2} t_{a_3} t_{b} \quad \text{in}\, \operatorname{Mod}(\Sigma_1^3) ,\end{equation}

where the monodromy curves are as shown in Figure 4.

For each n, we can produce three Stein fillings $\{X_i\}$ with different second Betti numbers. Each Stein filling will have $H_1(X_i)=0$ , as is easily seen by the monodromy curves generating the first homology of the page in each case, so $b_2(X_i)=\chi(X_i)-1$ is determined by the number of Dehn twist curves in the factorization.

The filling $X_1$ is given by the allowable Lefschetz fibration with monodromy factorization as above. We have $b_2(X_1)= n+4$ , which matches $b_2$ of the minimal resolution. Next is the filling $X_2$ , given by the allowable Lefschetz fibration whose monodromy factorization is obtained from (4.2) by the star relation (the elliptic pencil monodromy with three base points): we replace the $t_{\delta_1} t_{\delta_2} t_{\delta_3}$ factor by $(t_{a_1} t_{a_2} t_{a_3} t_{b})^3$ . Then, $b_2$ increases by 9, so $b_2(X_2)= n+13$ , which matches the $b_2$ of a Milnor fiber (here we computed $Z_K^2=-3$ and used (2.8)). Finally, we obtain the filling $X_3$ by applying a lantern substitution to (4.2): we replace the Dehn twists around the curves $\{a_1, a_3, \delta_2, \delta_3\}$ by three other positive Dehn twists. Here the presence of the boundary component $\delta_1$ ensures that all three of the new curves are also homologically essential on $\Sigma_1^3$ . The corresponding Stein filling has $b_2(X_3)=n+3$ and thus cannot be a Milnor filling or the resolution.

4.2 Triangle singularities

A special class of minimal elliptic singularities are the triangle singularities $D_{p,q,r}$ , also known as Dolgachev singularities. These are characterized by three rational curves transversally intersecting at one point, with a resolution graph as shown in Figure 5, where ${1}/{p}+{1}/{q}+{1}/{r} \lt1$ and $r \geqslant q \geqslant p \geqslant 2$ .

Figure 5.

A family of smoothable triangle singularities and the monodromy curves for the open book on the link.

A triangle singularity $D_{p,q,r}$ is known to be smoothable if and only if $p+q+r \leqslant 22$ and $(p,q,r) \neq (2,10,10)$ (see [Reference LooijengaLoo83, Reference PinkhamPin83]). We consider the family $D_{4,4,4+k}$ , for $k=0, \ldots, 6$ , where $k=0$ is the hypersurface singularity $\{x^4-y^3-z^3=0\} \subset \mathbb{C}^3$ .

Label the vertices in the resolution graph in Figure 5 so that the central vertex comes first and the one with weight $4+k$ comes last. The fundamental cycle is $Z= (3, 1, 1, 1)$ . Applying Bhupal’s algorithm, we see that the link $(Y_k, \xi_k)$ is supported by an open book with monodromy

(4.3) \begin{equation}\phi= t_{\delta_1} t_{\delta_2} \cdots t_{\delta_{3+k}} t_{a_1}t_{a_2} t_{a_3} t_b \quad \text{in}\, \operatorname{Mod}(\Sigma_1^{3+k}) ,\end{equation}

where the monodromy curves are as shown in Figure 5. Here, the subsurfaces that make up page $\Sigma_1^{3+k}$ of the open book are $\Sigma_1^3$ corresponding to the central vertex (as a 3-fold cover of $\Sigma_0^3$ in the algorithm), two copies of $\Sigma_0^2$ corresponding to the vertices with weights $(-4)$ , and $\Sigma_0^{k+2}$ corresponding to the last vertex. The resulting page is $\Sigma_1^{3+k}$ . The algorithm yields a monodromy expression with ${2\pi}/{3}$ -fractional multitwist along the boundary components of the subsurface $\Sigma_1^3$ , which can be rewritten as shown using the star relation above.

Once again, we obtain three Stein fillings for each link $(Y_k, \xi_k)$ with distinct second Betti numbers. As before, we easily see that $H_1(X_i)=0$ and compute $b_2$ from the number of Dehn twist curves in the factorization.

Let the filling $X_1$ be given by the allowable Lefschetz fibration with monodromy factorization (4.3), which has $b_2(X_1)=3$ , matching $b_2$ of the minimal resolution. Next is the filling $X_2$ , given by the allowable Lefschez fibration whose monodromy factorization we derive from (4.3) by substituting the factor $t_{\delta_1} t_{\delta_2} \cdots t_{\delta_{3+k}}$ with 12 Dehn twists along homologically essential curves, coming from the monodromy of an elliptic pencil with nine base points [Reference Baykur, Monden and Van Horn-MorrisBMV17]. (For $k \gt 6$ , there is no such pencil.) Here, $b_2(X_2)= 12-k$ , which agrees with $b_2$ of the Milnor fiber(s). Lastly, we obtain the filling $X_3$ by applying a lantern substitution to (4.3) as before: we replace the Dehn twists around the curves $\{a_1, a_3, \delta_1, \delta_2\}$ by three other positive Dehn twists. So, $b_2(X_3)=2$ , and this filling is unexpected.

4.3 Non-smoothable cusp singularities

Rational surface singularities are always smoothable, and the Milnor fiber for one of the smoothings is diffeomorphic to the minimal resolution. In the minimally elliptic case, a new phenomenon appears: some of the singularities in this class are non-smoothable.

The following dual-resolution graphs correspond to cusp singularities. The graph is a cycle with r vertices corresponding to exceptional curves of genus 0, and the self-intersections of the exceptional curves are the integers $-a_i$ , where $a_i\geqslant 2$ for all $i=1, \dots, r$ , and $a_j\geqslant 3$ for some j. See Figure 6.

Figure 6.

The minimal resolution graph for a cusp singularity.

By [Reference WahlWah81], a singularity in this family is not smoothable whenever $m\gt r+9$ , where the multiplicity m is given by $\sum_{i=1}^r (a_i-2)$ . We consider non-smoothable cusp singularities with

(4.4) \begin{equation} \sum_{i=1}^r (a_i-3) \gt9. \end{equation}

The non-smoothable cusp singularities are taut, that is, there is only one analytic structure compatible with the given resolution graph. For this analytic structure, there are no smoothings and thus no Milnor fibers. The only Milnor filling in this case is given by the minimal resolution. We can construct unexpected fillings for many of these links.

Proof. A compatible open book for the contact link is constructed from the resolution graph of Figure 6 using [Reference Gay and MarkGM13, Theorem 1.1]. (Also see [Reference Bhupal and OzbagciBO14].) Indeed, [Reference Gay and MarkGM13] describes a compatible Lefschetz fibration for the symplectic plumbing given by the minimal resolution. The fiber of the Lefschetz fibration is a torus, with subsurfaces given by punctured spheres that correspond to the exceptional curves. The monodromy is given by the product of positive Dehn twists around the necks connecting these subsurfaces, together with positive boundary twists around each puncture. The total number of vanishing cycles on the boundary of the subsurface corresponding to the ith exceptional curve equals $a_i$ . If $a_i=2$ , then the corresponding subsurface is an annulus with no additional punctures (resulting in two parallel vanishing cycles). See Figure 7.

Figure 7.

An open book for the link of a cusp singularity has genus 1; each exceptional curve of self-intersection $-a_i$ corresponds to a cylindrical subsurface of the page with $a_i-2$ punctures. When $a_i=2$ , the subsurface is an annulus with no punctures. The monodromy is the product of the positive Dehn twists around the punctures and the meridianal curves separating the subsurfaces. Left: an open book for the cusp with $a_1=4, a_2=6, a_3=4, a_4=5, a_5=4, \dots$ . Right: an open book for the cusp with $a_1=8, a_2=a_3=2, a_4=5, a_5=a_6=\dots=a_{9}=2$ .

Under favorable numerical conditions on $\{a_i\}$ , the monodromy admits different positive factorizations, which allows us to generate different Stein fillings. If $a_i=4$ , then the page contains a subsurface given by a sphere with four punctures, with boundary Dehn twists around each puncture; we can apply the lantern substitution to the product of these four Dehn twists. Suppose that $a_i=4$ for all i, with $r\gt\max\{2N+1,9\}$ . Then at least N lantern substitutions can be performed on pairwise disjoint subsurfaces; different numbers of such substitutions yield Stein fillings with different $b_2$ since each lantern substitution lowers $b_2$ by one.

Additional lantern substitutions are often possible, and under other numerical hypotheses, one may be able to use generalized lantern relations (e.g. the monodromy factorization illustrated on the right of Figure 7 allows for daisy substitutions [Reference Baykur, Monden and Van Horn-MorrisBMV17]).

5.1 Smoothings of sandwiched singularities

We closely follow [Reference de Jong and van StratendJvS98]. Let us fix a possible topological type of a sandwiched singularity. It is characterized by its minimal resolution graph $\Gamma$ , which is a ‘sandwiched graph’. This means that $\Gamma$ is connected and it can be completed by several $(-1)$ vertices $\{F_i\}_{i=1}^r$ (and the supporting edges) such that the completed graph is smooth: it can be blown down to the empty graph. Let us regard the completed graph as the dual graph of an exceptional curve, and add to each $F_i$ a generic curvetta $\widetilde{C}_i$ , which is a small complex disc transverse to $F_i$ . In the language of graphs, each curvetta is represented by an arrowhead supported by the $F_i$ ; in this way we get a graph $\widetilde{\Gamma}$ . (For the correspondence between dual graphs and exceptional curves of resolutions, see e.g. [Reference NémethiNém99, Reference NémethiNém22].) Then, in this geometric presentation, if we blow down all the exceptional curves, the image of $\bigcup_{i=1}^r \widetilde{C}_i$ will be a plane-curve singularity $(C,0)=\bigcup_i (C_i,0)$ , with r local irreducible components, such that $\widetilde{\Gamma}$ is its embedded resolution graph. There can be several combinatorial ways to complete the graph $\Gamma$ by the $(-1)$ vertices as above. We fix one such combinatorial choice; this will provide the graph $\widetilde{\Gamma}$ . Then, the equisingularity type of (C,0) (i.e. its embedded topological type) is independent of all the other choices; it depends only on $\widetilde{\Gamma}$ .

One can proceed in the opposite order as well. We start with a decorated isolated curve singularity (C,l). Here, (C,0) is an isolated plane curve singularity, and each component $(C_i,0)$ of (C,0) is decorated by an integer $l_i$ , $i=1, \dots, r$ . We consider the minimal embedded resolution graph $\widetilde{\Gamma}_{\rm min}$ of (C,0) and then, for every i, we blow up several times the intersection point of the strict transform of $(C_i,0)$ with the exceptional curve (the number of blowups is determined from $l_i$ and $\widetilde{\Gamma}_{\rm min}$ ; for details, see [Reference de Jong and van StratendJvS98]). In this way, we obtain $\widetilde{\Gamma}=\widetilde{\Gamma}(l)$ . Then, the non- $(-1)$ -vertices form the resolution graph $\Gamma$ of the constructed sandwiched singularity. In particular, the equisingularity type of (C,0) and l determines the topological type of the sandwiched singularity (X,0), and the analytic type of (C,0) and l determines the analytic type of (X,0).

This shows that, if we consider all the decorated analytic plane-curve germs with fixed equisingularity type, as determined by $\widetilde{\Gamma}(l)$ , then the de Jong–van Straten construction provides all the analytic sandwiched singularities with topological type fixed by their common resolution graph $\Gamma$ .

Next, we connect smoothings of (X,0) with deformations of (C,l): picture deformations of (C,l) provide smoothing components of (X,0). Let us fix a decorated germ (C,l). A one-parameter deformation of (C,l) consists of a one-parameter $\delta$ -constant deformation $\{C_t\}_t$ of (C,0) and a flat deformation $\{l_t\}_t$ of the scheme associated to l. This deformation is a ‘picture deformation’ if, for $t\not=0$ , the divisor $l_t$ is reduced. In such a case, $C_t$ for $t\not=0$ can have only ordinary m-tuples as singularities. A part of the (reduced) points of $l_t$ must sit in these m-tuples $(m\geqslant 2$ ), and there might be some other free reduced points supported on smooth points of $C_t$ . Such a picture deformation generates a smoothing of (X,0), hence a Milnor fiber.

Let us recall an important property of a deformation $(\Sigma,0) \stackrel{\pi}{\rightarrow} (T,0)$ of (C,0) over a smooth one-dimensional germ (T,0). Let $\widetilde{C}$ be the normalization of (C,0) (as a smooth multigerm, which can be identified with r small discs). By a characterization of Teissier [Reference TeissierTei80] (see also [Reference Chiang-Hsieh and LipmanCL04]), a deformation $\pi$ is $\delta$ -constant if and only if the normalization morphism of $\Sigma$ has the form $\tilde{C}\times T\to \Sigma$ , and it is the simultaneous normalization of the fibers of $\pi$ . In particular, if one has a $\delta$ -constant embedded deformation of a germ of plane curve (C,0) such that the general fiber $C_t$ has only ordinary multiple points, then after a choice of a Milnor representative of the morphism, the general fibers are immersed discs in the 4-ball (in $\mathbb{C}^2$ ); their number is the number of local irreducible components of (C,0) (see e.g. [Reference Némethi and Popescu-PampuNP10a, 2.2]).

5.2 Fixing the topological type and combinatorial data

We consider the family of all the decorated analytic plane-curve germs with fixed equisingularity type and constant decorations. Since the picture deformations of any fixed member of the family provide all the smoothings of the corresponding sandwiched singularity, we will obtain all the Milnor fibers of all analytic structures (supported by a fixed topological type) when we consider all the deformations of all the members.

If we consider a deformation of (C,0) and fix a small Milnor ball $B^\epsilon \subset \mathbb{C}^2$ centered at the origin, then for $|t|$ small, the singularities of $C_t $ and the support of $l_t$ sit in $B^\epsilon$ , and $C_t$ intersects $\partial B^\epsilon$ transversely. For $t\not=0$ , they provide a topological picture: the union of immersed discs $C_{t,i}$ in $B^\epsilon$ , which intersect each other and themselves along certain m-tuples. From this embedded data $(C_{t,i}, l_{t,i})_i \subset B^\epsilon$ one constructs the Milnor fiber of the smoothing.

Since we are interested in fillings of the link $(Y, \xi)$ , we usually consider the compact version of the Milnor fiber, which is the complement of small tubular neighborhoods of the strict transforms in a closed Milnor ball. (Technically, this construction produces a manifold with corners; we will assume that the corners are smoothed.)

Combinatorially, the intersections of the immersed discs $C_{t,i}$ and the positions of the points $l_{t,i}$ are encoded by an incidence matrix of size $r \times \#\{$ points $\}$ with non-negative entries $v_{ij}$ , where $v_{ij} $ is the multiplicity of the branch $C_{t,i}$ at the point $p_j$ . These entries satisfy several identities (cf. [Reference de Jong and van StratendJvS98, p. 484]), e.g. for all i:

(5.1) \begin{equation}\sum_j v_{ij}(v_{ij}-1)=2\delta(C_i,0) \quad \mbox{or} \quad \sum_j v_{ij}=l_i.\end{equation}

Denote the set of possible decorated embedded topological types $(C_{t,i}, l_{t,i})_i \subset B^\epsilon$ by $\mathcal{E}$ , the set of incidence matrices by $\mathcal{I}$ , and the set of Milnor fibers by $\mathcal{F}$ (all realized by analytic deformations). Then, the construction [Reference de Jong and van StratendJvS98, Proposition 5.1] provides a map $\beta : \mathcal{E}\to \mathcal{F}$ , and $\alpha:\mathcal{E}\to \mathcal{I}$ associates the combinatorial incidence matrix $(v_{ij})_{ij}$ to a decorated embedded topological type $(C_{t,i}, l_{t,i})_i \subset B^\epsilon$ (i.e. to a picture deformation), cf. [Reference de Jong and van StratendJvS98, 4.3].

Let us make the following observation regarding the maps $\alpha$ and $\beta $ above. Since the graph $\widetilde{\Gamma}(l)$ is fixed, and hence the embedded topological type of (C,0) is fixed, its delta invariant $\delta$ and Milnor number $\mu$ are fixed as well. Furthermore, each $l_i$ is also fixed. This shows that the set $\mathcal{I}$ is finite. Indeed, we can define the set of possible combinatorial incidence matrices $\mathcal{I}_{\rm comb}$ satisfying (say) the identities as in (5.1), then $\mathcal{I}_{\rm comb}$ is finite by (5.1), and the analytically realized matrices $\mathcal{I}$ form a subset of $\mathcal{I}_{\rm comb}$ .

What is not immediate is that the cardinality of $\alpha^{-1}(I)$ ( $I\in\mathcal{I}$ ) is finite. In principle, we can imagine many different decorated immersed discs with the very same combinatorial incidence matrix. Even more, starting from a concrete immersion, one can even try to construct, by surgeries, many others with unmodified incidence matrix and unmodified boundary of the filling.

On the other hand, it is worth pointing out that the main result of [Reference Némethi and Popescu-PampuNP10a] implies that for $I_1, \, I_2\in\mathcal{I}$ , $I_1\not=I_2$ , $\beta(\alpha^{-1}(I_1))\cap \beta(\alpha^{-1}(I_2))=\emptyset$ . Hence, Milnor fibers corresponding to different incidence matrices are different (up to diffeomorphism fixing certain boundary data). In particular, $\#\mathcal{F} \geqslant \#\mathcal{I}$ . We wish to complete this lower bound by proving that $\mathcal{F}$ , in fact, is finite. (It will follow from the proof that the set $\mathcal{E}$ is also finite.)

5.3 A parameter space of curves

We would like to consider possible analytic structures of decorated curve germs (C,l) with fixed topology $\widetilde{\Gamma}(l)$ . To construct a parameter space, it is helpful to globalize the local curves into projective ones. It is known that for any plane-curve singularity (C,0), there exists $d_0$ sufficiently large (depending on the topological type of (C,0)) such that, for any $d\geqslant d_0$ , there exists a projective plane curve of degree d with only one singular point analytically equivalent to (C,0), see e.g. [Reference Greuel, Lossen and ShustinGLS18, Corollary 4.5.15] or [Reference Greuel, Lossen and ShustinGLS18, § 1.1.3.5]. In fact, even the irreducibility of the projective curve is guaranteed [Reference Greuel, Lossen and ShustinGLS18, Theorem 4.6.4].

Next, note that if (C,0) has Milnor number $\mu$ (determined by its embedded topological type), then the equation of (C,0) is $(\mu+1)$ -determined. This means that for any $k\geqslant \mu+1$ , the analytic type is determined by the k-jet of the equation, see e.g. [Reference Arnold, Goryunov, Lyashko and Vasil’evAGL⁺98, p. 14]. Furthermore, the universal deformation space of $(f=0,0) =(C,0)$ has finite dimension $\tau\leqslant \mu$ , and it has the form $f+\sum_{j=1}^\tau t_jg_j $ , where $t_j\in(\mathbb{C},0)$ and $\{g_j\}_j$ induce a basis of the Tjurina algebra $T(f):=\mathbb{C}\{x,y\}/(f, \partial f/\partial x, \partial f/\partial y)$ of dimension $\tau$ . Note that if we add to $g_j$ a monomial of degree larger than $\mu$ , then its class in T(f) will not change. This shows that, for any analytic type (C,0), one can find $ \tau$ polynomials $g_j$ , all with degrees bounded by $\mu$ , such that $f+\sum_{j} t_jg_j $ is a versal deformation of $f=0$ , hence it ‘contains’ all the deformations of (C,0).

Let us consider now the space $\mathcal{C} =\mathbb{C}\mathbb{P}^M$ of all projective curves in $\mathbb{C}\mathbb{P}^2$ of degree d, for d sufficiently large ( $M=d(d+3)/2$ ). By the above discussion, any local analytic singularity with graph $\widetilde{\Gamma}(l)$ appears as a singularity of a projective curve $\bar{C}\in \mathcal{C}$ ; also, the neighborhood of $\bar{C}$ in $\mathcal{C}$ serves as a versal deformation of $\bar{C}$ . In this way, all the local analytic types of plane singular germs with topology identified by $\widetilde{\Gamma}(l)$ , and all their deformations, can be realized inside $\mathcal{C}$ .

Let $\mathcal{C}(\Gamma)$ be the set of all curves $\bar{C}\in \mathcal{C}$ such that $\bar{C}$ has exactly one singular point of embedded topological type $\widetilde{\Gamma}(l)$ . Then, $\mathcal{C}(\Gamma)$ is a non-empty quasiprojective variety in $\mathcal{C}$ (see e.g. [Reference Greuel, Lossen and ShustinGLS18, pp. 59--61]). By [Reference WahlWah74], for any fixed $(C,0)\subset (\mathbb{C}^2,0)$ , the strata ES parameterizing the equisingular deformations of (C,0) inside the semiuniversal deformation space is a smooth subvariety. Since our parameter space is versal for any (C,0), the same property holds: $\mathcal{C}(\Gamma)$ is a smooth quasiprojective subvariety of $\mathcal{C}$ (see also [Reference Greuel, Lossen and ShustinGLS18, p. 61]). Moreover, by [Reference Greuel, Lossen and ShustinGLS18, Theorem 4.6.4], $\mathcal{C}(\Gamma)$ is even irreducible (a fact which will not be used here).

Consider the universal family $\mathcal{U}= \{(\bar{C},p)\in\mathcal{C}\times \mathbb{C}\mathbb{P}^2\,:\, p\in \bar{C}\}$ , the first projection $q:\mathcal{C}\times \mathbb{C}\mathbb{P}^2\to \mathcal{C}$ given by $(\bar{C},p)\mapsto \bar{C}$ , and its restriction to $\mathcal{U}$ , $q_{\mathcal{U}}:\mathcal{U}\to\mathcal{C}$ . Set $\mathcal{U}(\Gamma):= q_{\mathcal{U}}^{-1}(\mathcal{C}(\Gamma))\subset \mathcal{U}$ . Then, the restriction $q_{\mathcal{U}}:\mathcal{U}(\Gamma)\to \mathcal{C}(\Gamma)$ admits an algebraic section, $\bar{C}\mapsto (\bar{C},p_{\bar{C}})$ , where $p_{\bar{C}}$ is the unique singular point of $\bar{C}$ . Let us denote by $\mathcal{P}(\Gamma)$ this image $(\bar{C}, p_{\bar{C}})_{\bar{C}\in \mathcal{C}(\Gamma)}$ in $\mathcal{U}(\Gamma)$ . Then, $\mathcal{P}(\Gamma)\to \mathcal{C}(\Gamma)$ is an isomorphism and $\mathcal{P}(\Gamma)$ is a smooth subvariety of $\mathcal{C}\times \mathbb{C}\mathbb{P}^2$ .

5.4 Uniform Milnor radius

Now, a decorated-curve germ (C,l) with fixed topology $\widetilde{\Gamma}(l)$ corresponds to a point $C \in \mathcal{C}(\Gamma)$ with appropriate decoration. Given a choice of picture deformation $(C_t, l_t)$ of (C, l), the Milnor fiber of the corresponding smoothing is obtained from a Milnor ball $B^\epsilon$ of C as in Proposition 5.2. The radius $\epsilon$ of the Milnor ball a priori depends on C. We will need to be able to choose a uniform radius in a neighborhood of $C \subset \mathcal{C}(\Gamma)$ ; then, a uniform positive radius can be found over any subset of $\mathcal{C}(\Gamma)$ with compact closure. The existence of a uniform radius is guaranteed by the following local argument (it is ‘classical’, but far from trivial).

Hence, constancy of the $\mu^*$ -sequence implies the existence of the uniform Milnor radius. For more details, see also [Reference Buchweitz and GreuelBG80, p. 261 and Theorem 5.3.1] and the corresponding references therein.

It may be helpful to note that the lemma admits an elementary proof if all the irreducible components of the germ C are smooth (in the de Jong–van Straten setting, such decorated germs correspond to rational surface singularities with reduced fundamental cycle). In this case, the existence of a uniform Milnor radius follows simply from smoothness of individual branches and transversality.

In the subsequent steps, we will consider picture deformations of the curves C inside these Milnor balls. Different combinations of m-tuple singularities of picture deformations give a stratification of the corresponding space of curves. We will show that the corresponding Milnor fibers look the same within a component of a fixed stratum (more precisely, they form a $C^{\infty}$ locally trivial fibration). Finiteness of the set of Milnor fillings will then follow from finiteness of the related stratification.

5.5 Curves with m-tuple singularities: stratification

Consider

\begin{align*}\mathcal{S}=\{\bar{C}\in\mathcal{C}\,:\, \bar{C} \ \text{has only } \textit{m} \text{-tuple type singularities}\}.\end{align*}

The space $\mathcal{S}$ has an interesting stratification: it decomposes into several disjoint strata $\mathcal{S}_{\overline{m}}$ according to the type $\overline{m}=(m_1, m_2, \dots)$ of the singular points. In fact, we consider only those strata of $\mathcal{S}$ which appear in the neighborhood of $\mathcal{C}(\Gamma)$ , that is, their closure intersects $\mathcal{C}(\Gamma)$ non-trivially. Furthermore, we assume, for any such component, that they represent $\delta$ -constant deformations of certain points of $\mathcal{C}(\Gamma)$ . In other words, there exists a real-analytic path $\gamma:[0, \tau]\to\mathcal{C}$ , $\gamma(0,\tau]\in \mathcal{S}_{\overline{m}}$ , $\gamma(0)\in \mathcal{C}(\Gamma)$ , such that $\gamma(t)$ is a $\delta $ -constant deformation of $\gamma(0)$ , with m-tuple singularities of type $(m_1,m_2,\ldots)$ , so that the type is constant for $t\not=0$ . This shows that the constant delta value of the strata $\mathcal{C}(\Gamma)$ bounds the possible collection $(m_1, m_2, \ldots)$ , since $\sum_i m_i(m_i-1)/2=\delta$ .

Since all the $m_i$ values are bounded, if we choose d (the degree of the curves from $\mathcal{C}$ ) sufficiently large, then each $\mathcal{S}_{\overline{m}}$ is quasiprojective, and in fact it is smooth, cf. [Reference Greuel, Lossen and ShustinGLS18, Chapter 4]. Moreover, the number of irreducible, or connected, components of any $\mathcal{S}_{\overline{m}}$ is finite.

If (C,0) itself is an m-tuple, then $\mathcal{C}(\Gamma)$ is such a stratum $\mathcal{S}_{\overline{m}}$ , with only one entry $m_i$ which is $m=r$ .

In general, the varieties $\{\mathcal{S}_{\overline{m}}\}_{\overline{m}}$ form an interesting stratification, e.g. $\mathcal{S}_{(3)}$ sits in the closure of $\mathcal{S}_{(2,2,2)}$ . But we emphasize again: m-tuple points parameterized by a fixed component of $\mathcal{S}_{\overline{m}}$ do not merge, do not split, and their number is constant.

It can happen that the closure of a certain $\mathcal{S}_{\overline{m}}$ contains all $\mathcal{C}(\Gamma)$ . This means that any analytic germ of $\mathcal{C}(\Gamma)$ admits a $\delta$ -constant deformation of that type $\overline{m}$ . But it can also happen that the closure of a certain $\mathcal{S}_{\overline{m}}$ contains only a proper subspace of $\mathcal{C}(\Gamma)$ . This means that only certain restricted analytic germs of $\mathcal{C}(\Gamma)$ admit $\delta$ -constant deformations of that type $\overline{m}$ . See [Reference de Jong and van StratendJvS98, p. 501] for a concrete example.

5.6 Local triviality over strata

We focus on a stratum $\mathcal{S}_{\overline{m}}$ for a fixed choice of $\overline{m}$ . We will consider two cases, as the setup is slightly different: (1) $\mathcal{S}_{\overline{m}} = \mathcal{C}(\Gamma)$ and (2) $\mathcal{S}_{\overline{m}}\not= \mathcal{C}(\Gamma)$ . In the first case, (C,0) must be an m-tuple, and $\overline{m}=(m)$ , where $m=r$ . In the second case, $\mathcal{S}_{\overline{m}}\cap \mathcal{C}(\Gamma)=\emptyset$ , but the intersection of the closure of $\mathcal{S}_{\overline{m}}$ with $\mathcal{C}(\Gamma)$ must be non-empty (see the definition of the strata $\mathcal{S}_{\overline{m}}$ ).

Take $C \in \mathcal{C}(\Gamma)$ , and assume for convenience that $p_{C}=0$ . Let $\epsilon\gt0$ and let a neighborhood $V\subset \mathcal{C}(\Gamma)$ of C be as in Lemma 5.3, so that $B^\epsilon= B^\epsilon(0)$ is a Milnor ball for every curve in V. We fix this $B^\epsilon \subset \mathbb{C}^2 \subset \mathbb{C}\mathbb{P}^2$ .

In Case (1), we consider the projection

(5.2) \begin{equation} (V \times B^\epsilon, (V \times B^\epsilon) \cap \mathcal{U}) \stackrel{q}{\to} V.\end{equation}

In Case (2), we can find a neighborhood $W= W_{C}$ of C in the space $\mathcal{C}$ such that for every $\bar{C} \in W$ , all singular points of $\bar{C}$ are contained in the interior of $B^\epsilon$ , and the intersection of $\bar{C}$ and $\partial B^\epsilon$ is transverse. (Here and below, $B^\epsilon$ is the closed ball.) We then consider the projection

(5.3) \begin{equation} ((W \cap \mathcal{S}_{\overline{m}}) \times B^\epsilon, ((W \cap \mathcal{S}_{\overline{m}}) \times B^\epsilon) \cap \mathcal{U}) \stackrel{q}{\to} W \cap \mathcal{S}_{\overline{m}}.\end{equation}

In each case, the fiber over the curve $C_t$ representing a point in V respectively $W\cap \mathcal{S}_{\overline{m}}$ is $(B^\epsilon, C_t \cap B^\epsilon)$ . We would like to show that the fibration (5.2) respectively (5.3) is locally trivial in the appropriate sense. More precisely, for a picture deformation $C_t$ endowed with a decoration, we need to understand the fibration obtained by the procedure of constructing smoothings and Milnor fibers of the corresponding surface singularities, as in Proposition 5.2. Case (1) is the easier of the two, and we will comment on it after treating Case (2) in detail. We now assume that $\mathcal{S}_{\overline{m}}\not= \mathcal{C}(\Gamma)$ , and W is chosen as above.

By our assumptions, every $C_t \in W \cap \mathcal{S}_{\overline{m}}$ has the same $\delta$ as (C,0), all its singular points are m-tuples, these singular points are in $B^\epsilon$ , and $C_t$ intersects $\partial B^\epsilon$ transversely.

Then, $C_t$ can be completed to a picture deformation $(C_t,l_t)$ . The points of $l_t$ supported at the singular points are determined by $C_t$ ; the other (free) points, still sitting in the interior of $B^\epsilon$ , are distributed on the regular part of the components $C_{i,t}$ . The number of free points on each $C_{i,t}$ , say $l_{i,t}'$ , is fixed from the combinatorics. Indeed, if $p_j$ are the singular points of $C_t$ and $v_{ji}'$ is the multiplicity of $C_{t,i}$ at $p_j$ , then $l_{i,t}'=l_i-\sum_j v_{ij}'$ .

Each completion provides a picture deformation, hence an element of $\mathcal{E}$ (decorated immersed discs in $B^\epsilon$ ). By Proposition 5.2, if we blow up $B^\epsilon$ at the points of the support of $l_t$ and delete the strict transform of the curves $\{C_{i,t}\}_i$ , we get the Milnor fiber of the corresponding picture deformation (or of the corresponding smoothing).

Regarding these blowups, we proceed in two steps. First, we blow up the singular points of $C_t$ . As a byproduct of this step, we will also prove the appropriate local triviality of the map q from (5.3). Then, as a second step, we blow up the free points as well. Denote the set of singular points of $((W \cap \mathcal{S}_{\overline{m}}) \times B^\epsilon) \cap \mathcal{U}$ by $\rm{Sing}_{\overline{m}}$ .

In order to prove the local triviality of q from (5.3), for any point $\bar{C}_0$ of the base $W \cap \mathcal{S}_{\overline{m}}$ we have to find a small ball (or a contractible open set) $W' \subset W \cap \mathcal{S}_{\overline{m}}$ in the base space which contains $\bar{C}_0$ , such that q is trivial over W’. We choose W’ as follows.

By construction, the number of singular points in any fiber $q^{-1}(\bar{C})\cap B^\epsilon$ is the same, say k. In particular, the restriction $ q:\mathrm{ Sing}_{\overline{m}}\to W \cap \mathcal{S}_{\overline{m}}$ is an unramified k-covering. This can be seen as follows. First, note that the restriction of q to $\mathrm{Sing}_{\overline{m}}$ is quasi-finite (i.e. any fiber is a finite set). Since this restriction is proper, it is even a finite morphism. Therefore, it is a branched covering. However, since all the fibers have the same cardinality, the critical (branched) locus is empty; for details, see [Reference Dethloff and GrauertDG94, p. 186]. Hence the restriction of q to $\mathrm{Sing}_{\overline{m}}$ is a regular unbranched covering. From this also follows that Sing_m itself is smooth (see also below).

We choose W’ contractible and so small that this regular (analytic) covering over W’ is trivial.

We wish to show the triviality of

(5.4) \begin{equation} (W' \times B^\epsilon, (W' \times B^\epsilon) \cap \mathcal{U}) \stackrel{q}{\to} W'.\end{equation}

From the choice of W’, the covering $(W' \times B^\epsilon) \cap \mathrm{Sing}_{\overline{m}}$ over W’ is trivial, hence it has k regular sections $s_i: W' \to (W' \times B^\epsilon) \cap \mathrm{Sing}_{\overline{m}}$ , $1\leqslant i\leqslant k$ , so that $q\circ s_i=\mathrm{id}_{W'}$ .

Since W’ is smooth, the image $s_i(W') $ is also smooth, and $s_i$ and q realize complex analytic isomorphisms between W’ and $s_i(W')$ . Now, $\bigcup_i s_i(W')$ is a smooth complex analytic subspace of complex codimension two in a smooth complex analytic space $W' \times B^\epsilon$ . We blow up this subspace and denote the total space by $\mathcal{B}(W')$ , the blowup projection by b, $q\circ b=\widetilde{q}$ , the exceptional divisor by $\widetilde{E}$ , and the strict transform of $(W' \times B^\epsilon) \cap \mathcal{U}$ by $\widetilde{St}(W')$ . Since all singular points of the curves from W’ are m-tuples, the strict transform $\widetilde{St}(W')$ is smooth, and it intersects the smooth subspace $\widetilde{E}$ transversely.

Over a point $p\in W'$ representing a curve with m-tuple singularities, $\widetilde{q}^{-1}(p)$ is the blowup of $B^\epsilon$ at the singular points of the curve p, $\widetilde{q}^{-1}(p)\cap \widetilde{E}$ consists of k copies of $\mathbb{C}{\mathbb P}^1$ which are the exceptional curves of this blowup of $B^\epsilon$ , $\widetilde{q}^{-1}(p)\cap \widetilde{St}(W')$ is the strict transform of the curve $p \cap B^\epsilon$ , and $\widetilde{q}^{-1}(p)\cap \widetilde{St}(W')\cap \widetilde{E}$ consists of $\sum_jm_j$ points corresponding to the type of the m-tuples. Furthermore, $\widetilde{St}(W')$ intersects $\partial\mathcal{B}(W')$ transversely as well.

Then, using the transversality and an Ehresmann-type argument (see below), we obtain that

(5.5) \begin{equation}(\mathcal{B}(W'), \partial \mathcal{B}(W'), \widetilde{E}, \partial \mathcal{B}(W')\cap \widetilde{St}(W'), \widetilde{E}\cap \widetilde{St}(W'))\to W'\end{equation}

is a $C^\infty$ locally trivial fibration of 5-tuples of spaces over W’. But W’ being contractible, this is in fact a trivial smooth fibration.

By an Ehresmann-type argument, we mean the following. The classical Ehresmann theorem asserts that a proper submersion between two closed smooth manifolds is a $C^\infty$ locally trivial fibration [Reference DundasDun18, § 8.5]. There are versions of this result for manifolds with boundary (the restriction of the map to the boundary is required to be a submersion), as well as for the relative case (see e.g. [Reference Neumann and PichonNP20, Theorem 1.2.16], [Mat20]). Because the construction involves the projection $\tilde{q}$ which is tautologically a submersion, and transversality ensures that all intersections are smooth submanifolds, it is not hard to see that the Ehresmann hypotheses are satisfied for the 5-tuple of (5.5), as well as in our subsequent applications of a similar argument. We leave the details to the reader.

A similar argument works in Case (1), but since each curve $\bar{C} \in V$ is an m-tuple, there is only one singular point to blow up in each fiber. We obtain a $C^\infty$ locally trivial fibration of 5-tuples of spaces over V,

(5.6) \begin{equation}(\mathcal{B}(V), \partial \mathcal{B}(V), \widetilde{E}, \partial \mathcal{B}(V)\cap \widetilde{St}(V), \widetilde{E}\cap \widetilde{St}(V))\to V.\end{equation}

Blowing down the locally trivial fibrations (5.6) and (5.5), we can get local triviality statements for (5.3) and (5.2). The local triviality for (5.2) means the following. The first entry of the total space of the pair is smooth (and the fiber, the ball, as well), so in this case the usual definition works (of course, this is just a product fibration). The fiber of the second entry consists of $m=r$ embedded discs with m-tuple singularities. Here, at the smooth points it behaves as a usual smooth locally trivial fibration, and at the singular points all the branches (automatically smooth) behave as smooth locally trivial fibration, and additionally, the tangent lines at the singular points remain distinct. (This follows from local triviality of (5.6) and properties of blowdown.) Local triviality of (5.3) holds in a similar sense: the fiber of the second entry consists of the union of immersed discs, we have the usual $C^\infty$ local triviality at smooth points, and at each of the m-tuple singularities each branch individually behaves as a smooth locally trivial fibration, and the tangent lines to the branches are distinct. In particular, this implies that the fibrations (5.3) and (5.2) are topologically trivial, and away from the singularities they are smoothly trivial. Note that they cannot be smoothly trivial everywhere, in general: it is well known that even in the case of a 4-tuple of lines, there is an additional smooth invariant, the cross ratio.

5.7 Back to the decorations of the picture deformation

To complete the construction of Milnor fibers from a picture deformation $C_t$ , we need to perform the additional blowups at the free points. We proceed as above, blowing up a family of free points in a fibration (we explain Case (2), and Case (1) is similar).

Let us fix $W' \subset W \cap \mathcal{S}_{\overline{m}}$ as above (and the fibration over it with all its properties already proved). Let $C_t$ and $C_{t'}$ be the curves corresponding to two points of W’. Let us complete these curves with decorations, that is with free points $\{a_{t,j}\}_j\subset C_t$ and $\{a_{t',j}\}_j\subset C_{t'}$ . (The other decoration points are the singular points, discussed in the previous subsection.) The number of free points on $C_{i,t}$ and $C_{i,t'}$ agree: it is $l'_{i,t}$ .

We claim that there exist smooth sections

\begin{align*}s_{i,j}:W' \to (W' \times B^\epsilon) \cap \mathcal{U}, \quad 1\leqslant j\leqslant l'_{i,t}\end{align*}

such that $s_{i,j}(t'')$ are free decoration points on the curve $q^{-1}(t'')\cap \mathcal{U}$ , and over the points t and t’ we have $\{s_{i,j}(t)\}_j=\{a_{t,j}\}_j\subset C_t$ and $\{s_{i,j}(t')\}_j=\{a_{t',j}\}_j\subset C_{t'}$ .

This follows from the fact that, for any i, the space $(C_{i,t}\cap B^\epsilon) \setminus \mathrm{Sing }[C_{i,t}]$ is a smooth connected manifold embedded smoothly in the smooth ambient space, and small tubular neighborhoods of these manifolds do not intersect each other. ‘Moving’ the free points on this smooth curve can be realized by a smooth flow which is identity in the complement of these tubular neighborhoods, and flows non-trivially in the tubes (similarly to e.g. [Reference MilnorMil97, pp. 22--23]). Let $s_{i,j}(W')$ denote the smoothly embedded image of $s_{i,j}$ .

We wish to make the following comment. In the case of singular points of $C_t$ , the sections $s_i$ ( $1\leqslant i\leqslant k$ ) are complex analytic (or algebraic), but the above construction of the section of $s_{i,j}$ provides ‘only’ a smooth embedding. In fact, in the theory of de Jong and van Straten, the moving free points also appear as the support of an analytic/algebraic scheme, hence if we repeat their construction over the parameter space $\mathcal{S}_{\overline {m}}$ , we could even obtain that the sections $s_{i,j}$ are complex analytic, or algebraic. However, we will avoid this technical part and stay with the smooth case, which is still perfectly enough for smooth local triviality of the obtained fibrations and smooth identification of the Milnor fiber.

Hence, in this case, $W' \times B^\epsilon$ is a smooth complex analytic space and $\bigcup_{i,j} s_{i,j}(W')$ are smooth $C^\infty$ (disjoint) subspaces of real codimension four. We blow up $W' \times B^\epsilon$ along $\bigcup_is_i(W') $ (analytically) as in the previous subsection, and also along $\bigcup_{i,j}s_{i,j}(W') $ smoothly. (Recall that the blowup of a complex surface S can be performed at a point in the smooth category as well, e.g. as a smooth surgery, resulting in $S\#\overline{\mathbb{C}\mathbb{P}^2}$ . This can be done in smooth families as well.) Let us denote the total space of this additional blowup by $\mathcal{B}_b(W')$ instead of the previous $\mathcal{B}(W')$ , and adopt all the previous notation. Then, by an identical argument, we get that

(5.7) \begin{equation}\widetilde{q}_b\,:\, (\mathcal{B}_b(W'), \partial \mathcal{B}_b(W'), \widetilde{E}_b, \partial \mathcal{B}_b(W')\cap\widetilde{St}_b(W'), \widetilde{E}_b\cap \widetilde{St}_b(W'))\to W' \end{equation}

is a $C^\infty$ (locally) trivial fibration of 5-tuples of spaces over W’.

5.8 The Milnor fibers of picture deformations

We are ready to prove finiteness of the set of Milnor fillings of the link of a sandwiched singularity (X, 0). We fix a decorated germ (C, l) that corresponds to (X, 0) as in § 5.2, and consider the space of curves $\mathcal{C}(\Gamma)$ as in § 5.3. Fix a stratum $\mathcal{S}_{\overline{m}} \neq \mathcal{C}(\Gamma)$ whose closure intersects $\mathcal{C}(\Gamma)$ . (Again, we consider Case (2) of § 5.6; Case (1) is similar but easier.) Suppose that $(C_t, l_t)$ is a picture deformation of a curve $C_0 \in \mathcal{C}(\Gamma)$ , such that $C_t \in \mathcal{S}_{\overline{m}}$ . If we fix $\epsilon\gt0$ , the ball $B^\epsilon$ centered at the singular point $p_{C_0}$ of $C_0$ , and $W_0= W_{C_0}$ as in § 5.6, then $C_t \in W_0 \cap \mathcal{S}_{\overline{m}}$ for small t, so that all the singular points of $C_t$ lie in $B^\epsilon$ , and $C_t$ is transverse to $\partial B^\epsilon$ . Applying Proposition 5.2 to $(B^\epsilon, C_t \cap B^\epsilon)$ with the decoration $l_t$ , we get the Milnor fiber of the smoothing corresponding to $(C_t, l_t)$ . This Milnor fiber, associated to the picture deformation $C_t$ and the free decorations $\{a_{t,j}\}_j$ , can be constructed from the fibration $\widetilde{q}_b$ of (5.7), assuming that $C_t\in W'$ : we delete $\widetilde{St}_b(W') \cap (\widetilde{q}_b)^{-1}(C_t)$ (or its small tubular neighborhood) from $(\widetilde{q}_b)^{-1}(C_t)$ . The (local) triviality of (5.7) implies that if we apply the same construction in the fiber $(\widetilde{q}_b)^{-1}(C_{t'})$ where $C_{t'}\in W'$ for a decorated curve $(C_{t'}, l_{t'})$ , then the resulting 4-manifold with boundary $M_{(C_{t'}, l_{t'})}$ will be diffeomorphic to the Milnor fiber associated to $(C_t, l_t)$ . Here, we need to point out a technicality: $M_{(C_{t'}, l_{t'})}$ is constructed by applying the procedure of Proposition 5.2 to $(B^\epsilon, C_{t'} \cap B^\epsilon)$ , where the ball $B^\epsilon = B^\epsilon(p_{C_0})$ depends on the curve $C_0$ . Strictly speaking, Proposition 5.2 says that if $(C_{t'}, l_{t'})$ is a picture deformation of another germ $(C_0', l')$ , then we need to choose a Milnor ball $B^{\epsilon'}(p_{C'_0})$ and apply the procedure to this ball (assuming that t’ is small enough so that all singularities of $C_{t'}$ are contained in $B^{\epsilon'}(p_{C'_0})$ , and $C_{t'}$ is transverse to the boundary of this ball). However, since the same assumption is satisfied for $B^\epsilon(p_{C_0})$ and $B^{\epsilon'}(p_{C'_0})$ , it is easy to see that the resulting 4-manifolds are diffeomorphic (use an interpolating family of balls satisfying the same condition). Therefore, $M_{(C_{t'}, l_{t'})}$ is indeed diffeomorphic to the Milnor fiber associated to the picture deformation $(C_{t'}, l_{t'})$ of $(C_0', l')$ .

We can now finish the proof as follows. For each $\bar{C} \in \mathcal{C}(\Gamma)$ , we choose an open set $W_{\bar{C}}$ as in § 5.6, then $\mathcal{N=}\bigcup_{\bar{C} \in \mathcal{C}(\Gamma)} W_{\bar{C}}$ is an open tubular neighborhood of $\mathcal{C}(\Gamma)$ . For a fixed one-parameter picture deformation $(C_t, l_t)$ , all values of t produce smoothings with diffeomorphic Milnor fibers, so we know that every Milnor fiber has a representative that comes from a curve in some connected component of $\mathcal{S}_{\overline{m}}\cap\mathcal{N}$ , for some stratum $\mathcal{S}_{\overline{m}}$ , such that the closure of this component intersects $\mathcal{C}(\Gamma)$ . Taking a smaller $\mathcal{N}$ of the same form, we can assume the closure of each component of $\mathcal{S}_{\overline{m}}\cap\mathcal{N}$ intersects $\mathcal{C}(\Gamma)$ , for all the strata $\mathcal{S}_{\overline{m}}$ . For each fixed stratum $\mathcal{S}_{\overline{m}} \neq \mathcal{C}(\Gamma)$ and two decorated curves $(C_t, l_t)$ and $(C_{t'}, l_{t'})$ that lie in the same connected component $\mathcal{S}_{\overline{m}}\cap \mathcal{N}$ , the Milnor fillings that correspond to $(C_t, l_t)$ and $(C_{t'}, l_{t'})$ are diffeomorphic by a local triviality argument as above. Now, we observe that for each $S_{\overline{m}}$ there are only finitely many connected components of $S_{\overline{m}} \cap \mathcal{N}$ . Indeed, since every ${\mathcal S}_ {\overline{m}}$ is quasiprojective, it has finitely many irreducible (or connected) components. Assume that ${\mathcal S}_ {\overline{m}}\not= {\mathcal C}(\Gamma)$ . Let ${\mathcal S}'_ {\overline{m}}$ be one of the irreducible components of ${\mathcal S}_ {\overline{m}}$ . Write $X= \overline {{\mathcal S}'_ {\overline{m}}} \cap {\mathcal C}(\Gamma) $ . It is a quasiprojective space; let $X=\bigcup_i X_i$ be its irreducible decomposition (which is finite). Set $ \dim X_i= d_i$ and intersect $X_i$ with a generic $d_i$ -codimensional linear germ (cut) $L_i$ and write $L_i\cap X_i=p_i$ . Then, the union of germs $ (\overline {{\mathcal S}'_ {\overline{m}}}\cap L_i,p_i) $ intersects each irreducible component of $ {\mathcal S}_ {\overline{m}}\cap {\mathcal N} $ (for ${\mathcal N}$ small). But each germ $ (\overline {{\mathcal S}'_ {\overline{m}}}\cap L_i,p_i) $ has finitely many analytic local irreducible components, because the local analytic ring ${\mathcal O}_{{\mathcal C},p_i} $ is Noetherian. Now, since there are only finitely many possible $\overline{m}$ types and therefore finitely many strata $\{S_{\overline{m}}\}_{\overline{m}}$ , we conclude that the number of possible Milnor fibers (up to diffeomorphism) is finite.