On the topology of elliptic singularities

TL;DR

This paper introduces a new elliptic sequence for elliptic normal surface singularities with rational homology sphere links, linking its length to geometric genus and topological invariants like the Seiberg--Witten invariant.

Contribution

It proposes a novel elliptic sequence that differs from previous ones but shares the same length, connecting it to key topological and geometric invariants.

Findings

The new elliptic sequence's length matches that of Laufer and Yau's sequence.

The sequence's length relates to the geometric genus.

The sequence's length correlates with the Seiberg--Witten invariant.

Abstract

For any elliptic normal surface singularity with rational homology sphere link we consider a new elliptic sequence, which differs from the one introduced by Laufer and S. S.-T. Yau. However, we show that their length coincide. Using the properties of both sequences we succeed to connect the common length with the geometric genus and also with several topological invariants, e.g. with the Seiberg--Witten invariant of the link.

Full text

Dedicated to Gert–Martin Greuel

1. Introduction

1.1.

The most important analytic invariant of a complex normal surface singularity $(X,o)$ is its geometric genus $p_{g}$. Even if we fix a topological type — usually identified by the link $M$ of the germ, or by a resolution graph —, and even if we assume that the link $M$ is a rational homology sphere, the geometric genus might vary when we vary the analytic structure. Hence, it is natural to find topological bounds for it. In the literature there are several topological invariants, which are related with $p_{g}$ in this sense.

One of them is ${\mathrm{Path}}^{\uparrow}$, cf. [trieste, NS16, NO17], see subsection LABEL:s:Pathi below. It is a topological upper bound for $p_{g}$, that is, for any analytic structure one has $p_{g}\leq{\mathrm{Path}}^{\uparrow}$. However, usually it is hard to verify whether the inequality is optimal or not for a certain topological type, that is, whether a special analytic structure realizes the equality. (One knows topological types when the inequality is not sharp, see Example LABEL:ex:pGpathno.)

Another topological invariant is the (modified) Seiberg–Witten invariant $\overline{{\mathfrak{sw}}}_{0}(M))$ of the link (associated with the canonical $spin^{c}$–structure). It is related with the geometric genus via the Seiberg–Witten Invariant Conjecture (SWIC) $p_{g}(X,o)=\overline{{\mathfrak{sw}}}_{0}(M))$, cf. [trieste, NCL, NO08, NOk, NS16, NWCasson], which is expected to be true for certain special analytic structures. But, again, the verification of this identity usually is hard (and in some cases it is not even true).

In the case of elliptic singularities there is another topological numerical invariant, the length of the elliptic sequence introduced by Laufer and S. S.-T. Yau [Yau5, Yau1]. In the numerically Gorenstein case (when a Gorenstein structure exist) it is easier to connect with $p_{g}$ and ${\mathrm{Path}}^{\uparrow}$, however in the general case the Yau’s elliptic sequence is rather complicated (and it is also hard to connect with possible analytic realizations).

In order to eliminate these difficulties, we consider a new elliptic sequence, which in the non–numerically Gorenstein case is different than the one studied by Yau, and which fits much better in such comparisons. It was motivated (and introduced) in the author’s study of the Abel map of surface singularities [NNIII], and it has several advantages compared with the earlier approaches. E.g., it identifies the support of a numerically Gorenstein subgraph with the following property. If the analytic type supported on this subgraph is Gorenstein, that $p_{g}(X,o)$ is maximal, and it satisfies the identity $p_{g}(X,o)={\mathrm{Path}}^{\uparrow}$ (and the statement (3) from below).

In this note first we prove that the length $\ell+1$ of the Yau’s elliptic sequence coincides with the length $m+1$ of our elliptic sequence. Then using properties of both sequences we prove the following statements for any elliptic germ with rational homology sphere link:

(1) $m+1=\ell+1$;

(2) $m+1={\mathrm{Path}}^{\uparrow}$;

(3) there exists an analytic structure (characterized precisely) supported on the fixed elliptic topological type such that $p_{g}=m+1$;

(4) $m+1=\overline{{\mathfrak{sw}}}_{0}(M))$, in particular, for any analytic structure from (2) the SWIC holds.

Strictly speaking, in the proof of (2) we use an additional assumption, namely that the minimal resolution is good. The main reason for this assumption is that the elliptic sequences are defined (and have nice properties) in the minimal resolution, while the invariant ${\mathrm{Path}}^{\uparrow}$ is defined in via good resolutions. We expect that the statement remains valid in any case, but in this note we did not check the compatibility of the two resolutions (the minimal one and the minimal good one) from the point of view of these two set of invariants (and we didn’t carry out the pathological cases either).

1.2.

The structure of the article is the following. In section 2 we review the standard notations related with resolution of normal surface singularities, and we recall some facts regarding ${\mathrm{Path}}^{\uparrow}$. In the next section we discuss elliptic singularities (we always assume that the link is a rational homology sphere). We recall the definition of the elliptic sequence according to Yau, we establish several properties which will be needed later. Then we discuss the special case of numerically Gorenstein graphs, and finally we provide the definition and several properties of the ‘new’ elliptic sequence. Finally in Theorem LABEL:th:main we prove (1) and (2).

Section LABEL:s:surg reviews several results regarding surgery properties of the Seiberg–Witten invariants (based on some coefficient counting of the topological Poincaré series), and in the last section we prove (3) via such a surgery formula.

2. Preliminaries and notations

2.1. Notations regarding a resolution

[Nfive, trieste, NCL, LPhd, NN1] Let $(X,o)$ be the germ of a complex analytic normal surface singularity. We denote by $p_{g}$ the geometric genus of $(X,o)$. We will assume that the link $M$ of $(X,o)$ is a rational homology sphere.

Let $\phi:\widetilde{X}\to X$ be a resolution of $(X,o)$ with exceptional curve $E:=\phi^{-1}(0)$, and let $\cup_{v\in{\mathcal{V}}}E_{v}$ be the irreducible decomposition of $E$.

$L:=H_{2}(\widetilde{X},\mathbb{Z})$, endowed with a negative definite intersection form $(\,,\,)$, is a lattice. It is freely generated by the classes of $\{E_{v}\}_{v\in\mathcal{V}}$. The dual lattice is $L^{\prime}={\rm Hom}_{\mathbb{Z}}(L,\mathbb{Z})=\{l^{\prime}\in L\otimes\mathbb{Q}\,:\,(l^{\prime},L)\in\mathbb{Z}\}$. It is generated by the (anti)dual classes $\{E^{*}_{v}\}_{v\in\mathcal{V}}$ defined by $(E^{*}_{v},E_{w})=-\delta_{vw}$ (where $\delta_{vw}$ stays for the Kronecker symbol). $L^{\prime}$ is also identified with $H^{2}(\widetilde{X},\mathbb{Z})$.

All the $E_{v}$–coordinates of any $E^{*}_{u}$ are strict positive. We define the Lipman cone as {\mathcal{S}}^{\prime}:=\{l^{\prime}\in L^{\prime}\,:\,(l^{\prime},E_{v})\leq 0\ \mbox{for all v}\}. As a monoid it is generated over ${\mathbb{Z}}_{\geq 0}$ by $\{E^{*}_{v}\}_{v}$. Write also ${\mathcal{S}}:={\mathcal{S}}^{\prime}\cap L$.

$L$ embeds into $L^{\prime}$ with $L^{\prime}/L\simeq H_{1}(M,\mathbb{Z})$, which is abridged by $H$. The class of $l^{\prime}$ in $H$ is denoted by $[l^{\prime}]$.

There is a natural (partial) ordering of $L^{\prime}$ and $L$: we write $l_{1}^{\prime}\geq l_{2}^{\prime}$ if $l_{1}^{\prime}-l_{2}^{\prime}=\sum_{v}r_{v}E_{v}$ with all $r_{v}\geq 0$. We set $L_{\geq 0}=\{l\in L\,:\,l\geq 0\}$ and $L_{>0}=L_{\geq 0}\setminus\{0\}$.

The support of a cycle $l=\sum n_{v}E_{v}$ is defined as $|l|=\cup_{n_{v}\not=0}E_{v}$.

Since $H_{1}(M,\mathbb{Q})=0$, each $E_{v}$ is rational, and the dual graph of any good resolution is a tree.

2.1.1**.**

Minimal cycles in $L^{\prime}_{\geq 0}$ and in ${\mathcal{S}}^{\prime}$. Consider the semi-open cube $\{\sum_{v}l^{\prime}_{v}E_{v}\in L^{\prime}\ |\ 0\leq l^{\prime}_{v}<1\}$. It contains a unique representative $r_{h}$ for every $h\in H$ so that $[r_{h}]=h$. Similarly, for any $h\in H$ there is a unique minimal element of $\{l^{\prime}\in L^{\prime}\ |\ [l^{\prime}]=h\}\cap\mathcal{S}^{\prime}$, which will be denoted by $s_{h}$ (cf. Lemma LABEL:lem:cs2 below). One has $s_{h}\geq r_{h}$; in general, $s_{h}\neq r_{h}$.