Lattice cohomology and subspace arrangements: the topological and analytic cases

Abstract

In this paper we consider the (topological) lattice cohomology $\mathbb {H}^*$ of a surface singularity with rational homology sphere link. In particular, we will be studying two sets of (topological) invariants related to it: the weight function $\upchi$ that induces the cohomology and the topological subspace arrangement $T(\ell ,I)$ at each lattice point $\ell$ — the latter of which is the weaker of the two. We shall prove that the two are in fact equivalent by establishing an algorithm to compute $\upchi$ from the subspace arrangement. Replacing the topological arrangements with the analytic, we get another formula — one that connects them with the analytic lattice cohomology introduced and studied in our earlier papers. In fact, this connection served as the original motivation for the definition of the latter. Aside from the historical interest, this parallel also provides us with tools to study more easily the connection between the two cohomologies.

1 Introduction

In this paper we show the equivalence of two topological, and analogously a pair of analytic invariants of a normal surface singularity.

1.1. The lattice cohomology of a singularity was originally defined for a complex normal surface singularity (X, o), whenever the link is a rational homology sphere. The link of such a singularity is a plumbed (graph) 3-manifold, where the plumbing graph can be chosen as one of the resolution graphs of the germ. In particular, it is connected and negative definite. In such a situation, using the combinatorics of the graph, a lattice cohomology $\mathbb {H}^*_\textrm{top}(X,o)$ was constructed, with an extra grading (the spin$^c$-structures of the link), with the Euler characteristic of the cohomology being the Seiberg–Witten invariant of the link [12].

In order to define a lattice cohomology, one usually needs a lattice $\mathbb {Z}^s$, or more generally a CW complex $\mathfrak {X}$, and a suitable integer-valued weight function on the set of cells (see Sect. 2.1). In different geometric situations the lattice itself appears rather naturally, however the weight function can be hidden in the geometrical structure.

In the present scenario, the lattice is generated by the vertices of the graph (i.e., integral divisors supported on the exceptional curve) and the weight function is a Riemann–Roch expression.

It was verified that the output $\mathbb {H}^*_{\textrm{top}}(X,o)$ is independent of the resolution [12, 14]. In [12] it was conjectured (and in some cases verified) and in [16] proved that it coincides with the Heegaard Floer cohomology of the link. For some more connections with topology (e.g., applications of the graded roots) see [5,6,7,8].

The analytic version $\mathbb {H}^*_{\textrm{an}}(X,o)$ of the lattice cohomology associated with a normal surface singularity was defined in [1, 2], and even a graded $\mathbb {Z}[U]$-module morphism $\mathbb {H}^*_{\textrm{an}}(X,o)\rightarrow \mathbb {H}^*_\textrm{top}(X,o)$ was provided. In this analytic case the Euler characteristic equals the geometric genus of the germ.

Later, in [3], for any n-dimensional complex isolated singularity $(n\ge 2)$ an analytic lattice cohomology was constructed. In this higher dimensional case its topological pair is (still) missing (conjecturally it is the Embedded Contact Homology of the link).

In any dimension $n\ge 2$ the construction of $\mathbb {H}^*_{\textrm{an}}$ is based on the multivariable divisorial filtration provided by a resolution: the lattice is the free $\mathbb {Z}$-module generated by the exceptional divisors, and the weight function was given by some sheaf-cohomology ranks.

1.2. The other invariants we consider are the topological and analytic subspace arrangements, as defined in chapter 8.3 of [14], and their basic properties summarized here in Sect. 2.2. We associate a system of vector spaces $\bigl (T(\ell ,I)\bigr )_{I\subseteq \mathcal {V}}$ and $\bigl (A(\ell ,I)\bigr )_{I\subseteq \mathcal {V}}$ to each lattice point $\ell \in L$ — or more generally, to each point in the dual lattice $L'\subset L\otimes \mathbb {Q}$ — where the systems are containment reversing with respect to subsets $I\subseteq \mathcal {V}$.

The codimensions in these arrangements can be computed directly from the weight functions for the topological and analytic lattice cohomologies, respectively. We show that the reverse is also possible, i.e., it is possible to reconstruct the weight functions given the codimensions, and that the means of doing that are directly analogous in the topological and analytic cases. This helps to further understand how the lattice cohomology $\mathbb {H}^*_{\textrm{an}}$ fits into a set of known topological and analytic invariants in position corresponding to $\mathbb {H}^*_{\textrm{top}}$.

As a historical note, it was this connection between the topological arrangements and the topological weight function that first inspired the definition of the weight function of $\mathbb {H}^*_{\textrm{an}}$ as seen in [1], leading to further generalizations down the line.

2 Preliminary

2.1 The lattice cohomology associated with a system of weights. [12]

The construction of the lattice cohomology associates a graded $\mathbb {Z}[U]$-module to a CW complex $\mathfrak {X}$ endowed with a set of weights. In the original definition as seen in [12,13,14], $\mathfrak {X}$ is a cubical complex generated by a free $\mathbb {Z}$-module, i.e., a lattice (or a subcomplex of it). For the more general construction, see [4] (though in the present paper we are only dealing with the original version).

Now let us summarize the definitions, first setting some notations regarding $\mathbb {Z}[U]$-modules.

Notation 2.1

Let $\mathfrak {X}$ be a CW complex (for definitions and properties see, e.g., [15]). Let $\left\{ \mathop {\textrm{sk}}\nolimits _q\mathfrak {X}\right\} _{q\ge 0}$ be the skeleton decomposition of $\mathfrak {X}$. The q-dimensional cells of $\mathfrak {X}$, i.e., the images of the characteristic maps $\kappa _{q,\alpha }:D^q\rightarrow \mathfrak {X}$, constitute the set $\mathcal {Q}_q=\mathcal {Q}_q(\mathfrak {X})$, and this forms a basis in $\mathscr {C}_q=\mathscr {C}_q(\mathfrak {X})=\mathbb {Z}\left\langle \mathcal {Q}_q\right\rangle$, the free $\mathbb {Z}$-module generated by them.

The elements of $\mathcal {Q}_q$ will also be referred to as closed cells in $\mathfrak {X}$ to differentiate them from their relative interiors $\mathord {\square }^\circ _q:= \mathord {\square }_q{\setminus }\mathop {\textrm{sk}}\nolimits _{q-1}\mathfrak {X}$, which we call the open cells.

Using the above setting, in order to define an ‘interesting’ cohomology theory, we consider a set of compatible weight functions $\left\{ w_q\right\} _q$.

Definition 2.2

A set of functions $w_q:\mathcal {Q}_q\rightarrow \mathbb {Z}$ is called a set of compatible weight functions if

• (a) $w_0$ is bounded from below;

• (b) for any $\mathord {\square }_q\in \mathcal {Q}_q$ and any point $p\in \mathord {\square }_q{\setminus } \mathord {\square }^\circ _q$, consider $r<q$ such that $p\in \mathop {\textrm{sk}}\nolimits _r\mathfrak {X}\setminus \mathop {\textrm{sk}}\nolimits _{r-1}\mathfrak {X}$, with $\mathord {\square }^\circ _r$ being the unique open cell of $\mathop {\textrm{sk}}\nolimits _r\mathfrak {X}$ with $p\in \mathord {\square }_r^\circ$ — that is, $\mathord {\square }_r$ is a cell on the boundary of $\mathord {\square }_q$. Then in any such case we require $w_q(\mathord {\square }_q)\ge w_{r}(\mathord {\square }_{r})$.

The index q of $w_q$ may be omitted henceforth if it causes no confusion, i.e., we set $w=\cup _{q} w_q$. Such a pair $(\mathfrak {X},w)$ is called a weighted CW complex.

Remark 2.3

Given a function $w_0:\mathcal {Q}_0\rightarrow \mathbb {Z}$ that is bounded from below, a compatible choice for all $w_q$ ($q>0$) can always be obtained as

$$\begin{aligned} w_q(\mathord {\square }_q) = \max _{p\in \mathord {\square }_q\cap \mathcal {Q}_0} w_0(p). \end{aligned}$$

To a CW complex $\mathfrak {X}$ and compatible weight function w, we associate the lattice cohomology $\mathbb {H}^*(\mathfrak {X},w)$ (or when the pair $(\mathfrak {X},w)$ is clear from the context, $\mathbb {H}^*$ for short). Here we summarize a geometric realization of this cohomology. For a more complete picture and another construction as the homology of a cochain complex, see [4, 12].

Definition 2.4

For each $n\in \mathbb {Z}$ define $S_n=S_n(w)\subseteq \mathfrak {X}$ as the union of all the closed cells $\mathord {\square }_q$ (of any dimension) with $w(\mathord {\square }_q)\le n$. We have $S_n=\emptyset$, whenever $n<m_w=\min w_0$. For any $q\ge 0$, set

$$\begin{aligned} \mathbb {H}^q(\mathfrak {X},w):=\bigoplus _{n\ge m_w}\, H^q(S_n,\mathbb {Z}). \end{aligned}$$

Then $\mathbb {H}^q$ is $\mathbb {Z}$ (in fact, conventionally $2\mathbb {Z}$)-graded: the $d=2n$-homogeneous elements $\mathbb {H}^q_d$ consist of $H^q(S_n,\mathbb {Z})$.

Furthermore, $\mathbb {H}^q$ has a $\mathbb {Z}[U]$-module structure: the U-action is given by the restriction map $r_{n+1}:H^q(S_{n+1},\mathbb {Z})\rightarrow H^q(S_n,\mathbb {Z})$. Namely, $U*(\alpha _n)_n=(r_{n+1}\alpha _{n+1})_n$. Thus, we obtain a doubly graded $\mathbb {Z}[U]$-module

$$\begin{aligned} \mathbb {H}^*(\mathfrak {X},w) = \bigoplus _{q\ge 0}\mathbb {H}^q(\mathfrak {X},w). \end{aligned}$$

Moreover, for $q=0$, the choice of a fixed base point $x_w\in S_{m_w}\subseteq S_n$ provides an augmentation (splitting) $H^0(S_n,\mathbb {Z})=\mathbb {Z}\oplus \tilde{H}^0(S_n,\mathbb {Z})$, hence an augmentation of the graded $\mathbb {Z}[U]$-modules

$$\begin{aligned} \mathbb {H}^0= \Bigl (\bigoplus _{n\ge m_w}\mathbb {Z}\Bigr )\oplus \Bigl (\bigoplus _{n\ge m_w}\tilde{H}^0(S_n,\mathbb {Z})\Bigr )= \Bigl (\bigoplus _{n\ge m_w}\mathbb {Z}\Bigr )\oplus \mathbb {H}^0_{\textrm{red}}. \end{aligned}$$

In this paper we will be dealing with the “classical” case of this construction where, as mentioned at the beginning, we have the lattice $L = \mathbb {Z}^{\mathcal {V}}$ with the canonical basis denoted by $\left\{ E_v\right\} _{v\in \mathcal {V}}$, and $\mathfrak {X}$ is $L\otimes \mathbb {R}= \mathbb {R}^\mathcal {V}$ with the cubical decomposition. Furthermore, we will always assume that the higher-dimensional weights are chosen as specified in 2.3, so we need only define suitable weights on the lattice points. This particular set of notations comes from the context where lattice cohomology was first introduced, that of normal surface singularities (see [12]).

More concretely, given a normal surface singularity (X, o) with good resolution ${\Phi {:}}(\tilde{X},E)\rightarrow (X,o)$, let $\Gamma =(\mathcal {V},\mathcal {E})$ be the resolution graph, $L=H_2(\tilde{X})\simeq \mathbb {Z}^{\left| \mathcal {V}\right| }$ a lattice together with the intersection form — the adjacence matrix of $\Gamma$ with the respective Euler numbers written in the diagonal. Also let $L'=H_2(\tilde{X},\partial \!\mathop {}\tilde{X})\subset L\otimes \mathbb {Q}$ be the dual lattice (with respect to the inner product). $\ell$ and $\ell '$ will often denote elements of L and $L'$, respectively. We also introduce the following notation:

Notation 2.5

For elements $x,y\in L\otimes \mathbb {R}\simeq \mathbb {R}^s$ with $x\le y$, let R(x, y) denote the (hyper)rectangle $\left\{ z\in L\otimes \mathbb {R}\big \vert x\le z\le y\right\}$.

Then we will consider the topological and analytic lattice cohomology as defined in [12] and [1], respectively. We also assume from this point on that the singularity (X, o) has a rational homology sphere link.

Definition 2.6

The topological lattice cohomology $\mathbb {H}^*_{\textrm{top}}$ is induced by the weight function

$$\begin{aligned} w_{\textrm{top}}:L\rightarrow \mathbb {Z}, \qquad w_{\textrm{top}}(\ell ) = \upchi (x) = -\frac{1}{2}(x,x+k), \end{aligned}$$

where k is the canonical class. (By the adjunction formula, this is topological.)

Note that the function $\upchi :L\otimes \mathbb {R}\rightarrow \mathbb {R}$ is symmetric with respect to $\frac{1}{2}Z_K$ where $Z_K=-k$, so for numerically Gorenstein (X, o) (i.e., when $k\in L$), $w_{\textrm{top}}$ itself is symmetric as well.

Remark 2.7

In general, we actually have a set of weight functions corresponding to each torsion $\textrm{spin}^c$-structure on the link, where in the above definition, we replace the canonical class k with other elements of

$$\begin{aligned} \textrm{Char} = \left\{ \ell '\in L'\big \vert \forall \ell \in L: (\ell '+\ell ,\ell )\in 2\mathbb {Z}\right\} . \end{aligned}$$

The induced cohomology $\mathbb {H}^*$ only depends on the image of $k\in \textrm{Char}$ in $L'/L$: different choices from the same coset yield weight functions that are shifted copies of each other. In particular, in the case of an integral homology sphere link, we only have $\mathbb {H}^*_{\mathrm{top,\,can}}$. In any case, in the following discussion we only consider the above defined, canonical weight function.

Definition 2.8

The analytic lattice cohomology $\mathbb {H}^*_{\textrm{an}}$ is defined using the weight function

$$\begin{aligned} w_{\textrm{an}}:L_{\ge 0}\rightarrow \mathbb {Z}, \qquad w_{\textrm{an}}(\ell ) = \mathfrak {h}(\ell ) - \mathfrak {h}^1(\mathscr {O}_{\ell }) \end{aligned}$$

with $\mathfrak {h}(\ell ) = \dim H^0(\mathscr {O}_{\tilde{X}}) / H^0(\mathscr {O}_{\tilde{X}}(-\ell ))$ being the Hilbert function of the divisorial filtration.

In particular, whenever (X, o) is Gorenstein, we have

$$\begin{aligned} w_{\textrm{an}}(\ell ) = \mathfrak {h}(\ell ) + \mathfrak {h}(Z_K-\ell ) - \mathfrak {h}(Z_K) \qquad (0\le \ell \le Z_K) \end{aligned}$$

due to duality, so $w_{\textrm{an}}|_{L\cap R(0,Z_K)}$ is symmetric with respect to $\frac{1}{2}Z_K$.

It is further notable that (cf. Lemma 4.2.4 in [1] and Corollary 3.3.5, as well as the proof of Theorem 3.2.4 in [12]) the following holds:

Proposition 2.9

Both for the case of $w_{\textrm{top}}:L\rightarrow \mathbb {Z}$ and $w_\textrm{an}:L_{\ge 0}\rightarrow \mathbb {Z}$, the induced lattice cohomologies $\mathbb {H}^*_{\textrm{top}}$ and $\mathbb {H}^*_{\textrm{an}}$ are isomorphic to that obtained from restricting the respective weight functions to the rectangle $R(0,\left\lfloor Z_K\right\rfloor )$ as a sub-CW-complex.

In particular, for (X, o) Gorenstein, both $\mathbb {H}^*_{\textrm{top}}$ and $\mathbb {H}^*_{\textrm{an}}$ can be obtained from the cubical complex defined on $R(0,Z_K)$, and the (symmetric) weight functions $w_\textrm{top}|_{L\cap R(0,Z_K)}$ and $w_{\textrm{an}}|_{L\cap R(0,Z_K)}$.

We will show that these two share an analogous connection to another set of topological / analytic invariants.

2.2 Subspace arrangements. [14]

Using notations introduced above, for any $I\subseteq \mathcal {V}$, let $\displaystyle E_I=\sum _{v\in I}E_v$ (in particular, $E=E_{\mathcal {V}}$).

Recall that the topological and analytic subspace arrangements are defined in the following way in section 8.3. of [14]. Below, for any $\ell '\in L'$, $\mathscr {O}_{\tilde{X}}(-\ell ')$ denotes the corresponding ‘natural line bundle’ (cf. section 6.2 of [14]), as a generalization of the usual line bundles $\mathscr {O}_{\tilde{X}}(-\ell )$ defined for integral divisors $\ell \in L$. However, in this paper we will solely use the case $\ell \in L$ (see also Remark 2.19). Hence, the reader not familiar with the natural line bundles can consider this case only.

Definition 2.10

For $\ell '\in L'$ the exact sequence

$$\begin{aligned} 0\rightarrow \mathscr {O}_{\tilde{X}}(-\ell '-E)\rightarrow \mathscr {O}_{\tilde{X}}(-\ell ')\rightarrow \mathscr {O}_E(-\ell ')\rightarrow 0 \end{aligned}$$

induces via the homology exact sequence a map

Then for $I\subseteq \mathcal {V}$ we define

Here, by the identification

$$\begin{aligned} T(\ell ',I)=\mathop {\textrm{Ker}}\bigl (H^0(\mathscr {O}_E(-\ell '))\longrightarrow H^0(\mathscr {O}_{E_I}(-\ell '))\bigr )=H^0(\mathscr {O}_{E-E_I}(-\ell '-E_I)), \end{aligned}$$

we have the subspaces $T(\ell ',I)\subseteq T(\ell ',\emptyset )$ satisfying

$$\begin{aligned} A(\ell ',I)=A(\ell ',\emptyset )\cap H^0(\mathscr {O}_{E-E_I}(-\ell '-E_I))=A(\ell ',\emptyset )\cap T(\ell ',I). \end{aligned}$$

For any $I\subseteq \mathcal {V}$ we get

$$\begin{aligned} T(\ell ',I)=\bigcap _{v\in I}T(\ell ',\left\{ v\right\} ), \quad A(\ell ',I)=\bigcap _{v\in I}A(\ell ',\left\{ v\right\} ). \end{aligned}$$

In particular, for $I_1\subseteq I_2\subseteq \mathcal {V}$ we have $T(\ell ',I_1)\supseteq T(\ell ',I_2)$ and $A(\ell ',I_1)\supseteq A(\ell ',I_2)$.

We call $\displaystyle \bigl (T(\ell ',I)\bigr )_{I\subseteq \mathcal {V}}$ the topological, $\displaystyle \bigl (A(\ell ',I)\bigr )_{I\subseteq \mathcal {V}}$ the analytic subspace arrangement at $\ell '$. Note that these are finite-dimensional vector spaces.

We will be using the topological arrangement $T(\ell ',I)$ for now, but the existence of a direct analytic counterpart provided the main motivation for the presently discussed results.

In order to describe these subspace arrangements, we can use the values below:

Notation 2.11

For $I_1\subset I_2\subseteq \mathcal {V}$ write

$$\begin{aligned} \mathscr {C}^{T}_{\ell '}(I_1,I_2)&=\mathop {\textrm{codim}}\bigl (T(\ell ',I_2)\hookrightarrow T(\ell ',I_1)\bigr ), \\ \mathscr {C}^{A}_{\ell '}(I_1,I_2)&=\mathop {\textrm{codim}}\bigl (A(\ell ',I_2)\hookrightarrow A(\ell ',I_1)\bigr ). \end{aligned}$$

Remark 2.12

We often conceptualize this as assigning numbers — given by either $\mathscr {C}^{T}_{\ell '}$ or $\mathscr {C}^{A}_{\ell '}$ — to each edge $R(\ell '+E_I,\ell '+E_{I\cup \left\{ v\right\} })$ within the cube $R(\ell ',\ell '+E)$. (The codimensions for any subsets $I_1\subseteq I_2\subseteq \mathcal {V}$ can then be obtained from these via addition.)

As for understanding what these values actually are, we have

Proposition 2.13

$$\begin{aligned}{} & {} \mathscr {C}^{A}_{\ell '}(I_1,I_2)= \mathop {\textrm{codim}}\bigl (A(\ell ',I_2)\hookrightarrow A(\ell ',I_1)\bigr )\\{} & {} \quad = \mathop {\textrm{codim}}\bigl (H^0(\mathcal {O}_{\tilde{X}}(-\ell '-E_{I_2}))\hookrightarrow H^0(\mathcal {O}_{\tilde{X}}(-\ell '-E_{I_1}))\bigr ). \end{aligned}$$

Getting the topological codimensions requires a bit of preparation though. Namely, let us use the following definition from Lemma 8.3.7. in [14]:

Definition 2.14

For $\ell '\in L'$ and $I\subseteq \mathcal {V}$, let $J(\ell ',I)\subseteq \mathcal {V}$ denote the unique minimal subset J of $\mathcal {V}$ for which

$$\begin{aligned} J\supseteq I, \qquad \forall v\in \mathcal {V}\setminus J: (E_v,\ell '+E_{J})\le 0. \end{aligned}$$

(\(*\))

Assuming that such a unique minimum does exist — which we will show in a moment —, one can observe the following:

Lemma 2.15

If $I_1\subseteq I_2\subseteq \mathcal {V}$, then $J(\ell ',I_1)\subseteq J(\ell ',I_2)$.

Proof

We just need to verify that condition ($*$) from 2.14 for $I_1$ is satisfied by $J(\ell ',I_2)$. The first part is clear enough: $J(\ell ',I_2)\supseteq I_2\supseteq I_1$. As for the second part, it does not depend on I at all, whether it is $I_1$ or $I_2$, so it is automatically satisfied by $J(\ell ',I_2)$.

Crucially for us, there is a clear procedure we can use to find this set $J(\ell ',I)$, which also provides a proof for its existence:

Algorithm 2.16

The set $J(\ell ',I)$ can be obtained as follows. We set $J_0=I$, and we construct a sequence $(J_i)_{i=1}^t$ by adding distinct elements from $\mathcal {V}\setminus I$ one by one, for every i: if $J_i$ does not have the property ($*$), i.e., $(E_v,\ell '+E_{J_i})>0$ for some vertex $v\in \mathcal {V}\setminus J_i$, then we add that vertex: $J_{i+1}=J_i\cup \left\{ v\right\}$. We can see that this vertex would need to be added at some point anyway if we are to find a set containing $J_i$ and satisfying ($*$) since the value $(E_v,\ell '+E_J)$ is increasing with respect to $J\subset \mathcal {V}\setminus \left\{ v\right\}$. For some t, there will be no such vertices left (possibly upon reaching $J_t=\mathcal {V}$), at which point the algorithm ends: $J(\ell ',I)=J_t$ satisfies ($*$).

In each step from J to $J\cup \left\{ v\right\}$,

$$\begin{aligned} \upchi (\ell '+E_{J\cup \left\{ v\right\} })-\upchi (\ell '+E_J)=\upchi (E_v)-(E_v,\ell '+E_J)=1-(E_v,\ell '+E_J)\le 0 \end{aligned}$$

is equivalent to the property $(E_v,\ell '+E_J)>0$, and equality holds if and only if $(E_v,\ell '+E_J)=1$.

In other words, the previous procedure can be rephrased as a variant of Laufer’s algorithm: in the cube of lattice points $\ell '+E_J$ we move upwards starting from $E_I$, increasing one of the coordinates by 1 in every step in such a way that $\upchi$ does not increase. Eventually we arrive at a point where along all edges within the cube going upwards, the value of $\upchi$ increases: this will be $\ell '+E_{J(\ell ',I)}$. By the original definition of $J(\ell ',I)$, this point does not depend on the choice of individual steps. We will refer to this procedure as $(L)^{\ell '}_I$ from now on.

By the statement 8.3.11 in the same book,

Proposition 2.17

For any $I\subseteq \mathcal {V}$ we have

$$\begin{aligned} \dim T(\ell ',I) = \dim \bigcap _{v\in I}T(\ell ',\left\{ v\right\} ) = \upchi (\ell '+E) - \upchi (\ell '+E_{J(\ell ',I)}). \end{aligned}$$

Corollary 2.18

For any sets $I_1\subseteq I_2\subseteq \mathcal {V}$

$$\begin{aligned} \mathscr {C}^{T}_{\ell '}(I_1,I_2) = \mathop {\textrm{codim}}\left( T(\ell ',I_2)\hookrightarrow T(\ell ',I_1)\right) = \upchi (\ell '+E_{J(\ell ',I_2)})-\upchi (\ell '+E_{J(\ell ',I_1)}).\nonumber \\ \end{aligned}$$

(2.1)

In particular for $I\subseteq \mathcal {V}$ and $v\in \mathcal {V}{\setminus } I$,

$$\begin{aligned} \mathscr {C}^{T}_{\ell '}(I,I\cup \left\{ v\right\} ){} & {} = \mathop {\textrm{codim}}\left( T(\ell ',I\cup \left\{ v\right\} )\hookrightarrow T(\ell ',I)\right) = \upchi (\ell '+E_{J(\ell ',I\cup \left\{ v\right\} )})\nonumber \\{} & {} \quad -\upchi (\ell '+E_{J(\ell ',I)}). \end{aligned}$$

(2.2)

This connects the codimensions in the subspace arrangement with the function $\upchi$, in particular the former can be computed from the latter. From this, it is still unclear though if the converse is true as well. If it were, then we could consequently deduce the lattice cohomology from the subspace arrangement, and obtain a similar, analytic invariant by passing to the analytical subspace arrangement instead, using formally the very same type of formula. It will turn out that this invariant is, for Gorenstein singularities at least, the same as the analytic lattice cohomology introduced in [1], as we will show in Sect. 4.

Remark 2.19

As in [14], we introduced these arrangements for all $\ell '\in L'$. However, in the present discussion, we only really need it at points of L, not the dual lattice $L'$.

Analogous arguments could be made to connect the weight functions of the lattice cohomologies corresponding to all torsion $\textrm{spin}^c$-structures — as per Remark 2.7 — to the subspace arrangements considered at all points of $L'$, but we will not do so in this paper.

3 Getting back the weight function

Our aim is to show that, at least in some cases, the set of codimensions of topological subspace arrangements and the weight function $\upchi$ are in fact equivalent.

Throughout this section, we will use the following notation:

Notation 3.1

Given a vertex $v\in \mathcal {V}$, let $\mathcal {N}(v)\subset \mathcal {V}$ be the set of vertices adjacent to v in $\Gamma$ (i.e., those with an edge in $\mathcal {E}$ connecting them to v).

A key component for the proof of the equivalence is the following lemma:

Lemma 3.2

If $\ell \in L$, $I\subset \mathcal {V}$, $v\in \mathcal {V}\setminus I$, then the following are true:

• a) If $\upchi (\ell +E_{I\cup \left\{ v\right\} })\le \upchi (\ell +E_I)$ then

  $$\begin{aligned} \upchi (\ell +E_{J(\ell ,I\cup \left\{ v\right\} )})-\upchi (\ell +E_{J(\ell ,I)})=0. \end{aligned}$$

• b) If $\upchi (\ell +E_{I\cup \left\{ v\right\} })>\upchi (\ell +E_I)$ then

  $$\begin{aligned} \upchi (\ell +E_{J(\ell ,I\cup \left\{ v\right\} )})-\upchi (\ell +E_{J(\ell ,I)}){} & {} = \max \Bigl (0,\left( \upchi (\ell +E_{I\cup \left\{ v\right\} })-\upchi (\ell +E_I)\right) \\{} & {} \quad -\left| (J(\ell ,I)\setminus I)\cap \mathcal {N}(v)\right| \Bigr ). \end{aligned}$$

Essentially, we consider the values of $\upchi$ at two adjacent lattice points, i.e., the endpoints of an edge $R(\ell +E_I,\ell +E_{I\cup \left\{ v\right\} })$ within the cube $R(\ell ,\ell +E)$: along the edge connecting them, $\upchi$ may either decrease, stay the same, or increase. Then, upon running Algorithm 2.16 from these starting points within the cube $(\ell ,\ell +E)$, we end up with two points $\ell +E_{J(\ell ,I)}$ and $\ell +E_{J(\ell ,I\cup \left\{ v\right\} )}$ (which are no longer necessarily adjacent of course, nor even distinct, but the latter is still greater than or equal to the former). We compare the values of $\upchi$ at these two pairs of points: more specifically we look at the differences

$$\begin{aligned} d&= \upchi (\ell +E_{I\cup \left\{ v\right\} })-\upchi (\ell +E_I), \\ d'&= \upchi (\ell +E_{J(\ell ,I\cup \left\{ v\right\} )})-\upchi (\ell +E_{J(\ell ,I)}) = \mathscr {C}^{T}_\ell (I,I\cup \left\{ v\right\} ), \end{aligned}$$

and we determine $d'$ from d.

If $\upchi$ was nonincreasing along the original edge ($d\le 0$) then the first choice in Algorithm 2.16 (as applied for this $\ell$ and I) can be v, thus in fact $J(\ell ,I\cup \left\{ v\right\} )=J(\ell ,I)$, and $d'=0$. If, on the other hand, $\upchi$ was increasing ($d>0$) then it turns out that $d'$ is likewise nonnegative, and often positive, but may be no bigger than d — with the exact value depending on how many vertices $w\in \mathcal {V}$ adjacent to v are added when we perform Algorithm 2.16 starting from I to get $J(\ell ,I)$.

Before proceeding to the proof, let us see how this statement can be used. We are going to make use of the symmetry of $\upchi$ with respect to $\frac{1}{2}Z_K$ mentioned in 2.6. For that purpose, we introduce the following notation:

Notation 3.3

For a point $x\in \mathbb {R}^{\left| \mathcal {V}\right| }$, let

$$\begin{aligned} x^*=Z_K-E-x. \end{aligned}$$

Additionally, for a set $I\subset \mathcal {V}$, let $I^*=\mathcal {V}\setminus I$.

Remark 3.4

Note that with this choice, the cubes $R(x,x+E)$ and $R(x^*,x^*+E)$ are mirror images of each other with respect to $\frac{1}{2}Z_K$. Furthermore, for all x and I we have

$$\begin{aligned} (x^*+E_{I^*}) = Z_K-(x+E_I), \end{aligned}$$

in particular

$$\begin{aligned} \upchi (x+E_I) = \upchi (x^*+E_{I^*}). \end{aligned}$$

Now we will consider a cube $R(\ell ,\ell +E)$ in the lattice and the topological codimension $\mathscr {C}^{T}_\ell$ along one of its edges — as well as its symmetric counterpart in $R(\ell ^*,\ell ^*+E)$. The idea behind this is that while 3.2 computes the codimension $\mathscr {C}^{T}_\ell (I,I\cup \left\{ v\right\} )$ from $\upchi (\ell +E_{I\cup \left\{ v\right\} })-\upchi (\ell +E_I)$, we would want to be able to do the converse: obtaining $\upchi (\ell +E_{I\cup \left\{ v\right\} })-\upchi (\ell +E_I)$ from $\mathscr {C}^{T}_\ell (I,I\cup \left\{ v\right\} )$, or at least from the set of such codimensions in the topological subspace arrangement.

But if the former is negative then, by 3.2, $\mathscr {C}^{T}_\ell (I,I\cup \left\{ v\right\} )$ provides no further information at all, it is always 0. We can, however, look at $\upchi (\ell ^*+E_{I^*})-\upchi (\ell ^*+E_{(I\cup \left\{ v\right\} )^*})$ instead then — which is its opposite — and apply the Lemma there.

Unfortunately, $R(\ell ^*,\ell ^*+E)$ is not always actually a cell in the cubical decomposition associated with the lattice L, so for now, we assume that $Z_K\in L=\mathbb {Z}^{\left| \mathcal {V}\right| }$, i.e., that the singularity is numerically Gorenstein — in which case, we do get an actual correspondence between cubes defined by the lattice $L=\mathbb {Z}^{\left| \mathcal {V}\right| }$.

Theorem 3.5

Assume the singularity is numerically Gorenstein. Let $\ell \in L$, $I\subset \mathcal {V}$, and $v\in \mathcal {V}\setminus I$. Using the notation $I_+=I\cup \left\{ v\right\}$ and keeping in mind Corrollary 2.18, consider the following expressions:

$$\begin{aligned} C&=\upchi (\ell +E_{I_+})-\upchi (\ell +E_I), \\ D_1&=\mathscr {C}^{T}_\ell (I,I_+)=\upchi (\ell +E_{J(\ell ,I_+)})-\upchi (\ell +E_{J(\ell ,I)}), \\ D_2&=\mathscr {C}^{T}_{\ell ^*}(I_+^*,I^*)=\upchi (\ell ^*+E_{J(\ell ^*,I^*)})-\upchi (\ell ^*+E_{J(\ell ^*,I_+^*)}), \\ D&=D_1-D_2=\mathscr {C}^{T}_\ell (I,I_+)-\mathscr {C}^{T}_{\ell ^*}(I_+^*,I^*). \end{aligned}$$

(Case 1):

If $C=0$ then $D=0$.

(Case 2):

If $C>0$ then

$$\begin{aligned} 0\le D=\max \Bigl (0,C-\left| \left( J(\ell ,I)\setminus I\right) \cap \mathcal {N}(v)\right| \Bigr )\le C. \end{aligned}$$

(Case 3):

If $C<0$ then

$$\begin{aligned} C\le D=\min \Bigl (0,C+\left| \left( J(\ell ^*,I_+^*)\setminus I_+^*\right) \cap \mathcal {N}(v)\right| \Bigr )\le 0. \end{aligned}$$

Proof

From 3.4, we get

$$\begin{aligned} \upchi (\ell ^*+E_{I^*})-\upchi (\ell ^*+E_{I_+^*}) =-\left( \upchi (\ell +E_{I_+})-\upchi (\ell +E_{I})\right) =-C. \end{aligned}$$

So for $C=0$, by 3.2, we indeed have $D_1=0$, and $D_2=0$, so $D=0$.

If $C>0$ then again by the lemma

$$\begin{aligned} D_1=\max \Bigl (0,C-\left| \left( J(\ell ,I)\setminus I\right) \cap \mathcal {N}(v)\right| \Bigr ), \end{aligned}$$

and $-C<0$ implies $D_2=0$, so $D=D_1$, and we are done in this case as well.

Finally for $C<0$, i.e., $-C>0$, the lemma gives us $D_1=0$, and

$$\begin{aligned} D_2=\max \Bigl (0,-C-\left| \left( J(\ell ^*,I_+^*)\setminus I_+^*\right) \cap \mathcal {N}(v)\right| \Bigr ), \end{aligned}$$

so

$$\begin{aligned} D=-D_2=\min \Bigl (0,C+\left| \left( J(\ell ^*,I_+^*)\setminus I_+^*\right) \cap \mathcal {N}(v)\right| \Bigr ), \end{aligned}$$

and this is what we needed to show.

Thus D, the difference of the “mirrored codimensions”, is equal to $C=\upchi (\ell +E_{I_+})-\upchi (\ell +E_I)$ plus a correctional term — in particular their sign is always the same (or 0) but the absolute value may be smaller. If from the dataset of all the topological codimensions we could deduce the differences of this type, i.e., $\upchi (x+E_v)-\upchi (x)$ for every $x\in L$ and $v\in \mathcal {V}$ then we would also get $\upchi$ since $\upchi (0)=0$ always.

The extra “correctional term” seems to be an obstacle to that. Note however that we have additional freedom of choice for how to apply 3.5: for a given x and v, we can choose the set $I\subseteq \mathcal {V}\setminus \left\{ v\right\}$ and apply the statement for $\ell =x-E_I$ — in which case $\ell +E_I=x$ and $\ell +E_{I_+}=x+E_v$.

If we could find and identify a choice when the extra term is 0 then we could in fact thus obtain $\upchi$ from the codimensions. This is the point of the next statement, where either S or the complement of $S\cup \left\{ v\right\}$ will stand for this choice of I:

Corollary 3.6

For a numerically Gorenstein singularity let $x\in L=\mathbb {Z}^{\left| \mathcal {V}\right| }$ and $v\in \mathcal {V}$. Let S be any subset of $\mathcal {V}$ such that $\mathcal {N}(v)\subseteq S\subseteq \mathcal {V}\setminus \left\{ v\right\}$, $S_+=S\cup \left\{ v\right\}$ and

$$\begin{aligned} \ell _{>0}=x-E_S, \qquad \ell _{<0}=x-E_{S_+^*}. \end{aligned}$$

If $\upchi (x+E_v)-\upchi (x)\ge 0$ then

$$\begin{aligned} \upchi (x+E_v)-\upchi (x) =\mathscr {C}^{T}_{\ell _{>0}}(S,S_+)-\mathscr {C}^{T}_{\ell _{>0}^*}\bigl (S_+^*,S^*\bigr ), \end{aligned}$$

while if $\upchi (x+E_v)-\upchi (x)\le 0$ then

$$\begin{aligned} \upchi (x+E_v)-\upchi (x) =\mathscr {C}^{T}_{\ell _{<0}}\bigl (S_+^*,S^*\bigr ) -\mathscr {C}^{T}_{\ell _{<0}^*}(S,S_+). \end{aligned}$$

Proof

In the first case we apply the previous theorem for $\ell =\ell _{>0}$ and $I=S$. Here $S\supseteq \mathcal {N}(v)$ implies

$$\begin{aligned} \left| (J(\ell ,I)\setminus I)\cap \mathcal {N}(v)\right| =\left| (J(\ell _{>0},S)\setminus S)\cap \mathcal {N}(v)\right| =0, \end{aligned}$$

and (with the notation used in 3.5)

$$\begin{aligned} C=\upchi (\ell _{>0}+E_{S_+})-\upchi (\ell _{>0}+E_S)=\upchi (x+E_v)-\upchi (x)\ge 0, \end{aligned}$$

so Case 2 applies and the statement indeed follows.

In the second case we set $\ell =\ell _{<0}$ and $I=S_+^*$ instead to apply the theorem. Then we have

$$\begin{aligned} \left| (J(\ell ^*,I_+^*)\setminus I_+^*)\cap \mathcal {N}(v)\right| =\left| (J(\ell _{<0}^*,S)\setminus S)\cap \mathcal {N}(v)\right| =0, \end{aligned}$$

and

$$\begin{aligned} C=\upchi (\ell _{<0}+E_{S^*})-\upchi (\ell _{<0}+E_{S_+^*})=\upchi (x+E_v)-\upchi (x)\le 0, \end{aligned}$$

so by Case 3 we once again get exactly what we wanted.

Remark 3.7

In particular, the choice $S=\mathcal {V}\setminus {v}$ always works.

To conclude and obtain the final formula for $\upchi (x+E_v)-\upchi (x)$, we introduce one more notation:

Notation 3.8

For a subset $A\subseteq \mathbb {R}$ let $\mathop {\textrm{absmax}}(A)$ denote the element of A with the maximal absolute value, if such an element exists and is unique.

Now applying 3.6 to together with 3.5, we get

Theorem 3.9

For a numerically Gorenstein singularity let $x\in L=\mathbb {Z}^{\left| \mathcal {V}\right| }$ and $v\in \mathcal {V}$. Then

$$\begin{aligned} \upchi (x+E_v)-\upchi (x)&=\mathop {\textrm{absmax}}\limits _{\overset{I\subseteq \mathcal {V}\setminus \left\{ v\right\} }{\ell =x-E_I}}\Bigl ( \mathscr {C}^{T}_\ell \bigl (I,I\cup \left\{ v\right\} \bigr )-\mathscr {C}^{T}_{\ell ^*}\bigl ((I\cup \left\{ v\right\} )^*,I^*\bigr ) \Bigr )= \\&=\mathop {\textrm{absmax}}\Bigl ( \mathscr {C}^{T}_x\bigl (\emptyset ,\left\{ v\right\} \bigr )-\mathscr {C}^{T}_{x^*}\bigl (\mathcal {V}\setminus \left\{ v\right\} ,\mathcal {V}\bigr ), \\&\quad \mathscr {C}^{T}_{x-E+E_v}\bigl (\mathcal {V}\setminus \left\{ v\right\} ,\mathcal {V}\bigr )-\mathscr {C}^{T}_{(x-E+E_v)^*}\bigl (\emptyset ,\left\{ v\right\} \bigr ) \Bigr ). \end{aligned}$$

In particular, we claim that the $\mathop {\textrm{absmax}}$ exists.

Proof

According to 3.5, the expressions within the $\mathop {\textrm{absmax}}$ have the same sign as $\upchi (x+E_v)-\upchi (x)$ so in particular $\mathop {\textrm{absmax}}$ is well defined. Furthermore the absolute value of these expressions is no more than that of $\upchi (x+E_v)-\upchi (x)$. And according to 3.6 this inequality is sharp for some I — and more specifically, as per 3.7, it is so either in the case of $I=\mathcal {V}\setminus \left\{ v\right\}$ or $I=\emptyset$. This gives us both equalities. $\square$

Corollary 3.10

For a numerically Gorenstein singularity with $\mathbb {Q}HS^3$ link the function $\upchi :L\rightarrow \mathbb {Z}$ and therefore the lattice cohomology $\mathbb {H}^*$ can be calculated from the codimensions $\mathscr {C}^{T}_\ell \bigl (I,I\cup \left\{ v\right\} \bigr )$ ($\ell \in L$, $I\subseteq \mathcal {V}$, $v\in \mathcal {V}\setminus I$).

All that is left now is for us to prove the main lemma, introduced at the beginning of this section:

Proof (Lemma 3.2)

Part a) follows from the fact that in the procedure $(L)^{\ell }_I$ — as described in 2.16 — we can just pick v as the first vertex to add, implying $J(\ell ,I\cup \left\{ v\right\} )=J(\ell ,I)$.

To prove b), consider a particular sequence of steps produced by $(L)^{\ell }_I$:

$$\begin{aligned} J_0=I,J_1=J_0\cup \left\{ v_1\right\} ,\ldots ,J_t=J_{t-1}\cup \left\{ v_t\right\} =J(\ell ,I), \end{aligned}$$

where $v_i\not \in J_{i-1}$ for all i. Construct also a “parallel” sequence $J_i'=J_i\cup \left\{ v\right\}$. Now either $J(\ell ,I)\ni v$, i.e., $v_s=v$ for some s, in which case the two sequences are the same after $J_s=J'_s=J'_{s-1}$, or the sequence $(J'_i)$ is likewise obtained via the recursion $J'_i=J'_{i-1}\cup \left\{ v_i\right\}$, adding one element at a time. Whichever case we may be dealing with, we will show that $(J'_i)$ is also a valid Laufer-type computation sequence (or at least the beginning of one), from $I'$ (after removing the one possible duplicate element $J'_s$ in the first case).

For the sake of treating all cases in a unified manner, set s, as above, to be the index for which $v_s=v$ if such exists (i.e., if $v\in J(\ell ,I)$), and $s=t+1$ otherwise — that is,

$$\begin{aligned} J_i'= {\left\{ \begin{array}{ll} J_i\sqcup \left\{ v\right\} ,&{}\text {\ if\ }i<s,\\ J_i,&{}\text {\ if\ }i\ge s. \end{array}\right. } \end{aligned}$$

In particular, $s\le t$ if and only if $J(\ell ,I)=J_t\ni v$.

For keeping track of how $\upchi$ changes along the sequences $(\ell +E_{J_i})$ and $(\ell +E_{J'_i})$, let us introduce the notations

$$\begin{aligned} \Delta _i&=\upchi (\ell +E_{J_i})-\upchi (\ell +E_{J_{i-1}}), \\ \Delta '_i&=\upchi (\ell +E_{J_i'})-\upchi (\ell +E_{J_{i-1}'}). \end{aligned}$$

We have $\Delta _i\le 0$ by the definition of the algorithm $(L)^{\ell }_I$. Also,

$$\begin{aligned} \Delta _i=\upchi (\ell +E_{J_i})-\upchi (\ell +E_{J_{i-1}})=\upchi (E_{v_i})-(E_{v_i},\ell +E_{J_{i-1}})=1-(E_{v_i},\ell +E_{J_{i-1}})\le 0, \end{aligned}$$

and for $i<s$

$$\begin{aligned} \Delta '_i&{=}\upchi (\ell +E_{J_i'}){-}\upchi (\ell {+}E_{J_{i-1}'}) {=}\upchi (E_{v_i}){-}(E_{v_i},\ell {+}E_{J_{i-1}'}) {=}1-(E_{v_i},\ell +E_{J_{i-1}}+E_v) \nonumber \\&=\Delta _i-(E_{v_i},E_v). \end{aligned}$$

(3.1)

So if $i<s$ then

$$\begin{aligned} \Delta _i-1\le \Delta '_i\le \Delta _i\le 0. \end{aligned}$$

In particular, we can conclude that $(J'_i)$ is indeed a valid Laufer-type computation sequence up to $J'_{s-1}$. Also, if $s\le t$ then afterwards it matches the corresponding part of the sequence $(J_i)$, so the elements in the sequence $(J'_i)$ ($0\le i\le t$) do give us a possible start for the procedure $(L)^\ell _{I\cup \left\{ v\right\} }$.

Note that if $s\le t$ then $J'_t=J_t$, meaning the algorithm in 2.16 ends here (since the $(L)^\ell _I$ also did): this is in fact the complete computation sequence, $J(\ell ,I\cup \left\{ v\right\} )=J(\ell ,I)$, and as such

$$\begin{aligned} \upchi (\ell +E_{J(\ell ,I\cup \left\{ v\right\} )})-\upchi (\ell +E_{J(\ell ,I)}) = 0. \end{aligned}$$

(3.2)

Regardless, $J(\ell ,I\cup \left\{ v\right\} )$ can be obtained from $J'_t$ by continuing the procedure until it stops at some $J'_\tau$: for all $t<i\le \tau$, we also have $J'_i=J'_{i-1}\sqcup \left\{ v_i\right\}$, while $\upchi (\ell +E_{J'_i})$ is nonincreasing in i.

As for how this helps us prove the statement, let

$$\begin{aligned} d_i = \upchi (\ell +E_{J_i'})-\upchi (\ell +E_{J_i}). \end{aligned}$$

The statement to prove can then be phrased as follows: if $d_0>0$ then

$$\begin{aligned} d_t+(\upchi (\ell +E_{J'_\tau })-\upchi (\ell +E_{J'_t})) = \max \left( 0,d_0-\left| (J_t\setminus J_0)\cap \mathcal {N}(v)\right| \right) . \end{aligned}$$

(3.3)

(In particular, for the case $s\le t$, the left-hand side is simply $d_t=0$.) Thus, it makes sense to investigate how $d_i$ — the difference between the values of $\upchi$ corresponding to the two “parallel” computation sequences — changes as it goes from 0 to t.

We can see that

$$\begin{aligned} d_i-d_{i-1}=\upchi (\ell +E_{J'_i})-\upchi (\ell +E_{J_i})-\upchi (\ell +E_{J'_{i-1}})+\upchi (\ell +E_{J_{i-1}})=\Delta '_i-\Delta _i. \end{aligned}$$

From (3.1) we also obtain that if $i<s$ then

$$\begin{aligned} \Delta _i'= {\left\{ \begin{array}{ll} \Delta _i-1, &{} \text {if }v_i\in \mathcal {N}(v), \\ \Delta _i, &{} \text {if }v_i\not \in \mathcal {N}(v), \end{array}\right. } \end{aligned}$$

and consequently

$$\begin{aligned} d_i-d_{i-1}=\Delta _i'-\Delta _i= {\left\{ \begin{array}{ll} -1, &{} v_i\in \mathcal {N}(v), \\ 0, &{} v_i\not \in \mathcal {N}(v). \end{array}\right. } \end{aligned}$$

(3.4)

In other words, as i increases, $d_i$ decreases one by one whenever an element of $\mathcal {N}(v)$ gets added to $J_i$ (and otherwise stays the same).

Now if $s\le t$ — that is, $J(\ell ,I)=J_t\ni v$ — then $v_s=v$, and $J_s=J'_s=J'_{s-1}$, so

$$\begin{aligned} d_{s-1}=\Delta _s\le 0. \end{aligned}$$

Together with $d_0>0$, by (3.4), this implies

$$\begin{aligned} \left| (J_t\setminus J_0)\cap \mathcal {N}(v)\right|{} & {} \ge \left| (J_{s-1}\setminus J_0)\cap \mathcal {N}(v)\right| =\left| \left\{ v_i\big \vert 1\le i\le s-1\right\} \cap \mathcal {N}(v)\right| =d_0-d_{s-1}\\{} & {} \ge d_0. \end{aligned}$$

Thus,

$$\begin{aligned} \max \left( 0,d_0-\left| (J_t\setminus J_0)\cap \mathcal {N}(v)\right| \right) =0, \end{aligned}$$

which along with (3.2) proves the statement.

What remains is the case of $s>t$, i.e., if $v\not \in J_t=J(\ell ,I)$. Then by (3.4) we have

$$\begin{aligned} d_t=d_0-\left| (J_t\setminus J_0)\cap \mathcal {N}(v)\right| . \end{aligned}$$

Here $d_t>0$ because otherwise after the end of $(L)^{\ell }_I$ we could set $J_{t+1}=J_t\sqcup \left\{ v\right\}$ with $\upchi (\ell +E_{J_{t+1}})\le \upchi (\ell +E_{J_t})$, and that contradicts the fact that $J_t$ ended the procedure. So

$$\begin{aligned} \max \left( 0,d_0-\left| (J_t\setminus J_0)\cap \mathcal {N}(v)\right| \right) =\max (0,d_t)=d_t, \end{aligned}$$

meaning according to (3.3) that it would be enough to show

$$\begin{aligned} \upchi (\ell +E_{J'_\tau })=\upchi (\ell +E_{J'_t}), \end{aligned}$$

(3.5)

i.e., that $\upchi (\ell +E_{J'_i})$ does not decrease further as we continue the procedure $(L)^\ell _{I\cup \left\{ v\right\} }$ from $J'_t$ to its end.

For any set $S\subset \mathcal {V}$ and $u\in \mathcal {V}{\setminus } S$ let

$$\begin{aligned} \Delta _u^S=\upchi (\ell +E_{S\cup \left\{ u\right\} })-\upchi (\ell +E_S), \end{aligned}$$

i.e., when executing Algorithm 2.16, we can choose to add a new vertex u to the set J exactly if $\Delta _u^J\le 0$.

We will make use of the following fact: for disjoint sets $S,T\subset \mathcal {V}$ and $u\in \mathcal {V}{\setminus }(S\cup T)$ we have

$$\begin{aligned} \Delta ^{S\cup T}_u&=\upchi (\ell +E_{S\cup T\cup \left\{ u\right\} })-\upchi (\ell +E_{S\cup T}) =\upchi (E_u)-(E_u,\ell +E_{S\cup T})= \nonumber \\&=1-(E_u,\ell +E_S)-(E_u,E_T) =\upchi (\ell +E_{S\cup \left\{ u\right\} })-\upchi (\ell +E_S)-(E_u,E_T)= \nonumber \\&=\Delta ^S_u-(E_u,E_T). \end{aligned}$$

(3.6)

The procedure $(L)^{\ell }_I$ ending in $J_t=J(\ell ,I)$ means that $\Delta ^{J_t}_u>0$ for all $u\in \mathcal {V}\setminus J_t$ (in particular, we have $\Delta ^{J_t}_v=d_t>0$ by definition).

If, after the initial segment of $J_0'=I\cup \left\{ v\right\} ,J_1',\ldots ,J_t'=J(\ell ,I)\cup \left\{ v\right\}$, the procedure $(L)^\ell _{I\cup \left\{ v\right\} }$ immediately ends, i.e., $\tau =t$, then (3.5) is trivially satisfied.

If, on the other hand, it does not then we add a vertex $v_{t+1}\in \mathcal {V}\setminus J_t'=\mathcal {V}\setminus (J_t\cup \left\{ v\right\} )$ such that $\Delta _{v_{t+1}}^{J_t'}\le 0$. Applying (3.6) we get the condition

$$\begin{aligned} 0\ge \Delta ^{J'_t}_{v_{t+1}}=\Delta ^{J_t\cup \left\{ v\right\} }_{v_{t+1}}=\Delta ^{J_t}_{v_{t+1}}-(E_{v_{t+1}},E_v), \end{aligned}$$

where $\Delta ^{J_t}_{v_{t+1}}>0$, so this holds precisely when $v_{t+1}\in \mathcal {N}(v)$ and $\Delta ^{J_t}_{v_{t+1}}=1$.

Similarly, whenever $t<i\le \tau$, i.e., the procedure has not ended after $i-1$ steps, in step i we add a vertex $v_i\in \mathcal {V}{\setminus } J'_{i-1}$ to $J'_{i-1}$ for which

$$\begin{aligned} 0\ge \Delta ^{J'_{i-1}}_{v_i} =\Delta ^{J_t}_{v_i}-(E_{v_i},E_{J'_{i-1}\setminus J_t}) =\Delta ^{J_t}_{v_i}-\left( E_{v_i},E_v+\sum _{j=t+1}^{i-1}E_{v_j}\right) , \end{aligned}$$

(3.7)

and we thus get $J'_i=J'_{i-1}\cup \left\{ v_i\right\}$. The procedure ends when such a $v_i$ can not be found, and then we will have arrived at $J'_\tau =J(\ell ,I\cup \left\{ v\right\} )$.

We will prove by induction that for all $t<i\le \tau$ the vertex set $\left\{ v,v_{t+1},v_{t+2},\ldots ,v_i\right\}$ spans a connected subgraph in $\Gamma$ and in (3.7) we have an equality. This would be sufficient because then

$$\begin{aligned} \upchi (\ell +E_{J'_t})=\upchi (\ell +E_{J'_{t+1}})=\cdots =\upchi (\ell +E_{J(\ell ,I\cup \left\{ v\right\} )}), \end{aligned}$$

and thus (3.5) is satisfied.

We already checked the case $i=t+1$. Assume that this property holds up to $i-1$. Then $\left\{ v,v_{t+1},v_{t+2},\ldots ,v_{i-1}\right\}$ spans a connected subgraph in $\Gamma$. The vertex $v_i$ is not in this set, and because of $\Delta ^{J_t}_{v_i}>0$ we need

$$\begin{aligned} \left( E_{v_i},E_v+\sum _{j=t+1}^{i-1}E_{v_j}\right) >0 \end{aligned}$$

so $v_i$ is adjacent to this subgraph, implying the connectivity part. But since $\Gamma$ is a tree, there can only be one vertex $w\in \left\{ v,v_{t+1},v_{t+2},\ldots ,v_{i-1}\right\}$ adjacent to $v_i$, and thus

$$\begin{aligned} \left( E_{v_i},E_v+\sum _{j=t+1}^{i-1}E_{v_j}\right) =1. \end{aligned}$$

This means that by (3.7) we get $\Delta ^{J_t}_{v_i}=1$ and $\Delta ^{J'_{i-1}}_{v_i}=0$.

Thus, the inductive step is done, finishing the proof of part b).

4 The analytic analogue

We know by Proposition 2.13 that, for $I_1\subseteq I_2\subseteq \mathcal {V}$,

$$\begin{aligned} \mathscr {C}^{A}_{\ell '}(I_1,I_2) =\mathop {\textrm{codim}}\bigl (H^0(\mathscr {O}_{\tilde{X}}(-\ell -E_{I_2}))\hookrightarrow H^0(\mathscr {O}_{\tilde{X}}(-\ell -E_{I_1}))\bigr ). \end{aligned}$$

Thus, if we swap out $T$ to $A$ in the formula of Theorem 3.9, we get that for all $I\subseteq \mathcal {V}{\setminus }\left\{ v\right\}$ and $\ell =x-E_I$, with the notation $I^*=\mathcal {V}{\setminus }(I\cup \left\{ v\right\} )$,

$$\begin{aligned} \mathscr {C}^{A}_\ell (I,I\cup \left\{ v\right\} )-\mathscr {C}^{A}_{\ell ^*}\bigl (I^*,I^*\cup \left\{ v\right\} \bigr )&=\mathop {\textrm{codim}}\bigl (H^0(\mathscr {O}_{\tilde{X}}(-x-E_v))\hookrightarrow H^0(\mathscr {O}_{\tilde{X}}(-x))\bigr )- \mathop {\textrm{codim}}\\&\bigl (H^0(\mathscr {O}_{\tilde{X}}(-(Z_K-x)))\hookrightarrow H^0(\mathscr {O}_{\tilde{X}}(-(Z_K-x-E_v)))\bigr ). \end{aligned}$$

Since the terms including I all fall out, the $\mathop {\textrm{absmax}}$ becomes redundant, hence the induced function $\upchi _{A}$ can be expressed simply as

$$\begin{aligned} \upchi _{A}(x+E_v)-\upchi _{A}(x)&=\mathop {\textrm{codim}}\bigl (H^0(\mathscr {O}_{\tilde{X}}(-x-E_v))\hookrightarrow H^0(\mathscr {O}_{\tilde{X}}(-x))\bigr )- \\&\quad -\mathop {\textrm{codim}}\bigl (H^0(\mathscr {O}_{\tilde{X}}(-(Z_K-x)))\hookrightarrow H^0(\mathscr {O}_{\tilde{X}}(-(Z_K-x-E_v)))\bigr ). \end{aligned}$$

Setting $\upchi _{A}(0)=0$ results in the following:

Proposition 4.1

For a numerically Gorenstein singularity and $\ell \in L$, the above described function $\upchi _{A}$ induced by the analytic arrangements is

$$\begin{aligned} \upchi _{A}(\ell )&=\mathop {\textrm{codim}}\bigl (H^0(\mathscr {O}_{\tilde{X}}(-x))\hookrightarrow H^0(\mathscr {O}_{\tilde{X}})\bigr ) -\mathop {\textrm{codim}}\bigl (H^0(\mathscr {O}_{\tilde{X}}(-Z_K))\hookrightarrow H^0(\mathscr {O}_{\tilde{X}}(-Z_K+x))\bigr ) \\&=\mathop {\textrm{codim}}\bigl (H^0(\mathscr {O}_{\tilde{X}}(-x))\hookrightarrow H^0(\mathscr {O}_{\tilde{X}})\bigr ) +\mathop {\textrm{codim}}\bigl (H^0(\mathscr {O}_{\tilde{X}}(-Z_K+x))\hookrightarrow H^0(\mathscr {O}_{\tilde{X}})\bigr ) \\&\quad -\mathop {\textrm{codim}}\bigl (H^0(\mathscr {O}_{\tilde{X}}(-Z_K))\hookrightarrow H^0(\mathscr {O}_{\tilde{X}})\bigr ). \end{aligned}$$

As per 2.8, this is in fact the weight function $w_{\textrm{an}}$ for the singularity’s analytic lattice cohomology (obtained from the divisorial filtration of $H^0(\mathscr {O}_{\tilde{X}})$, as per [1]) if the singularity is Gorenstein. (Otherwise, we get a symmetrized version of it.)

This way we establish a further parallel between the topological and analytic lattice cohomologies — though in fact originally, as mentioned in the introduction, it was this connection that served as the inspiration for the definition of the analytic lattice cohomology as introduced in [1] (and then further analogues and generalizations, see [2,3,4]).