The Abel map for surface singularities II. Generic analytic structure

We study the analytic and topological invariants associated with complex normal singularities. Our goal is to provide topological formulae for several discrete analytic invariants whenever the analytic structure is generic (with respect to a fixed topological type), under the condition that the link is a rational homology sphere. The list of analytic invariants include: the geometric genus, the cohomology of certain natural line bundles, the cohomology of their restrictions on effective cycles (in a resolution), the cohomological cycle of natural line bundles, the multivariable Hilbert and Poincar\'e series associated with the divisorial filtration, the analytic semigroup, the maximal ideal cycle. The first part contains the definition of `generic structure' based on the work of Laufer. The second technical part rely on the properties of the Abel map developed in a previous manuscript of the authors. The results can be compared with certain parallel statements from the Brill-Noether theory (and the theory of Abel map) associated with projective smooth curves, though the tools and machineries are very different.

1. Introduction 1.1. Our major objects in this note are the analytic and topological invariants associated with complex normal singularity germs. Our goal is to provide topological formulae for several discrete analytic invariants whenever the analytic structure is generic (with respect to a fixed topological type). Regarding this problem very little is known in the present literature. The progress in this direction was obstructed by two major obstacles.
The first one is related with the notion of the 'generic' analytic structure. Let us comment first what kind of difficulties appear in the definition of 'generic'. The point is that for a fixed topological type the moduli space of all analytic structures supported by that fixed topological type, is not yet described in the literature; hence, we cannot define our generic structure as a generic point of such a space. Laufer in [La73b] characterized those topological types which support only one analytic type, but about the general cases very little is known. Usually, generic structures -when they appeared -were introduced by certain ad-hoc definitions, or only in very particular situations. A huge progress was made by Laufer (see e.g. [La73]) when he defined local complete deformations of singularities. This parameter space will be the major tool in our working definition as well (see 1.2).
However, even if one defines some 'generalicity' notion by eliminating some discriminant from a parameter space (consisting of the pathological objects from the point of view of the discussion, cf. 1.2), a major obstruction remains: how to exploit this generalicity and what are the tools which make transparent the effects of the generalicity on numerical invariants. E.g., in the present article for general structures we prove the following key geometric property, cf. Theorem 4.1.1: for a fixed Chern class, if the Abel map is dominant then the corresponding natural line bundle is necessarily in the image of the Abel map as a regular value (for the needed definitions see below).
The second obstruction is the 'lack of experience' with respect to the concrete values of the invariants: there exist a very small number of concrete examples when for a 'generic' structure some (usually analytically non-constant) discrete invariant is computed. Laufer itself provided such an example in [La77] where he proved that a generic elliptic singularity has geometric genus p g = 1 (though. e.g., if the topological type is numerically Gorenstein, and the link is rational homology sphere, then for the very special Gorenstein structure its geometric genus is the length of the elliptic sequence ℓ [N99], which can be arbitrary high depending on the topological structure).
Usually one is not able even to guess what are the possible topological candidates for the invariants of the generic analytic structure. The expectation is that they should be certain sharp topological bounds, but even if some topological bound is known, usually there are no tools to prove its realization for the generic (or any) analytic structure. The situation is exemplified rather trustworthily already by the geometric genus. Wagreich already in 1970 in [Wa70] defined topologically the 'arithmetical genus' p a of a normal surface singularity and for any non-rational germ (that is, when p g = 0) he proved that p a ≤ p g (see [Wa70,p. 425]). Though in some (easy) cases was known that they agree, the proofs suggest (see e.g. the very short proof in [NO17]) that this inequality for germs with complicated topological types (resolution graphs) probably is extremely week. The point is that in the present note we prove that the geometric analytic structure realizes exactly this p a . (Just for comparison: the sharp topological upper bound for p g is not known, at least for the authors, for more see [NO17].) For some other invariants (e.g. for h 1 (O Z ) for Z > 0 arbitrary effective cycle, or for the maximal ideal cycle, or the cohomology cycle, etc) even the corresponding candidates were not on the table. (If we take a look at the topological formulae form below, which are the answers to these questions, we realize that even the type of such formulae were never considered, except maybe in lattice cohomology [N08b].) In fact, even in this article we make the selection of a package of analytic invariants (organized around the cohomology of natural line bundles), for which we present the corresponding 'package of topological expressions', and we will treat, say, the Hilbert-Samuel function/multiplicity/embedded-dimension package in a forthcoming manuscript (with rather different type of combinatorial answers).
1.2. The generic analytic type. Usually when we have a parameter space for a family of geometric objects, the 'generic object' might depend essentially on the fact that what kind of geometrical problem we wish to solve, or, what kind of anomalies we wish to avoid. Accordingly, we determine a discriminant space of the non-wished objects, and generic means its complement. In the present article all the discrete analytic invariants we treat are basically guided by the cohomology groups of the natural line bundles (for their definition see [N07], [O04] or 3.4 here), hence the discriminant spaces (sitting in the base space of complete deformation spaces of Laufer) are defined as the 'jump loci' of the cohomology groups of the natural line bundles. In section 3 we recall the needed results of Laufer and we introduce this working definition of general analytic types.
The definition of the natural line bundles is also obstructed, all the possible definitions are valid only if the link of the singularity is a rational homology sphere. Hence, in the article we also impose this topological restriction. (In fact this assumption is required also by the other main tool we will use, namely by the machinery of the Abel map, see below).
its applicability in a really difficult problem, with a priori unexpected answers which become totally natural and motivated from the perspective of this new approach.
Let us recall shortly this object (for details see [NN18] or §2 and 3.4 here). Let (X, o) be any fixed complex normal surface singularity germ, and let us fix one of its good resolutions X → X. Set the lattice L = H 2 ( X, Z) and its dual lattice L ′ := H 2 ( X, Z). Then for any effective cycle Z (whose support is the reduced exceptional curve E) and for any (possible) Chern class l ′ ∈ L ′ we consider the space ECa l ′ (Z) of effective Cartier divisors D supported on Z, whose associated line bundles O Z (D) have first Chern class l ′ . Furthermore, we also consider the space Pic l ′ (Z) ⊂ H 1 (O * Z ) of isomorphism classes of holomorphic line bundles with Chern class l ′ and the Abel map c l ′ (Z) : ECa l ′ (Z) → Pic l ′ (Z), D → O Z (D). Let χ(l) := −(l, l − Z K )/2 be the Riemann-Roch quadratic function (where Z K ∈ L ′ is the anticanonical cycle). Finally, for any Chern class l ′ ∈ L ′ set the analytically (and uniquely) defined natural line bundle O Z (l ′ ).
Using the Abel map, in [NN18,Th. 5.3.1] we show that for any analytic singularity and resolution with fixed resolution graph, and for any L ∈ Pic l ′ (Z), one has h 1 (Z, L) ≥ χ(−l ′ ) − min 0≤l≤Z, l∈L χ(−l ′ + l), and equality holds for a generic line bundle L gen ∈ Pic l ′ (Z). In particular, for any analytic type, L gen ∈ Pic l ′ (Z) can be expressed combinatorially from the resolution graph, or from the lattice. Now, the expectation and our guiding principle is the following: for a generic analytic structure the natural line bundle O Z (l ′ ) should have the same h 1 as the generic line bundle L gen ∈ Pic l ′ (Z) (associated with any analytic structure). The proof of this statement (Theorem 4.1.1) is the key part of the note. It goes simultaneously with another statement (with similar message): for a generic structure O Z (l ′ ) ∈ im(c l ′ ) ⇔ L gen ∈ im(c l ′ ). The proof is long and technical, it fills in all section 4. It uses the tangent map of c l ′ , which can be written explicitly via Laufer duality (integration of forms along divisors, cf. 4.2.9). In this section certain familiarity with [NN18] might help the reading.
1.4. The main applications. Let us fix a resolution graph (hence, in particular, a topological type). The list of analytic invariants which are described combinatorially form the resolution graph for a generic analytic type (with respect to the fixed graph) are the following: h 1 (O Z ), h 1 (O Z (l ′ )) (with certain restriction on the Chern class l ′ ), -this last one for Z ≫ 0 provides h 1 (O X ) and h 1 (O X (l ′ )) too -, the cohomological cycle of natural line bundles, the multivariable Hilbert and Poincaré series associated with the divisorial filtration, the analytic semigroup, the maximal ideal cycle. (See [CDGZ04,CDGZ08,Li69,N99b,N08,N12,O08,Re97] for the definitions of different analytic invariants of normal surface singularities.) For precise formulae and other consequences see section 5. The topological expressions use (merely) χ, and it is surprising how complicated combinatorial invariants this 'simple' (?) quadratic function might carry. (Definitely, this can happen due to the fact that we work over the lattice. It is a real challenge now to interpret these expressions in terms of lattice cohomology [N08b, N11] or other topological 3-manifold invariants.) In section 6 we determine the dimension of the image of c l ′ . It is based on a general (technical) algorithm which produces dim im(c l ′ ) for any singularity. However, for generic germs this can be transformed into a different concrete topological formula, again in terms of χ, see section 8. (In forthcoming works the general algorithm will be used in other concrete situations as well.) The importance of dim im(c l ′ ) resides in the following observation: since the dimension of ECa l ′ (Z) (when it is not empty) is known, it is (l ′ , Z), the dimension of the generic fiber also follows. But the dimension of each fiber (c l ′ ) −1 (L) is connected with h 1 (Z, L) (see [NN18, (3.1.8)]) hence as a byproduct we get the cohomology of the generic bundle from the image of c l ′ (that is, the cohomology of the generic bundle with given Chern class and without fixed components).
These are the first non-trivial steps in the determination of the stratification of Pic l ′ (Z) according to the value of L → h 1 (Z, L) (the analogue of the Brill-Noether stratification in the case of projective smooth curves, cf. [ACGH85,Fl10]), for more details see [NN18] again.

Preliminaries and notations
2.1. Notations regarding a good resolution. [N99b,N07,N12,L13,NN18] Let (X, o) be the germ of a complex analytic normal surface singularity, and let us fix a good resolution φ : X → X of (X, o). Let E be the exceptional curve φ −1 (0) and ∪ v∈V E v be its irreducible decomposition. Define E I := v∈I E v for any subset I ⊂ V.
We will assume that each E v is rational, and the dual graph is a tree. This happens exactly when the link M of (X, o) is a rational homology sphere.
L := H 2 ( X, Z), endowed with a negative definite intersection form ( , ), is a lattice. It is freely generated by the classes of where δ vw stays for the Kronecker symbol). L ′ is also identified with H 2 ( X, Z), where the first Chern classes live.
All the E v -coordinates of any E * u are strict positive. We define the Lipman cone as There is a natural (partial) ordering of L ′ and L: The support of a cycle l = n v E v is defined as |l| = ∪ nv =0 E v .
2.2. The Abel map. [NN18] Let Pic( X) = H 1 ( X, O * X ) be the group of isomorphic classes of holomorphic line bundles on X. The first Chern map c 1 : Similarly, if Z ∈ L >0 is an effective non-zero integral cycle supported by E, then Pic(Z) = H 1 (Z, O * Z ) denotes the group of isomorphism classes of invertible sheaves on Z. Again, it appears in the exact sequence 0 → Pic 0 (Z) → Pic(Z) Here L(|Z|) denotes the sublattice of L generated by the base element E v ⊂ |Z|, and L ′ (|Z|) is its dual lattice.
For any Z ∈ L >0 let ECa(Z) be the space of (analytic) effective Cartier divisors on Z. Their supports are zero-dimensional in E. Taking the class of a Cartier divisor provides the Abel map c : ECa(Z) → Pic(Z). Let ECal(Z) be the set of effective Cartier divisors with Chern classl ∈ L ′ (|Z|), i.e. ECal(Z) := c −1 (Picl(Z)). The restriction of c is denoted by cl : ECal(Z) → Picl(Z). We also use the notation ECa l ′ (Z) := ECa R(l ′ ) (Z) and Pic l ′ (Z) := Pic R(l ′ ) (Z) for any l ′ ∈ L ′ , where R : L ′ → L ′ (|Z|) is the cohomological restriction, dual to the inclusion L(|Z|) ֒→ L. (This By this definition (see (3.1.5) of [NN18]) ECal(Z) = ∅ if and only if −l ∈ S ′ (|Z|) \ {0}. It is advantageous to have a similar statement forl = 0 too, hence we redefine ECa 0 (Z) as {∅}, a set/space with one element (the empty divisor), and c 0 : ECa 0 (Z) → Pic 0 (Z) by c 0 (∅) = O Z . Hence the previous equivalence extends to thisl = 0 case too: It turns out that ECal(Z) is a smooth complex algebraic variety of dimension (l, Z). Furthermore, the Abel map is an algebraic regular map. For several properties and applications see [NN18]. In the body of the present paper several needed statements of [NN18] will be reviewed as well.
3. Resolutions with generic analytic structure 3.1. The setup. We fix a topological type of a normal surface singularity. This means that we fix either the C ∞ oriented diffeomorphism type of the link, or, equivalently, one of the dual graphs of a good resolution (all of them are equivalent up to blowing up/down rational (−1)-vertices). We assume that the link is a rational homology sphere, that is, the graph is a tree of rational vertices.
Any such topological type might support several analytic structures. The moduli space of the possible analytic structures is not described yet in the literature, hence we cannot rely on it. In particular, the 'generic analytic structure', as a 'generic' point of this moduli space, in this way is not well-defined. In fact, in order to run/prove the needed technical statements, instead of such theoretical definition it would be even much better to consider a definition based on a list of stability properties under certain concrete deformations (whose validity could be expected for the 'generic' analytic structure). Hence, for us in this note, a generic analytic structure will be a structure, which will satisfy such stability properties. In order to define them it is convenient to fix a resolution graph Γ and treat deformation of singularities together with their resolutions having dual graph Γ.
The type of stability we wish to have is the following. The topological type (or, the graph Γ) determines a lower bound for the possible values of the geometric genus (which usually depends on the analytic type). Let MIN(Γ) be the unique optimal bound, that is, MIN(Γ) ≤ p g (X, o) for any singularity (X, o) which admits Γ as a resolution graph, and MIN(Γ) = p g (X, o) for some (X, o). Then one of the requirements for the 'generic analytic structure' (X gen , o) is that p g (X gen , o) = MIN(Γ). (In the body of the paper MIN(Γ) will be determined explicitly.) However, we will need several similar stability requirements involving other line bundles as well (besides the trivial one, which provides p g ). For their definition we need a preparation.
3.2. Laufer's results. In this subsection we review some results of Laufer regarding deformations of the analytic structure on a resolution space of a normal surface singularity with fixed resolution graph (and deformations of non-reduced analytic spaces supported on exceptional curves) [La73].
First, let us fix a normal surface singularity (X, o) and a good resolution φ : ( X, E) → (X, o) with reduced exceptional curve E = φ −1 (o), whose irreducible decomposition is ∪ v∈V E v and dual graph Γ. Let I v be the ideal sheaf of E v ⊂ X. Then for arbitrary positive integers {r v } v∈V one defines two objects, an analytic one and a topological (combinatorial) one. At analytic level, one sets the ideal sheaf I(r) := v I rv v and the non-reduces space O Z(r) := O X /I(r) supported on E. The topological object is a graph with multiplicities, denoted by Γ(r). As a non-decorated graph coincides with the graph Γ without decorations. Additionally each vertex v has a 'multiplicity decoration' r v , and we put also the self-intersection decoration E 2 v whenever r v > 1. (Hence, the vertex v does not inherit the self-intersection decoration of v if r v = 1). Note that the abstract 1-dimensional analytic space Z(r) determines by its reduced structure the shape of the dual graph Γ, and by its non-reduced structure all the multiplicities {r v } v∈V , and additionally, all the selfintersection numbers E 2 v for those v's when r v > 1 (see [La73,Lemma 3.1]). We say that the space Z(r) has topological type Γ(r).
Clearly, the analytic structure of (X, o), hence of X too, determines each 1-dimensional nonreduced space O Z(r) . The converse is also true in the following sense.
Theorem 3.2.1. [La71,Th. 6.20], [La73,Prop. 3.8] (a) Consider an abstract 1-dimensional space O Z(r) , whose topological type Γ(r) can be completed to a negative definite graph Γ (or, lattice L). Then there exists a 2-dimensional manifold X in which Z(r) can be embedded with support E such that the intersection matrix inherited from the embedding E ⊂ X is the negative definite lattice L. In particular (since by Grauert theorem [GR62] the exceptional locus E in X can be contracted to a normal singularity), any such Z(r) is always associated with a normal surface singularity (as above).
(b) Suppose that we have two singularities (X, o) and (X ′ , o) with good resolutions as above with the same resolution graph Γ. Depending solely on Γ the integers {r v } v may be chosen so large that if O Z(r) ≃ O Z ′ (r) , then E ⊂ X and E ′ ⊂ X ′ have biholomorphically equivalent neighbourhoods via a map taking E to E ′ . (For a concrete estimate how large r should be see Theorem 6.20 in [La71].) In particular, in the deformation theory of X it is enough to consider the deformations of nonreduced spaces of type O Z(r) .
Fix a non-reduced 1-dimensional space Z = Z(r) with topological type Γ(r). Following Laufer and for technical reasons (partly motivated by further applications in the forthcoming continuations of the series of manuscripts) we also choose a closed subspace Y of Z (whose support can be smaller, it can be even empty). More precisely, Definition 3.2.2. [La73, Def. 2.1] A deformation of Z, fixing Y , consists of the following data: (i) There exists an analytic space Z and a proper map λ : Z → Q, where Q is a manifold containing a distinguished point 0.
(ii) Over a point q ∈ Q the fiber Z q is the subspace of Z determined by the ideal sheaf λ * (m q ) (where m q is the maximal ideal of q). Z is isomorphic with Z 0 , usually they are identified.
(iii) λ is a trivial deformation of Y (that is, there is a closed subspace Y ⊂ Z and the restriction of λ to Y is a trivial deformation of Y ).
(iv) λ is locally trivial in a way which extends the trivial deformation λ| Y . This means that for ant q ∈ Q and z ∈ Z there exist a neighborhood W of z in Z, a neighborhood V of z in Z q , a neighborhood U of q in Q, and an isomorphism φ : W → V × U such that λ| W = pr 2 • φ (compatibly with the trivialization of Y from (iii)), where pr 2 is the second projection; for more see [loc.cit.].
One verifies that under deformations (with connected base space) the topological type of the fibers Z q , namely Γ(r), stays constant (see [La73,Lemma 3.1]).
Laufer proved the following results. (c) There exists a deformation λ with ρ 0 bijective. In such a case in a neighbourhood U of 0 the deformation is essentially unique, and the fiber above q is isomorphic to Z for only at most countably many q in U .

Functoriality
3.3. The '0-generic analytic structure'. We wish to define when is the analytic structure of a fiber Z q (q ∈ Q) of a deformation 'generic'. We proceed in two steps. The '0-genericity' is the first one (corresponding to the Chern class l ′ = 0), which will be defined in this subsection.
It is rather advantageous to set a definition, which is compatible with respect to all the restrictions O Z → O Z ′ . In order to do this, let us fix the coefficientsr = {r v } v so large that for them Theorem 3.2.1 is valid. In this way basically we fix a resolution ( X, E) and some large infinitesimal neighbourhood Z(r) associated with it. Moreover, let us also fix a complete deformation λ(r) : Z(r) → Q whose fibers have the topological type of Γ(r). Next, we consider all the other coefficient sets r := {r v } v such that 0 ≤ r v ≤r v for all v, not all r v = 0. This choice automatically provides deformations λ(r) : Z(r) → Q. Then set is not constant in a neighbourhood of q for some i}.
Next, we wish to generalize this definition for all Chern classes l ′ ∈ L ′ , or, for all 'natural line bundles', instead merely for the structure sheaf.
3.4. Natural line bundles. Let us start again with a good resolution φ : ( X, E) → (X, o) of a normal surface singularity with rational homology sphere link, and consider the cohomology exact sequence associated with the exponential sequence of sheaves Here c 1 (L) ∈ H 2 ( X, Z) = L ′ is the first Chern class of L. Then, see e.g. [O04, N07], there exists a unique homomorphism (split) of c 1 , s : L ′ → Pic( X), with c 1 • s = id, such that s restricted to L is l → O X (l). The line bundles s(l ′ ) are called natural line bundles of X, and are denoted by O X (l ′ ).
For several definitions of them see [N07]. E.g., L is natural if and only if one of its power has the form O X (l) for some integral cycle l ∈ L supported on E. Here we recall another construction from [O04, N07], which will be extended later to the deformations space of singularities. Fix some l ′ ∈ L ′ and let n be the order of its class in L ′ /L. Then nl ′ is an integral cycle; its reinterpretation as a divisor supported on E will be denoted by div(nl ′ ). We claim that there exists a divisor D = D(l ′ ) in X such that one has a linear equivalence nD ∼ div(nl ′ ) and c 1 (O X (D)) = l ′ . Furthermore, D(l ′ ) is unique up to linear equivalence, hence l ′ → O X (D(l ′ )) is the wished split of (3.4.1). Indeed, since c 1 is onto, there exists a divisor D 1 such that c 1 (O X (D 1 )) = l ′ . Hence O X (nD 1 − div(nl ′ )) has the form ǫ(L) for some L ∈ Pic 0 ( X) = H 1 ( X, O X ) = C pg . Define D 2 such that O X (D 2 ) = 1 n L in H 1 ( X, O X ). Then D 1 − D 2 works. The uniqueness follows from the fact that Pic 0 ( X) is torsion free.
We wish to bring to the attention of the reader the following warning. Note that if X 1 is a connected small convenient neighbourhood of the union of some of the exceptional divisors (hence X 1 also stays as the resolution of the singularity obtained by contraction of that union of curves) then one can repeat the definition of natural line bundles at the level of X 1 as well. However, the restriction to X 1 of a natural line bundle of X (even of type O X (l) with l integral cycle supported on E) usually is not natural on is the natural restriction), though their Chern classes coincide.
In the sequel we will deal with the family of 'restricted natural line bundles' obtained by restrictions of O X (l ′ ). Even if we need to descend to a 'lower level' X 1 with smaller exceptional curve, or to any cycle Z with support included in E (but not necessarily E) our '(restricted) natural line bundles' will be associated with Chern classes l ′ ∈ L ′ = L ′ ( X) via the restrictions Pic( X) → Pic( X 1 ) or Pic( X) → Pic(Z) of bundles of type O X (l ′ ). This basically means that we fix a tower of singularities { X 1 } X1⊂ X , or {O Z } |Z|⊂E , determined by the 'top level' X, and all the (restricted) natural line bundles, even at intermediate levels, are restrictions from the top level.
We use the notations 3.5. The universal family of natural line bundles. Next, we wish to extend the definition of the line bundles O Z (l ′ ) to the total space of a deformation (at leats locally, over small balls in the complement of ∆ 0 (r)). We fix some Z = Z(r) with allr v ≫ 0, supported on E, such that Theorem 3.2.1 is valid (similarly as in 3.3). Fix also some Y ⊂ Z, and a complete deformation λ : Z(r) → Q of (Z, Y ) as in Definition 3.2.2 (such that all the fibers have the same fixed topological type Γ(r)). We consider the discriminant ∆ 0 (r) ⊂ Q, and we fix some q 0 ∈ Q \ ∆ 0 (r), and a small ball U , q 0 ∈ U ⊂ Q \ ∆ 0 (r). Above U the topologically trivial family of irreducible exceptional curves form the irreducible divisors {E v } v , such that E v above any point q ∈ U is the corresponding irreducible exceptional curve E v,q of X q . With the notations of the previous paragraph, if nl ′ has the form v n v E v write div λ (nl ′ ) := v n v E v for the corresponding divisor in λ −1 (U ). Since U is contractible, one has H 2 (λ −1 (U ), Z) = L ′ and H 1 (λ −1 (U ), Z) = 0, hence the exponential exact sequence on λ −1 (U ) gives Lemma 3.5.2. The first Chern class morphism c 1 in (3.5.1) is onto.
Proof. We show using the Leray spectral sequence that H 2 (λ −1 (U ), O λ −1 (U) ) = 0. Recall, see e.g. EGA III.2 §7, or [Os], that if q → h i (Z(r) q , O Z(r)q ) is constant over some open set U (and all i) Then, if in the above construction of the split of c 1 in (3.4.1) we replace X by λ −1 (U ) and div(nl ′ ) by div λ (nl ′ ), we get the following statement.
Let us summarize what we obtained: For any q 0 ∈ Q \ ∆ 0 (r), and small ball U with q 0 ∈ U ⊂ Q \ ∆ 0 (r), we have defined for each l ′ ∈ L ′ a line bundle O λ −1 (U) (D λ (l ′ )) in Pic(λ −1 (U )), such that its restriction to each fiber Z(r) q is the line bundle O Z(r)q (l ′ ). Let us denote it by O λ −1 (U) (l ′ ).
3.6. The semicontinuity of q → h 1 (Z q , O Zq (l ′ )). We fix a complete deformation λ : Z(r) → Q, and we consider the set of multiplicities r v ≤r v , not all zero, as in 3.3. Then, for each r, we have a restricted deformation λ(r) : Z(r) → Q of Z(r) as in 3.5.
(Note that if each r v > 1 then the intersection form on Γ(r) is well-defined. In particular, the semicontinuities of h 0 and h 1 are equivalent, since h 0 − h 1 = (Z(r), l ′ ) + χ(Z(r)) by Riemann-Roch.) Proof. We fix a small ball U in Q \ ∆ 0 (r) as in subsection 3.5, and we run q ∈ U .
Let us denote (as above) the exceptional curves in the fiber λ(r) −1 (q) by {E v,q } v , hence the cycle Z(r) q is v r v E v,q . Then one has the short exact sequence of sheaves where the sum in the last term runs over the edges (v, w) of Γ(r). This gives the Mayer-Vietoris exact sequence Let us fix an arbitrary q 0 ∈ U . Note that a singularity with a resolution consisting only one rational irreducible divisor is taut, see [La73b], hence the analytic family {Z(r) q } q over a small neighbourhood U ′ ⊂ U of q 0 can be trivialized. Furthermore, Pic 0 (r v E v,q ) = 0, hence the line bundle O λ −1 (U) (l ′ )| rvEv,q is uniquely determined topologically by l ′ and r. Hence, O λ −1 (U) (l ′ )| rv Ev,q also can be trivialised over a small U ′ . In particular, by these trivializations, , and the q-dependence is codified in the restriction morphism δ. Hence, there exists a morphism ). Since the rank of δ(q) is semicontinuous, the statement follows , and the second term in this last sum is also topological and constant (by the same argument as above), hence semicontinuity for h 1 follows as well.
3.7. The 'generic analytic structure'. Now we are ready to give the definition of the 'generic structure'. Let us fix a complete deformation λ(r) : Z(r) → Q as in 3.3 (withr v large) whose fibers have the topological type of Γ(r). Similarly as there, we consider all the other coefficient sets Then ∆(l ′ , r) is a closed (reduced) proper subspace of Q (for this use e.g. an argument as in the proof of Lemma 3.6.1, or [Ri74,Ri76]). Then set the countable union of closed proper subspaces ∆(r) := (∪ l ′ ∈L ′ ∪ rv ≤rv ∆(l ′ , r)) ∪ ∆ 0 (r). Clearly, ∆(r) Q.
Definition 3.7.3. (a) For a fixed Γ(r) and for any complete deformation λ(r) : Consider a singularity (X, o) and one of its resolutions X with dual graph Γ. We say that the analytic type on X is generic if there existsr ≫ 0, and a complete deformation λ(r) : Z(r) → Q with fibers of topological type Γ(r), and q ∈ Q \ ∆(r) such that λ(r) −1 (q) = O X | vr v Ev . Remark 3.7.4. (a) Fix any 1-dimensional space Z with fixed topology Γ(r) with allr v ≫ 0. Then in any complete deformation λ of Z there exists a generic structure arbitrary close to Z.
(b) Though the above construction does not automatically imply that Q \ ∆(r) is open, for any q 0 ∈ Q \ ∆(r) and for any finite set F L ′ ⊂ L ′ there exists a small neighbourhood U of q 0 such that (c) Fix a complete deformation λ : Z(r) → Q of some (Z, Y ) with some fixedr v ≫ 0 as above. Then, by Theorem 3.2.1(b) for any q ∈ Q the fiber Z(r) q determined uniquely a holomorphic neighborhood X q of E. (Some {r v } v very large works uniformly for all fibers, since a convenient {r v } v can be chosen topologically.) Furthermore, h 1 ( X q , O Xq ) can be recovered from λ as h 1 (Z(r) q , O Z(r)q ) by the formal function theorem. This is the geometric genus of the singularity (X q , o) obtained by contracting E in this X q . Since ∆(0,r) = {q ∈ Q : p g (X q , o) = MIN(Γ)} is part of the discriminant ∆(r) (and it is closed), for any 'generic' q ∈ Q \ ∆(r) there is a ball q ∈ U ⊂ Q \ ∆(0,r) such that λ simultaneously blows down to a flat family X → U . This follows from [Ri74,Ri76,Wa76] by the constancy of Γ and p g .
3.8. A special 1-parameter family of deformation. Next, we describe our special families of 1-parameter deformations of a fixed normal surface singularity (X, o), what will play a crucial role in the proof of the main Theorem 4.1.1.
We choose any good resolution φ : where C w is the affine line with coordinate w, and C * w = C w \ {0}. Next, fix any curve E w of φ −1 (o) and also a generic point P w ∈ E w . There exists an identification of the tubular neighbourhood of E w via T (e) such that u 1 = v 1 = 0 is P w . By blowing up P w ∈ X we get a second resolution ψ : X ′ → X; the strict transforms of {E v }'s will be denoted by E ′ v , and the new exceptional (−1) curve by E new . If we contract E ′ w ∪ E new we get a cyclic quotient singularity, which is taut, hence the tubular neighbourhood of E ′ w ∪ E new can be identified with the tubular neighbourhood of the union of the zero sections in T (e − 1) ∪ T (−1). Here we represent will be modified by the following 1-parameter family of spaces: the neighbourhood of ∪ v E ′ v will stay unmodified, however T (−1), the neighbourhood of E new will be glued along ) is a small holomorphic parameter. The smooth complex surface obtained in this way will be denoted by X ′ t , and the 'moved' (−1)-curve in X ′ t by E new,t . If we blow down E new,t we obtain the surface X t .
Remark 3.8.1. Such a deformation λ : ( X , X) → (C, 0), reduced to some Γ(r), say withr ≫ 0, is always the pullback of a complete deformation of O X |Z(r). Hence, if X is generic, then the base point q 0 corresponding to the fiber O X |Z(r) is in Q \ ∆(r). Since for such q 0 there is a ball q ∈ U ⊂ Q \ ∆(0,r) such that λ simultaneously blows down to a flat family X → U (cf. 3.7.4(c)), the deformation λ : ( X , X) → (C, 0) also blows down to a deformation X → (C, 0) of (X, o). In fact, λ is a weak simultaneous resolution of the (topological constant) deformation X → (C, 0), cf. [La83,KSB88]. The point is that along the deformation λ automatically we will have the h 1stabilities for any other finitely many restricted natural line bundles as well, cf. Remark 3.7.4(b) (that is, for the very same X and its deformation λ, the finitely many Chern classes -whose h 1 -stability we wish -can be chosen arbitrarily, depending on the geometrical situation we treat).

The cohomology of restricted natural line bundles
4.1. The setup. We fix a normal surface singularity (X, o) and one of its good resolutions X with exceptional divisor E and dual graph Γ. For any integral effective cycle Z = Z(r) whose support |Z| is included in E (not necessarily the same as E) write V(|Z|) for the set of vertices {v : E v ⊂ |Z|} and S ′ (|Z|) ⊂ L ′ (|Z|) for the Lipman cone associated with the induced lattice L(|Z|). As above, for any l ′ ∈ L ′ we denote the restriction of the natural line bundle Recall also that for any −l ∈ S ′ (|Z|) one has the Abel map cl : ECal(Z) → Picl(Z).
Theorem 4.1.1. Assume that X is generic in the sense of Definition 3.7.3. Fix also some Z = Z(r) as above.
Then the following facts are equivalent:  If |V(|Z|)| = 1 then Pic 0 (Z) = 0 and all line bundles with the same Chern class are isomorphic, hence all the statements are trivially true for any Z and any l ′ . Hence let us fix some virtual support |Z| and assume that all the statements are valid for any cycle with support smaller than |Z| and for any l ′ with the corresponding restrictions.
Next, we run induction over v∈V(|Z|) r v . Assume that r v ≤ 1 for all v. Then Pic 0 (Z) = 0 again and both (I) and (II) hold. Hence, we assume that (I) and (II) hold for all cycles with v r v < N (and any l ′ with the required restrictions) and we consider some 4.1.4. The first part of the proof of Theorem 4.1.1(I). First we verify the 'easy' implications.
Since cl is an algebraic morphism and its image contains a small analytic ball of top dimension, cl is dominant.
The proof of (a) ⇒ (c) in (I) is much harder and longer, and it is the core of the present theorem.
4.2. The proof of (a) ⇒ (c). Fix any l * ∈ L ′ and write l ∈ L ′ (|Z|) for its restriction. Then there is a canonical identification of Pic l (Z) with Pic 0 (Z) by L → L ⊗ O Z (−l * ). Also, Pic 0 (Z) identifies with H 1 (Z, O Z ) by the inverse of the exponential map such that O Z is identified by 0. In particular, c l (Z) : ECa l (Z) → Pic l (Z) can be identified with its composition with the above two maps, namely with c l * (Z) : ECa l (Z) → H 1 (O Z ). In the sequel l * will stay either for l ′ or for different cycles of type E * u with E u ∈ |Z|. In this latter case, the restriction of E * u ∈ L ′ is E * u , where this second dual is considered in L ′ (|Z|). We use the same notation for both of them, from the context will be clear which one is considered.
Therefore, the wished statement (a) ⇒ (c) transforms into the following: Assume that this is not the case for some D. Then there exists a linear functional ω ∈ H 1 (O Z ) * , ω = 0, such that ω| im(TD c l ′ ) = 0.
During the proof we fix such a D and ω. First, we concentrate on ω.
Laufer duality (see [La72], [La77, p. 1281], or [NN18, 7.1]), hence any functional on H 1 (O X ) (that is, on classes of Cartier divisors with zero Chern class) is induced (up to a constant shift) by integration (locally in a neighborhood of the support of the divisor) of the representative of such 2-classform multiplied by a function obtained from the equation of the divisor (this will be explicitly described in 4.2.9). Since ω = 0 the form necessarily must have a pole along some E w . By combination of Theorems 6.1.9(d) and 8.1.3 of [NN18] we know that the kernel of ρ is dual with the subspace of forms which have no pole along |Z|. Therefore, the form must have a pole along some E w ⊂ |Z|. Since ECa −E * w (Z) is the space of effective Cartier divisors of X (up to the equation of Z), which intersect only E w (and the intersection is transversal), again by local nature of the integration formula (see [NN18,7.1] 4.2.2. Let Z, ω and E w ⊂ |Z| be as in Lemma 4.2.1. We wish to modify the resolution X (and the space Z) dictated by certain property of ω. For this we blow up X at generic point of E w creating the new exceptional divisor F 1 , then we blow up a generic point of F 1 creating the new exceptional divisor F 2 , etc. The sequence of n such blow ups will be denoted by b n : X n → X, which has is an isomorphism (use Leray spectral sequence). We define ω n by the composition Lemma 4.2.3. For n sufficiently large the next morphism is constant: Consider Z and the notations of the proof of Lemma 4.2.1, and the composition ω n • c −F * n (b * n ( Z)), similar to (4.2.4), but with Z instead of Z. This for any n gives the diagram (4.2.5) Recall that ω n corresponds to an integration of a 2-form (see the proof of Lemma 4.2.1 and subsection 4.2.9 below). Since the pole order along F n of the 2-form corresponding to ω n decreases by one after each blow up, after some steps n it will have no pole along F n , hence g. [NN18, Theorem 3.1.10]), the statement follows.
4.2.6. In the sequel, let k ≥ 1 be the smallest integer such that ω k • c −F * k (b * k (Z)) is constant. Consider again Z, a cycle with very large coefficients, as in the proof of Lemmas 4.2.1 and 4.2.3. The functionals ω k−1 and ω k associated with Z (as in 4.2.2) induce the functionals ω k−1 and ω k (as in diagram (4.2.5)) and they form the following commutative diagram: is constant. Therefore, the differential 2-form on X k−1 which induces ω k−1 (and also its pullback via the blow up on X k which indices ω k ) has a pole of order exactly one along F k−1 . Therefore, the maps (four of them) where V is either Z or Z, depend only on the reduced structure of b * k−1 (V ) and b * k (V ) along F k−1 , and they all can be identified with the map represented by Laufer's integration pairing (this will be explicitly written down below). 4.2.8. In Lemma 4.2.3 and in the discussion from 4.2.6 one can replace in ECa −F * k−1 and in ECa −F * k the terms F * k−1 and F * k by any multiple of them: ℓF * k−1 and ℓF * k respectively, for any ℓ ∈ Z >0 . Indeed, the space of divisors has a natural 'additive' structure, namely a dominant map s l ′ is constant. Furthermore, the discussion from 4.2.6 can be repeated for any ℓ, the composed maps depend only on the reduced structure of b * k (Z), hence Z can be replaced by any large Z, in which case the composition can be computed by Laufer's integration duality formula.
4.2.9. Let us write down Laufer's integration formula more precisely. The very same formulae a valid at both level of b * k ( Z) and b * k−1 ( Z), we make explicit here the case of b * k ( Z). The peculiar equation of the divisor (multiple of an irreducible one) is chosen since this is the case what we will need later, the general case is identical (see [La72], [La77, p. 1281 We assume that C = ℓC red , where C red intersects the exceptional curve at only one regular point of F k−1 , say P , and the intersection is transversal. Let us introduce local coordinates (u, v) at this intersection point P : we assume that (4.2.10) This reads as follows: This expression depends only on the intersection point P t := {v + t = 0} of C t with F k−1 , taken with multiplicity ℓ (that is, by the equation of C t modulo u). This shows that in (4.2.11) one can replace Z by Z, and provides a factorization (where V = Z or Z, and ω V,k = ω k or ω k ) of type (4.2.12) Furthermore, by the local equation ofω with pole of order one along F k−1 , in a generic point P of F k−1 one has a −1,0 = 0. In particular, this pairing starts with a nontrivial linear term and Though in (4.2.12) this factorization through ECa −ℓF * k−1 (F k−1 ) exists, a factorization through exists, a fact which will be crucial later. This is what we explain next.
4.2.14. In the space of resolution X k we consider the exceptional curve E k−1 := E ∪ (∪ k−1 i=1 F i ), and let U ⊂ X k be a small tubular neighbourhood of E k−1 inside X k . (Note that contracting E k−1 in U provides a singularity with different topological type than Γ.) One can restrict sheaves/bundles from X k to U ; at cycle level we denote this restriction by Then, for both V = Z or Z, one has the natural injection (which, for V = Z and Z fit in a commutative diagram): . (The second set contains divisors supported on F k−1 \ {F k−2 } with total multiplicity ℓ, while in the first set we have to eliminate those divisors which are supported at F k−1 ∩ F k .) On the other hand, the natural epimorphism ρ V : Proof. We concentrate first on the map c −F * k : , and let A 0 be the parallel linear subspace of the same dimension. As above, we denote the sum A + · · · + A (m times) by mA, clearly all of these affine subspaces have the same dimension, and are parallel to each other. Next, consider also the 'multiples' , or see 4.2.8). Therefore, im( c −mF * k ) ⊂ mA, and in fact, by [NN18, Theorem 6.1.9], for m ≫ 0, they agree. Furthermore, by the same theorem, A 0 = ker(ρ V ).
By the choice of k, ω V,k restricted on the image of This Lemma has the following geometric interpretation: If ω k is represented by a 2-form from ω ∈ H 0 ( X k \ b * k (E), Ω 2 ), then its restriction to U represents ω U Z,k =: ω U k . This has also pole of order one along F k−1 and all the local integration formulas are the same.
Indeed, all the statements of subsection 4.2.9 can be rewritten for with Z large, and we use the duality H 1 (O U ) ⊗ H 0 ( U \ E k−1 ), Ω 2 )/H 0 ( U , Ω 2 ) → C. Then, for a path of cycles of type C t as in 4.2.9 we get the analogue of (4.2.11) with a −1,0 = 0 whenever the point P is generic. This also shows the presence of a factorization of type (4.2.12) 4.2.17. Next, we concentrate on the divisor D ∈ ECal(Z) and on the line bundle O Z (l ′ ). As the center of blow up of b 1 is generic on E w , we can assume that it is not in the support of D. This guarantees that the divisor D lifts canonically into any of the spaces ECa b * n (l) (b * n (Z)) (still denoted by D), and the germs (ECal(Z), D) and (ECa b * n (l) (b * n (Z)), D) are canonically isomorphic. For any n we have the commutative diagram, where all the vertical arrows are isomorphism: Proof. A bundle is natural if one of its power has the form O(l) for some integral cycle l. In this case the Chern classes of the two bundles agree. Furthermore, if nl ′ is integral for certain n ∈ Z >0 , After all these preparations, we start with the key construction of the proof. We will construct a path in with the following properties. Firstly, by the choice of ω, ω • c • γ must have zero derivative at t = 0. On the other hand, the restrictions O b * k (Z) (D t )| E k−1 (in fact, the n-multiples of them) will be equivalent with a family of line bundles associated with divisors of type {C t } t as in 4.2.9, and the computation from (4.2.11) will show that the linear part of ω • c • γ at t = 0 is nonzero, a fact which leads to a contradiction.
The path of divisors will be constructed via a deformation, based on 3.8.

A special deformation of the analytic structure of
be the resolution as in 4.2.2, with the choice of k as in 4.2.6. Here we concentrate on the exceptional components F k−1 and F k , where F k is obtained by blowing up a generic point P . (If k = 1 then F k−1 = E w .) Then for the pair (F k−1 , F k ) we apply the construction of 3.8, that is, we move F k and its intersection point with F k−1 locally along F k−1 . In this way we obtain a 1-parameter family of deformations of the resolution X k , denoted by λ k : ( X k , X k ) → (C, 0), with fibers X k,t . In X k,t the exceptional curve has components E ∪ ∪ k−1 i=1 F i ∪ F k,t . If we blow down the F -type curves in X k,t we get a resolution X t , they form a family ( X , X). If we contract all the exceptional curves we get a family of singularities {(X t , o)} t . Since the analytic structure we started with is generic, the geometric genus h 1 (O X k,t ) stays constant and the deformation blows down to a deformation (X , X) → (C, 0) with fibers X t (cf. 3.8). We denote the contraction X k → X by the same symbol b k .
We assume that the base space of λ is so small that the universal map (C, 0) → Q to the base space of a complete deformation omits the discriminant ∆(r); this fact is guaranteed by the choice of the generic structure of the singularity.
Therefore, for the very same l ′ ∈ L ′ (which provided the bundle O Z (l ′ )) we can consider the universal line bundles constructed in Lemma 3.5.3, namely O X k (b * k (l ′ )) ∈ Pic( X k ) and O X (l ′ ) ∈ Pic( X ). By similar argument as in Lemma 4.2.
). The restriction to the fibers of the deformations are the natural line bundles of the fibers.
Corresponding to the irreducible exceptional curves ) has a section s whose divisor is D (by the definition of D from 4.2 and identification (4.2.18)).
Proof. Since X is generic, 0 (the center of the deformation space (C, 0)) does not sit in the union of the discriminant spaces considered in 3.7. In that subsection we considered all the discriminants associated with all the Chern classes and the 'tower', hence, in particular, we had countably many discriminant obstructions. By assumption, 0 is not contained in any of these. In this proof we have to concentrate on the Chern class l ′ and the tower level Z, hence only one discriminant. In particular, 0 ∈ C has a small neighbourhood which does not intersect it. Therefore, decreasing the representative of the (C, 0) we get the stability of the corresponding h 1 -cohomology sheaves. Furthermore, λ is proper, Let D t be the restriction of the divisor of s to the fiber over t.
Since the support of D = D 0 is disjoint with the center of b 1 , the same is true for each D t (for |t| ≪ 1). Note also that the exceptional components {E v } v are stable (t-independent) for all t. Hence, in this way we get a path germ γ : Note also that in the cycles b * k,t (Z) the curve F k,t (with its stable multiplicity) is 'moving' along the deformation, the other components with their multiplicities are stable, and the divisors D t are supported by this stable part (but they might move).
By the choice of ω and D (compare also with (4.2.18) and (4.2.19): The same is valid for any 'multiples' of the divisors D t . Indeed, choose e.g. o ∈ Z >0 such that n ·l is integral.

Let us restrict the bundles
Hence, by (4.2.25) we get Next we compute the left hand side of (4.2.25) in a different way. O Z k−1 (nD t ) is the restriction of the natural line bundle O X k,t (nl) to Z k−1 . Note that nl is an integral cycle, supported on the exceptional divisor of X k,t ; let us write in the form A + ℓF k,t , where A is supported on E k−1 and it is t-independent. Furthermore, ℓ = nl ′ w , which by the assumption of the theorem is strict positive. Therefore, where P t is the reduced intersection point F k−1 ∩ F k (the moving point on F k−1 ). But, by (4.2.16), Clearly, (4.2.27) and (4.2.28) contradict to each other. Hence, the assumption that T D c l ′ is not surjective cannot be true. This ends the proof of part (a) ⇒ (c) of part (I).
4.3. The proof of part (II). Note that the equalities for i = 0 and i = 1 are equivalent by Riemann-Roch. We will prove (II) in three steps. Then one has the cohomology exact sequence . Since T D (cl) is onto by (I)(c), h 1 (Z, O Z (l ′ )) = 0 follows. Next, assume that the equivalent assumptions of (I) are not satisfied. That is,  In the sequel we assume that h 1 (O Z ) = 0 and h 0 (O Z (l ′ )) = 0. Assume that H 0 (Z, O Z (l ′ )) reg = ∅, that is, O Z (l ′ ) has a section without fixed components. But, then by Chern class computation, this section has no zeros, hence O Z (l ′ ) = O Z , see also (2.2.1).
We claim that this identity O Z (l ′ ) = O Z cannot happen for generic (X, o). The argument runs similarly as the proof of (a) ⇒ (c) in (I).
Since h 1 (O Z ) = 0 we can choose a nonzero functional ω ∈ H 1 (O) * for which we can repeat the arguments from 4.2.1 to (4.2.11). In particular, there exists E w ⊂ |Z| which satisfies Lemma 4.2.1, we can consider the sequence of blow ups as in 4.2.2, and we can choose k as in 4.2.6. Finally we consider the deformation of singularities as in 4.2.22, and as in (4.2.28) we get that t → ω U k,Z (O Z k−1 (nD t )) is not constant. This implies that the path t ) cannot be constant either since its restriction to Z k−1 is not constant. Hence O Zt (l ′ ) cannot equal O Zt for all t-values, since the restriction to Z k−1 of b * k,t O Zt (l ′ ) is nonconstant, while the restriction of b * k,t O Zt is constant. In particular, for generic t we have the claim.
Therefore, if in the definition of the 'genericity' we add this criterion regarding the nonconstant family O Zt (l ′ ), requiring that its generic element is not the specific trivial bundle, we are done. But, in fact, one can prove that the assumptions of the original definition 3.7.3 suffice as well.
This fact can be seen as follows: we prove that h 1 (O Zt (l ′ )) < h 1 (O Zt ) for generic t (though the Chern classes agree). Since O Zt (l ′ ) = O Zt for generic t (and H 1 (O Zt ) is constant nonzero), O Zt (l ′ ) must have fix components, cf. (2.2.1). Let E u ∈ |Z| be a fix component.
Since their kernels have the same h 0 by the inductive step, the inequality follows. This proves the claim.

Applications. Analytic invariants
5.1. In this section we will fix a resolution graph Γ (hence, the lattice L associated with it as well), and we treat singularities (X, o), together with their resolution X whose dual graph is Γ. The goal is to list some consequences of Theorem 4.1.1: hence we will assume that X is generic, and we will provide combinatorial expressions for several analytic invariants in terms of L. We will use the notations from the setup of 4.1.
The first group of results provides topological formulae for the cohomology of certain natural line bundles over an arbitrary Z > 0. In particular, if l ′ = v∈V l ′ v E v ∈ L ′ satisfies l ′ v < 0 for any v ∈ V(|Z|) and X is generic then Theorem 4.1.1 gives the following topological characterization for the cohomology of O Z (l ′ ) (c) Recall that if −l ′ ∈ S ′ \ {0} then all the coefficients l ′ v of l ′ are strict negative. However, if the support of |Z| is strict smaller than E, then −R(l ′ ) ∈ S ′ (|Z|) \ {0} does not necessarily imply that l ′ v < 0 for v ∈ V(|Z|). (Take e.g. Z = E v a (−2)-curve, choose E u an adjacent vertex with it and set l ′ = E v + 3E u . Then −R(l ′ ) ∈ S ′ (E v ) \ {0} however l ′ v = 1.) 5.1.4. The setup for generalization. We construct the following 'Laufer type computation sequence' (see e.g. [La72] or [N07, Prop. 4.3.3]). We start with a class l ′ ∈ L ′ and an effective cycle Z with |Z| ⊂ E. Letl ∈ L ′ (|Z|) be the restriction of l ′ as in Theorem 4.1.1.
Assume that −l ∈ S ′ (|Z|). Then there exists E w ⊂ |Z| so that (l ′ , E w ) < 0. Then, for both line bundles L = L gen and L = O Z (l ′ ) of Picl(Z) one can consider the exact sequence Let us construct the following sequence (l ′ k , Z k ) t k=0 . By definition, (l ′ 0 , Z 0 ) = (l ′ , Z) the objects we started with. If −l = −R(l ′ ) ∈ S ′ (|Z|), then define (l ′ 1 , Z 1 ) : we repeat the procedure, otherwise we stop. After finitely many steps necessarily −l t := −R(l ′ t ) ∈ S ′ (|Z t |) (here Z t = 0 is also possible). (The choice of the sequence is not unique, however by similar argument as in [La72]  Theorem 5.1.5. Assume that X is generic with fixed dual graph Γ, and we choose an effective cycle Z and l ′ ∈ L ′ . Assume that the last term (l ′ t , Z t ) of the Laufer type computation sequence Example 5.1.6. Let X be generic, Z an effective cycle and l ′ ∈ L ′ . Assume that l ′ v ≤ 0 for all v ∈ V(|Z|) and for any connected component Z con of Z there exists v ∈ V adjacent with Z con with l ′ v < 0. (The adjacent condition is |Z con | ∩ E v = ∅.) Then the conditions from Theorem 5.1.5 are satisfied, hence h i (Z, O Z (l ′ )) = h i (Z, L gen ) and (5.1.3) holds.
Indeed, first note that if for some vertex with l ′ v = 0 one has (l ′ , E v ) ≥ 0 then l ′ u = 0 for all adjacent vertices u of v. Hence, (l ′ , E v ) ≥ 0 for all vertices v with l ′ v = 0 contradicts the assumption. That is, there exists v ∈ V(|Z|) so that l ′ v = 0 and (l ′ , E v ) < 0. Then we construct the computation sequence as follows. At the first part of the computation sequence, at step (l ′ k , Z k ) we choose E w(k) so that E w(k) ⊂ |Z k |, the E w(k) -coefficient of l ′ k is zero, and (E w(k) , l ′ k ) < 0. After finitely many such steps we arrive to the situation when along the support of Z k ′ all the coefficients of l ′ k ′ will be strict negative. Then we can continue the algorithm arbitrarily.
Corollary 5.1.7. If X is generic with dual graph Γ and |Z| is connected then Since δ is onto ι is an isomorphism. Corollary 5.2.4. Assume that X is generic with dual graph Γ. Choose any l ′ ∈ L ′ and consider O X (l ′ ), the natural line bundle on X. Then Proof. For any effective cycle Z (with |Z| = E) and l ′ ∈ L ′ let us write ∆(Z, In order to compute h 1 ( X, O X (l ′ )) let us fix some Z with all its coefficients very large. Then, if we start with the pair (l ′ , Z), the Laufer sequence from 5.1.4 ends with some (l ′ t , Z t ) with Z t ≥ E (still with large coefficients), and −l ′ t ∈ S ′ . We claim that ∆(Z k , l ′ k ) is constant along the computation sequence. Indeed, from the cohomological exact sequence used in 5.1.4 (for . Then, we compare min 0≤l≤Z χ(−l ′ + l) and min 0≤l≤Z−Ew χ(−l ′ + E w + l). Since for any x ≥ 0 with E w ∈ |x| we have χ(−l ′ + E w + x) ≤ χ(−l ′ + x), these two minima agree. Hence the claim follows. Now, for the pair (l ′ t , Z t ), with −l ′ t ∈ S ′ , we distinguish two cases. The case l ′ t = 0 occurs exactly . In this case ∆(Z t , l ′ t ) can be computed from (5.2.2). Or, l ′ t = 0. In this case all the coefficients of l ′ t are strict negative (use e.g. Remark 5.1.1(c)), and ∆(Z t , l ′ t ) = 0 by (5.1.3).
Example 5.2.6. For any h ∈ H define k h := K + 2r h and It is known (use e.g. the algorithm from [N07,Prop. 4.3.3]) that for any h ∈ H one has min l∈L ≥0 χ(r h + l) = min l∈L χ(r h + l). Therefore, for h = 0 one has Remark 5.2.8. (a) Let (X ab , o) be the universal abelian covering of (X, o). Then 5.5. The Hilbert series. Fix X generic and let H(t) be the multivariable (equivariant) Hilbert series associated with the divisorial filtration of the local algebra of the universal abelian covering of (X, o) associated with divisors supported on all irreducible exceptional divisors of X; for details see e.g. [CDGZ04,CDGZ08,N12]. Write .) It is known that for any l ′ there exists a unique s(l ′ ) ∈ S ′ such that s(l ′ ) − l ′ ∈ L ≥0 , and s(l ′ ) is minimal with these properties. Furthermore, for any l ′ ∈ L ′ one has h(l ′ ) = h(s(l ′ )). Hence it is enough to determine h(l ′ ) for the (closed) first quadrant (because S ′ ⊂ L ′ ≥0 ). Write l ′ as r h + l 0 for some l 0 ∈ L ≥0 (and h = [ [N12, (2.3.3)]. Therefore, for l 0 = 0 we get h(r h ) = 0.
Corollary 5.6.1. Write P (t) = l ′ ∈S ′ p(l ′ )t l ′ . Then for p(0) = 1 and for l ′ > 0 one has 5.7. The analytic semigroup. The analytic semigroup is defined as Corollary 5.7.4. Assume that X is generic with non-rational graph Γ. Then M = {Z ∈ L >0 : χ(Z) = min l∈L χ(l)} has a unique maximal element and Z max = max M.
Proof. For the first part see the second paragraph of 5.3. max M ∈ S an by the right hand side of 5.7.1, but min S an cannot be smaller than max M by the very same identity.
5.8. The O (X,o) -multiplication on H 1 ( X, O X ). Assume that p g > 0. On H 1 ( X, O X ) the O (X,o)module multiplication transforms on the dual vector space H 1 ( X, O X ) * = H 0 ( X\E, Ω 2 X )/H 0 ( X, Ω 2 X ) into the multiplication of forms by functions. The filtration on H 1 ( X, O X ) induced by the powers of the maximal ideal agrees with the filtration associated by the nilpotent operator determined by multiplication by a generic element of m (X,o) . For details see e.g. [To86].
The poles of forms are bounded by Z coh . Indeed, by the exact sequence 0 → Ω 2 → Ω 2 (Z coh ) → O Z coh (Z coh + K X ) → 0 and from the vanishing h 1 (Ω 2 ) = 0 (and from Serre duality) we have dim H 0 (Ω 2 (Z coh ))/H 0 (Ω 2 ) = h 0 (O Z coh (Z coh + K X )) = h 1 (O Z coh ) = p g . Hence the subspace H 0 (Ω 2 (Z coh ))/H 0 (Ω 2 ) ⊂ H 0 ( X \ E, Ω 2 )/H 0 (Ω 2 ) has codimension zero, hence the spaces agree. Proof. Since Z max ≥ Z coh , cf. 5.7.6, m (X,o) · H 0 (Ω 2 (Z coh )) ⊂ H 0 (Ω 2 (−Z max + Z coh )) ⊂ H 0 (Ω 2 ). 5.9. Generic Q-Gorenstein singularities. Recall that a singularity (X, o) is Gorenstein if the anticanonical cycle Z K is integral, and Ω 2 . Hence in this case O X (K X ) is natural. Recall, that more generally, a line bunlde L is natural if and only if one of its powers has the form O X (l) for some l ∈ L, or equivalently, if and only if its restriction L| X\E ∈ Pic( X \ E) = Cl(X, o) has finite order (that is, it is Q-Cartier). In particular, (X, o) is Q-Gorenstein if and only if O X (K X ) is a natural line bundle, which automatically should agree with O X (−Z K ).
Proposition 5.9.1. If a Q-Gorenstein singularity (X, o) admits a resolution X with generic analytic structure, then (X, o) is either rational of minimally elliptic. Proof.
Step 1. Let us fix a resolution X of a normal surface singularity (X, o). We claim that if (X, o) is neither rational nor minimally elliptic then there exists an effective cycle Z > 0, |Z| ⊂ E, with Z ≥ Z K and with h 1 (O Z ) > 0.
Assume that ⌊Z K ⌋ = Z K . Then Z K ∈ L and Z K > 0 (since p g > 0) hence necessarily Z K ≥ E (see [La87]). For any v ∈ V consider the exacts sequence 0 Since h 1 (O Ev (−Z K + E v )) = 1 we get that p g = 1 and Z K = Z coh . Then the geometric genus of the singularities obtained by contracting any E \ E v is rational, hence (X, o) is minimally elliptic (for details see [La77] or [Re97]).
Finally, let X be arbitrary and let π : X → X min be the corresponding modification of the minimal one. Let 0 < Z < Z K be the cycle obtained previously for X min . Then π * (Z) works in X.
Corollary 6.1.7. Let l ′ ∈ −S ′ with E * -support I and Z ≥ E. Assume that L is a regular value of c l ′ in im( c l ′ ) such that the bundle L has no base points, and L = c l ′ (D), where D is the O Zreduction of a reduced divisor D of X, which intersects E transversally. Then T L (im c l ′ ) = A Z (l ′ ). In particular, d Z (l ′ ) = e Z (l ′ ).
Proof. Since L is a regular value, L is a smooth point of im( c l ′ ) and T L im( c l ′ ) = im(T D c l ′ ) for any D ∈ ( c l ′ ) −1 (L) (cf. [NN18, 3.3.2]). Hence, we have to prove that im(T D c l ′ ) (the same subspace for any D ∈ (c l ′ ) −1 (L)) is A Z (l ′ ) . We will prove the dual identity in the space of forms, namely, Ω Z (I) = Ω Z ( D) (cf. (6.1.4)). Hence, we need Ω Z ( D) ⊂ Ω Z (I). Since L has no base points, and some section provides reduces transversal divisor D, there exists a family of sections with reduced and transversal divisors {D t } t whose intersection points with E have no fixed points. Therefore, if ω ∈ H 0 (Ω 2 X (Z)) has the property that Res Dt (ω) = 0 for all such D t , then necessarily ω ∈ Ω Z (I).
In this section we provide an algorithm, valid for any analytic structure, which determines d Z (l ′ ) in terms of a finite collection of invariants of type e Z (l ′ ), associated with a finite sequence of resolutions obtained via certain extra blowing ups from X.
6.2. Preparation for the algorithm. Fix a resolution X of (X, o) and −l ′ = v∈V a v E * v ∈ S ′ (hence each a v ∈ Z ≥0 ). Similarly as in 4.2.2 we consider a finite sequence of blowing ups starting from X. For every v ∈ V with a v > 0 we fix a v generic points on E v , say p v,kv , 1 ≤ k v ≤ a v . Starting from each p v,kv we consider a sequence of blowing ups of length s v,kv (s v,kv ≥ 0): first we blow up p v,kv and we create the exceptional curve F v,kv ,1 , then we blow up a generic point of F v,kv ,1 and we create F v,kv ,2 , and we do this all together s v,kv times. If s v,kv = 0 then we do not blow up p k,v k (this is 'chain of length zero'). We proceed in this way with all points p v,kv , hence we get v a v chains of modifications.
If a v = 0 we do no modification along E v . We can extend the above notation for a v = 0 too by k v = ∅ and s v,kv = 0. (In the sequel, in order to avoid aggregation of indices, we simplify k v into k.) Let us denote this modification by π s : X s → X. In X s we find the exceptional curves At each level s we set the next objects: Z s := π * s (Z), and e s := e Zs (I s ) (both associated with X s ).
Again, d s ≤ e s for any s. If a = 0 then k v = ∅ for all v, and s = 0, hence X s = X and d 0 = d Z (0) = 0 and e 0 = e Z (0) = 0. Moreover, if s = 0 (but l ′ not necessarily zero), then I s ⊂ V is the E * -support of l ′ , l ′ s = l ′ , X s = X, e 0 = e Z (l ′ ) and d 0 = d Z (l ′ ).
6.2.1. Let m v be the E v -multiplicity of max{0, ⌊Z K ⌋}. Then we define a very special index set s as follows. Let s v,kv = m v for any k v whenever a v > 0, and s v,kv = 0 if a v = 0. The definition is motivated by the fact that for any representative ω in H 0 ( X \ E, Ω 2 X )/H 0 ( X, Ω 2 X ) the order of pole of ω along some E v is less than or equal to m v (see e.g. [NN18, 7.1.3]).

6.2.2.
There is a natural partial ordering on the set of s-tuples. Some of the above invariants are constant with respect to s, some of them are only monotonous. E.g., by Leray spectral sequence one has h 1 (O Zs ) = h 1 (O Z ) for all s. One the other hand, because Ω Zs 1 (I s1 ) ⊂ Ω Zs 2 (I s2 ). In fact, for any ω, the pole-order along F v,k,s+1 of its pullback is at least one less than the pole-order of ω along F v,k,s . Hence, for s = m, hence all the possible poleorders along I m automatically vanish, one has Ω Zm (I m ) = H 0 ( X m , Ω 2 Xm (Z m ))/H 0 (Ω 2 Xm ). Hence e m = 0. In particular, necessarily d m = 0 too.
More generally, for any s and (v, k) with a v > 0, let s v,k denote that tuple which is obtained from s by increasing s v,k by one. By the above discussion, if no form has pole along F v,k,s , then Ω Zs (I s ) = Ω Z s v,k (I s v,k ), hence e s = e s v,k . Furthermore, under the same assumption, by integral presentation of the Abel map, d s = d s v,k as well. Therefore e s = e min{s,m} and d s = d min{s,m} .
The next theorem relates the invariants {d s } s and {e s } s .
Theorem 6.2.3. (Algorithm) With the above notations the following facts hold. ( (2) If for some fixed s the numbers {d s v,k } v,k are not the same, then In the case when all the numbers {d s v,k } v,k are the same, then if this common value d s v,k equals e s , then Proof. (1) Assume first that either s v,k ≥ 1 or a v = 1. Then divisors from ECa l ′ s (Z s ) intersect F v,k,s v,k by multiplicity one, hence the intersection (supporting) point gives a map q : ECa l ′ s (Z s ) → F v,k,s v,k , which is dominant. Moreover, ECa l ′ s v,k (Z s v,k ) is birational with a generic fiber of q (the fiber over the point which was blown up), hence (1) follows. Note also that d s = d s v,k if and only if the generic fiber of the Abel map c l ′ s is not included in a q-fiber. That is, for fixed v and k, On the other hand, d s = d s v,k + 1 whenever the generic fiber of the Abel map is included in a q-fiber. If s v,k = 0 and a v > 1 then write l ′ − := l ′ s − E * v and consider the 'addition map' s : Next, assume that the numbers {d s v,k } are the same, say d.  7. Towards a different dim im(c l ′ (Z))-formula Though in Theorem 6.2.6 we proved a d Z (l ′ )-formula in terms of e s -invariants, our goal is to find a different one, involving more topological terms, and which hopefully will show more intrinsic structure of the Abel maps.
In this section (X, o) is again an arbitrary singularity (unless stated otherwise). Example 7.1.5. The difference h 1 (Z, L im gen ) − h 1 (Z, L gen ) can be arbitrary large. Indeed, let us start with a singularity with an arbitrary analytic structure, we fix a resolution X with dual graph Γ, and we distinguish a vertex, say v 0 , associated with the irreducible divisor E 0 . Let k (k > 0) be the number of connected components of Γ \ v 0 , and we assume that each of them is non-rational. Furthermore, we choose Z ≫ 0, hence h 1 (O Z ) = p g . Let X| V\v0 be a small neighbourhood of be its connected components, and set p g,i = h 1 (O Xi ) for the geometric genus of the singularities obtained from X i by collapsing its exceptional curves. Write also Γ \ v 0 = ∪ i Γ i . We also assume that −l ′ = nE * 0 with n ≫ 0.
7.1.6. It is convenient to introduce the following notation whenever |Z| is connected: We wish to estimate h 1 (O Z ) − d Z (l ′ ) = codim(im c l ′ (Z)) = h 1 (Z, L im gen ). Note that the estimate given by the second line of (7.1.3), that is, h 1 (Z, L im gen ) ≥ T (Z, l ′ ) sometimes is week, see the previous example. However, surprisingly, if we replace Z by a smaller cycle Z ′ ≤ Z, then we might get a better bound. More precisely, first note that if L im gen is a generic element of im(c l ′ (Z)), and 0 < Z ′ ≤ Z, then its restriction r(L im gen ) (via r : , then for each Z ′ i we can apply (7.1.3). Therefore, we get (Here there is no need to restrict l ′ , cf. Remark 7.1.4.) Hence (7.1.8) reads as (7.1.10) h 1 (Z, L im gen ) ≥ t Z (l ′ ).
Example 7.1.11. (Continuation of Examle 7.1.5) The last computation of Example 7.1.5 shows that the maximum of χ(nE * 0 ) − min l≥0 χ(nE * 0 + l) is obtained for l 0 = 0 and T (Z, l ′ ) = 1 + i (− min χ(Γ i )). Hence, taking Z ′ = i Z ′ i , each Z ′ i supported on Γ i and large, we get that the restriction of l ′ is zero and i T (Z ′ i , l ′ ) = i (1 − min χ(Γ i )) = T (Z, l ′ ) + k − 1. Summarized (also from Example 7.1.5) for any analytic type one has i p g, . However, if X is generic then p g,i = 1 − min χ(Γ i ) (cf. Corollarty 5.1.7), hence, all the inequalities transform into equalities. Hence, for generic analytic structure h 1 (Z, L im gen ) = t Z (l ′ ), that is, (7.1.10) provides the optimal sharp topological lower bound. Note also that both t Z (l ′ ) and i (1−min χ(Γ i )) are topological, hence if they agree for X generic, then they are in fact equal. Since p g,i − 1 + min χ(Γ i ) for arbitrary analytic type can be considerably large, for arbitrary analytic types the inequality (7.1.10) can be rather week.
(2) If the analytic structure of X is generic then Proof. Part (1) follows from (7.1.3) and (7.1.10). For (2), let I be the E * -support of l ′ , and set Z ′ := Z| V\I . Then, cf. (6.1.6), Note that in this case the image of the Abel map is a point, hence 7.2. The Z-stability of t Z (l ′ ). In Lemma 7.1.12 t Z (l ′ ) appears together with the terms h 1 (O Z ), e Z (l ′ ) and d Z (l ′ ). These three terms stabilises when Z becomes large. Indeed, let us start with  In order to obtain a similar statement for Z → D(Z, l ′ ), it would be more convenient to assure that if |Z| is connected then the support of the reduced cycle, say, min{⌊Z K ⌋, Z}, is also connected. This would follow from the connectivity of | max{0, ⌊Z K ⌋}|, however this is not automatic at allfor some similar statement for special singularities and minimal (good) graphs see e.g. [V04, 2.10]. Therefore, we will replace max{0, ⌊Z K ⌋} by Z := max{E, ⌊Z K ⌋} (in which case |Z| = | min{Z, Z}|). Then for any Z with |Z| connected D(Z, l ′ ) = D(min{Z, Z}, l ′ ), proved similarly as for d Z (l ′ ) and using the stability of the targets of the Abel maps.
Everything is valid if we replace Z by the components of some Z ′ ≤ Z, hence Lemma 7.2.1. Set Z := max{E, ⌊Z K ⌋}. Then t Z (l ′ ) = t min{Z,Z} (l ′ ).
This fact will be used as a technical reduction in the proof of Lemma 7.4.12.
7.3. Our goal is to prove in section 8 that under some generality condition d Z (l ′ ) = h 1 (O Z ) − t Z (l ′ ) (compare with the inequality from Lemma 7.1.12(1)). The proof runs over induction. It needs several inductive identities for t Z (l ′ ), of type already proved for d Z (l ′ ) or d s (e.g., the stability with respect to blowing up, or the analogue of d s − d s v,k ∈ {0, 1} from Theorem 6.2.3(1)) They will be treated in the next subsection. Though the proofs of some of these statements for d Z are rather straightforward due to the geometry of the Abel map, the corresponding proofs for t Z will have their technical specific nature due to the 'max − min' shape of the defining expression of t Z . 7.4. Inductive properties of t Z (l ′ ). All the statements of this subsection 7.4 are valid for any analytic type and any good resolution X of a singularity (X, o). 7.4.1. First we prove a stability property of the invariant t Z (l ′ ) with respect to a blowing up.
Let X, Z, l ′ = − v a v E * v ∈ L ′ as usual, and assume that we blow up a generic point of E w , written as ψ : X + → X. We denote the strict transform of any E v by the same E v , and the new exceptional curve by E + . Write also Z + := ψ * (Z), l ′ Proof. First note that D(ψ * (Z ′ 1 ), l ′ + ) = D(Z ′ 1 , l ′ ), since the two Abel maps are birationally equivalent. Next, we claim that min 0≤l+≤ψ * Z ′ (χ(l + ) + (l + , l ′ + )) = min 0≤l≤Z ′ (χ(l) + (l, l ′ )). Indeed, if the minimum from the right hand side is realized by some l, then for l + := ψ * (l) one has χ(l + ) + (l + , l ′ + ) = χ(l) + (l, l ′ ), hence the left hand side cannot be larger. If the minimum from the left is realized for l + ≤ ψ * Z ′ , then write l + = ψ * l + tE + and observe that l ≤ Z ′ too. Furthermore, t → χ(l + ) + (l + , l ′ + ) = χ(l) + (l, l ′ ) + χ(tE + ) should take its minimum, hence necessarily t ∈ {−1, 0}, and the left-minimum is realised by the right hand side too. The fact that besides the l ↔ ψ * l correspondence l ↔ ψ * l − E + also works will be exploited in the proof of Lemma 7.4.12.
7.4.5. Recall that in 6.2 we started with −l ′ = v a v E * v ∈ S ′ , identified by the E * -vector a = {a v } v , and we defined for any s = {s v,kv } v,kv , 0 ≤ s v,kv ≤ m v (with s v,kv = 0 whenever a v = 0) invariants e s and d s . In order to emphasize the a-dependence let us denote them by e s (a) and d s (a).
Recall also the notations {π s : X s → X} s , Z s and l ′ s from 6.2.
Similarly to d s (a) for a and s we set (the reinterpreted (7.1.9)) (7.4.6) t s = t s (a) := max Next we establish the analogue of Theorem 6.2.3(1).
Lemma 7.4.7. Let s and s v,k be as in Theorem 6.2.3. Then t s v,k − t s ∈ {0, 1}.
Proof. The proof has several analogies with the proof of Lemma 7.4.3. However, there is a key difference on the definition of the lifted Chern classes; so we 'repeat/modify' the arguments.
Let ψ : X s v,k → X s be the blowing up. We abridge the vertex w v,k,s v,k into w (hence ψ is the blowing up of a generic point of the w-exceptional curve), and let the new exceptional curve be E + .
First we prove that t s v,k ≥ t s . Assume that t s , as the maximum from the right hand side of (7.4.6), is realized by If w ∈ |Z ′ | then the same cycle can be considered as a possible Z ′ in the computation of t s v,k too, with the very same value, hence t s v,k ≥ t s . If w ∈ |Z ′ 1 |, then set Z ′ 3). Next we verify the inequality − min 0≤l+≤φ * Z ′ 1 (χ(l + ) + (l + , l ′ s v,k )) ≥ − min 0≤l≤Z ′ 1 (χ(l) + (l, l ′ s )). Indeed, if the minimum from the right hand side is realized by some l, then for l + := φ * (l) one has χ(l + ) + (l + , l ′ s v,k ) = χ(l) + (l, l ′ s ), hence the left hand side cannot be smaller. Hence, in any case, t s v,k ≥ t s .
7.4.10. Assume next that s u,ku > m u for some (u, k u ). In this case no form has pole along F = F u,ku,s u,ku , thus the Abel map is not effected by the divisors supported on F . Hence, if we replace Lemma 7.4.12. If s u,ku > m u for some (u, k u ) then t s (a) = t s− (a − ).
Proof. Since along the modification π s the cycle Z is replaced by Z s , and π * s Z X ≥ Z Xs , by Lemma 7.2.1 we can assume that (7.4.13) Z ≤ Z = max{E, ⌊Z K ⌋}.
Let us recall that in the case of (a − , s − ) the point p u,ku is not blown up, while for (a, s) the point p u,ku is blown up consecutively more than m u times and its last exceptional curve F = F u,ku,s u,ku supports the contribution −F * of l ′ s := l ′ s (a). Let us blow up X s− consecutively s u,ku times (at the very same points as in the (u, k u )-chain of X s , starting with p u,ku ), in this way in fact we create the very same space X s . We define the supporting cycle (the 'Z-cycle') by consecutive pull-backs, hence in X s we create the very same cycle Z s . However, we do not define the Chern class by the procedure how l ′ s was defined, but by pull-back (as in Lemma 7.4.3), and we denote it still by l ′ s− (hence the (u, k u ) tower supports no contribution from l ′ s− ). Then, by Lemma 7.4.3 we have t Zs − (l ′ s− ) = t Zs (l ′ s− ) (the right hand side computed in this new situation). Hence, we need to show that t Zs (l ′ s− ) = t Zs (l ′ s ). Both these objects are supported on Z s , in one case −F * is present in the Chern class while in the other case not. We will compare the defining expressions (the right hand side of (7.1.9)) always for the very same Z ′ -cycles. Since s u,ku > m u , no differential form has pole along F , hence −F * has no effect of any Abel map of any restriction on 0 < Z ′ ≤ Z s . Therefore, D(Z ′ , l ′ s− ) = D(Z ′ , l ′ s ) for any 0 < Z ′ ≤ Z s . Hence we need to compare the expressions of type χ(−l ′ ) − min l χ(−l ′ + l) = − min l (χ(l) + (l ′ , l)) (0 ≤ l ≤ Z ′ i ) for the two different Chern classes l ′ = l ′ s and l ′ = l ′ s− . Since for any l ≥ 0 one has (l ′ s , l) = (l ′ s− , l) − (F * , l) ≥ (l ′ s− , l), we get t Zs (l ′ s ) ≤ t Zs (l ′ s− ). In order to prove the opposite inequality we construct a system 0 < Z ′ ≤ Z s , Z ′ = i Z ′ i , 0 < l i ≤ Z ′ i , which realises t Zs (l ′ s− ) and (l i , F * ) = 0 for each i. This system applied in the computation of t Zs (l ′ s ) provides the same value, hence t Zs (l ′ s ) ≥ t Zs (l ′ s− ) follows. We construct the system {Z ′ i , l i } i via the isomorphism t Zs − (l ′ s− ) = t Zs (l ′ s− ). So, let us start with a system {Z ′ −,i , l −,i } i (i.e. 0 < Z ′ − ≤ Z s− , Z ′ − = i Z ′ −,i , 0 < l −,i ≤ Z ′ −,i ), which realises t Zs − (l ′ s− ). Let ψ : X s → X s− be the sequence of blow ups ψ = ψ s • · · · • ψ 1 along the (u, k u )-chain (s = s u,ku ). Then first we set Z ′ i := ψ * (Z ′ −,i ). If p u,ku ∈ |l −,i | then we take l i := ψ * (l −,i ). However, if p u,ku belongs to say |l −,1 | then the cycles l 1 is defined inductively by the sequence {ℓ s } s as follows: ℓ 0 := l −,1 , ℓ s = ψ * s (ℓ s−1 ) − F u,ku,s if F u,ku,s ⊂ |ψ * s (ℓ s−1 )|, otherwise ℓ s = ψ * s (ℓ s−1 ). And finally, the last ℓ s is the wished l 1 . Since this tower is s u,ku long, s u,ku > m u , and any coefficient of l −,i , is not larger that the corresponding coefficient of Z s− , then (7.4.13) shows that the E u -coefficient of l −,i is ≤ m u + 1. Since we have at least m u + 1 blowing ups, F is not in the support of l 1 , or of i l i . This combined with the proof of Lemma 7.4.2 (which shows that this i l i realises t Zs (l ′ s− ) similarly as i ψ * l −,i ), we get that {Z ′ i , l i } i has the needed properties.
8. dim im(c l ′ (Z)) for generic singularities 8.1. Preliminaries for section 8. Our goal is to determine d Z (l ′ ) = dim im(c l ′ (Z)) combinatorially, whenever the analytic structure on X is 'sufficiently generic'.
Theorems 6.2.3 and 6.2.6 show that d Z (l ′ ) is computable from a sequence of invariants of type e Z (l ′ ) = h 1 (O Z ) − h 1 (O Z| V\I ) for any analytic structure. Moreover, from section 5 we know that the invariants of type h 1 (O Z ) are combinatorially computable from a resolution graph whenever the resolution has a generic analytic structure, for the concrete expression see Corollary 5.1.7.
However, the sequence of e Z (l ′ ) type invariants (and h 1 (O Z ) type invariants) used in Theorems 6.2.3 and 6.2.6 are not associated with the fixed resolution X of (X, o), but to an additional tower of resolutions obtained from X by a sequence of finitely many blowing ups. Via definition 3.7.3 the genericity of X is guaranteed by omitting several discriminant subspaces of the basespace of a complete deformation, their index set is associated with L ′ of X and Z. However, in order to guarantee the topological characterizations of e Z (l ′ ) type invariants used in Theorems 6.2.3 and 6.2.6 we need to omit more discriminant space (indexed by the larger lattice L ′ m associated with the resolution X m ). Therefore, in this subsection we need a stronger notion of genericity.
Let us formulate this restricted condition more precisely. We fix our resolution X and also −l ′ = v a v E * v ∈ S ′ . Set also max{0, ⌊Z K ⌋} = v m v E v . We construct the new resolution X m (with v m v a v new generic blowing ups) as in 6.2.
Definition 8.1.1. For fixed X and any −l ′ ∈ S ′ we say that the analytic type of X is l ′ -generic if the resolution X m (for some generic choices of the points what we blow up) is generic in the sense of Definition 3.7.3.
By the above discussion, if X is l ′ -generic, then (with the notations of 6.1.5) each e s is combinatorial via Corollary 5.1.7 applied for the lattice of X s .
Corollary 8.1.2. If the analytic type of X is l ′ -generic then d Z (l ′ ) is combinatorial, computable via the algorithm described in Theorems 6.2.3 and 6.2.6.
The second formula promised in section 7 is the following. In particular, h 1 (Z, L im gen ) = t Z (l ′ ) and h 1 (Z, L) ≥ t Z (l ′ ) for any L ∈ im(c l ′ (Z)).
Proof. We run double induction: increasing induction over |a| (with all allowed values of s), and for any fixed a decreasing induction over the entries of s. Assume that a = 0. Then by 6.2 s = 0 too and d 0 (0) = e 0 (0) = 0. Hence Lemma 7.1.12 gives t(0) = h 1 (O Z ) and Theorem 8.1.3 follows.
Assume next that the statement is valid for any a with |a| < N and for any allowed s. We choose some a with |a| = N . Then we run decreasing induction on the entries of s. Assume that s u,ku > m u for some (u, k u ) and define the objects a − and s − as in 7.4.10. Since |a − | < N , by induction (and by d s− (a − ) = d s (a), cf. For such cases we use Lemma 7.4.7. Let us fix some s. By the decreasing induction on the entries of s we can assume that for any (v, k) we have equality d s v,k = h 1 (O Z ) − t s v,k ( †), and we wish to prove the same identity for s. By Lemma 7.4.7 for any (v, k) one has t s v,k − t s ∈ {0, 1}.
If the numbers {t s v,k } v,k are not all the same, then the induction hypothesis ( †) the numbers {d s v,k } v,k are not all the same either, hence by the algorithm 6.2.3 one has d s = max v,k d s v,k . Therefore, since t s v,k − t s ∈ {0, 1} one also has t s = min v,k t s v,k , and necessarily d s = h 1 (O Z ) − t s . generic element of the 'new' strata O(l) ⊗ (im(c l ′ (Z))) of Pic l ′ +l (Z), unreachable directly by the previous results. Our hidden goal is to construct in this way line bundles with 'high' h 1 .
For simplicity we will assume that al the coefficients of Z are sufficiently large (even compared with l, hence the coefficients of Z −l are large as well). The monomorphism of sheaves L im gen | Z−l ֒→ L im gen (l) gives h 0 (Z − l, L im gen ) ≤ h 0 (Z, L im gen (l)), hence h 1 (Z − l, L im gen ) + χ(Z − l, L im gen ) ≤ h 1 (Z, L im gen (l)) + χ(Z, L im gen (l)).
Remark 9.2.3. By [NN18, Prop. 5.7.1] for Z ≫ 0, L ∈ Pic(Z) with c 1 (L) ∈ −S ′ one has h 1 (Z, L) ≤ p g whenever either H 0 (Z, L) = 0 or L ∈ im(c l ′ (Z)). For other line bundles a weaker bound is established (see [loc. cit.]), which does not guarantee h 1 (L) ≤ p g . However, it is not so easy to find singularities and bundles with h 1 (L) > p g in order to show that such cases indeed might appear. Our next goal is to provide such an examples (with a recipe to find many others as well) based partly on (9.2.2). In (9.2.2) with a convenient choice of l ′ one has −χ(−l ′ ) > 0, hence it gives sharper h 1 -bound than h 1 (Z, L im gen ) ≥ t Z (l ′ ) (cf. 7.1.12(1) and (7.1.3)).