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Advances in Mathematics
www.elsevier.com/locate/aim
The Abel map for surface singularities II. Generic analytic structure
✩János Nagya,András Némethib,c,d,∗
aCentralEuropeanUniversity,Dept.ofMathematics,Budapest,Hungary
bAlfrédRényiInstituteofMathematics,HungarianAcademyofSciences, Reáltanodautca13-15,H-1053,Budapest,Hungary
cELTE- UniversityofBudapest,Dept.ofGeometry,Budapest,Hungary
dBCAM- BasqueCenterforAppliedMath.,Mazarredo,14E48009Bilbao,Basque Country–Spain
a r t i c l e i n f o a b s t r a c t
Articlehistory:
Received7August2019 Receivedinrevisedform15May 2020
Accepted2June2020 Availableonline30June2020 CommunicatedbyKarenSmith
MSC:
primary32S05,32S25,32S50,57M27 secondary14Bxx,14J80
Keywords:
Normalsurfacesingularities Naturallinebundles Abelmap
Picardgroup Genericsingularity
Analyticandtopologicalinvariants
Westudytheanalytic andtopologicalinvariantsassociated with complex normal surface singularities. Our goal is to provide topological formulae for several discrete analytic invariants whenever the analytic structure is generic (with respecttoafixedtopologicaltype),undertheconditionthat thelink isa rational homology sphere. The listof analytic invariants includes: the geometric genus, the cohomology of certain natural line bundles, the cohomology of their restrictionsoneffectivecycles(supportedon theexceptional curve of a resolution), the cohomological cycle of natural line bundles, the multivariable Hilbert and Poincaré series associated with the divisorial filtration, the analytic semi- group,themaximalidealcycle.
The first part contains the definition of ‘generic structure’
based on the work of Laufer [14]. The second technical ingredientistheAbelmapdevelopedin[21].
Theresultscanbecomparedwithcertainparallelstatements fromtheBrill–Noethertheoryandfromthe theoryofAbel
✩ TheauthorsweresupportedbyNKFIHGrant“Élvonal(Frontier)”KKP126683.
* Correspondingauthor.
E-mailaddresses:nagy_janos@phd.ceu.edu(J. Nagy),nemethi.andras@renyi.mta.hu(A. Némethi).
https://doi.org/10.1016/j.aim.2020.107268 0001-8708/©2020ElsevierInc. Allrightsreserved.
map associated with projectivesmooth curves (see e.g. [1]
and[6]),thoughthetoolsandmachineriesareverydifferent.
©2020ElsevierInc.Allrightsreserved.
1. Introduction
1.1. Themainobjective.Ourmajorobjectsinthisnotearetheanalyticandtopological invariants associated with complex normal surface singularity germs. Our goal is to providetopologicalformulaeforseveraldiscreteanalyticinvariantswhenevertheanalytic structure isgeneric (withrespect to afixed topological type).Regarding this problem very little is known in the present literature. The type of formulae of the topological characterizationsofthepresentarticlearetotallynew,aswellasthemethods(basedon thenewlycreatedtheoryofAbelmap).
1.2. Discussionregardingthe‘generic analytictype’. Letus commentfirstwhatkindof difficultiesappearinthedefinitionandstudyof‘generic’analytictype.Thepointisthat forafixedtopologicaltypethemodulispaceofallanalyticstructuressupportedbythat fixedtopologicaltype,isnotyetdescribedintheliterature;hence,wecannotdefineour generic structure as ageneric point of suchaspace. Laufer in[15] characterized those topological types whichsupport only one analytic type (or, more generally,countably many analytictypes),butaboutthegeneralcasesverylittleisknown.Usually, generic structures — when theyappeared— wereintroduced bycertain ad-hocdefinitions,or only inparticular situations.Inaslightlydifferentdirectionaremarkableprogresswas madebyLaufer(seee.g.[14])whenhedefinedlocalcompletedeformationsof(resolution of) singularities.This parameterspace will be themajor tool inourworkingdefinition as well(see1.5).
However,evenifonedefinesacertain‘genericity’notionbyeliminatingadiscriminant from a parameter space (consisting of the pathological objects from the point of view of the discussion),the nexthard major taskis to exploitfrom the genericitysome key geometric/numerical/cohomological properties.(E.g.,inthepresent articlethisis done viaTheoremBbelow.)
Regarding the problem to find the valuesof the analytic invariants associated with thegenericanalytictype,acrucialobstruction was(beforethepresentnote)thelackof examplesandexperience.E.g.,Lauferin[16] provedthatagenericellipticsingularityhas geometric genuspg= 1,butexceptthis,almostnootherexampleisknown.Evenmore, using theknownstatementsof theliterature, itis almost impossibleto guess whatare the possible topological candidates forthe invariants of thegeneric analytic structure.
The expectation is that they should be certain sharp topological bounds, but even if sometopological boundis known,usuallythere are notoolsto proveitsrealization for thegeneric(orany) analyticstructure.
The situation is exemplified rather trustworthily already by the geometric genus.
Wagreich already in 1970 in [37] defined topologically the arithmetical genus pa of a normal surface singularity and for any non–rational germ (that is, when pg = 0) he provedthatpa≤pg(see[37,p.425]).Thoughinsome(easy)caseswasknownthatthey agree, analyzing the existing proofs of the inequality (see e.g. the very short proof in [29]),onemightthinkthatthis inequalityforgermswith complicatedtopologicaltypes probably is extremely week. However, the point is that in the present note we prove that (contrary to the first naive judgement) the geometric analytic structure realizes exactlythis pa.Fortheotherinvariants(seeTheoremAbelow)even thecorresponding candidates werenoton thetable(but we expect thattheywill havesomerelationship withlatticecohomology[26]).
Infact,eveninthis articlewemake theselectionofapackageof analyticinvariants (organized around the cohomology of natural line bundles), for which we present the corresponding ‘package oftopological expressions’,and we will treat, say,the Hilbert–
Samuelfunction/multiplicity/embedded-dimensionpackageinaforthcomingmanuscript (withratherdifferenttypeofcombinatorialanswers).
1.3. Thetechnicalpresentationoftheresults.Inordertoformulatetheinvariantsandthe topologicalcharacterizationsinamoreformalwayweneedsomenotation.LetX →X beagoodresolutionwithirreducibleexceptionalcurves{Ev}v∈V,withresolutiongraph Γ, negative definite intersection lattice L = H2(X, Z), dual lattice L = H2(X, Z) H2(X, ∂X, Z),and discriminant groupH =L/L(fordetailssee2.1).Weassumethat thelinkMof(X,o) isarationalhomologysphere,thatis,Γ isatreeofrationalEv’s.In suchacaseH =H1(M,Z) is finite.Usually Z will denote aneffective cyclesupported ontheexceptional curveE. Thedual latticeL isalso thetargetof thesurjective first Chern class map c1 : Pic(X) → L, set c−11 (l) = Picl(X). For any Chern class one definesthe‘naturallinebundle’OX(l)∈Picl(X),anditsrestrictionsOZ(l),cf.3.4.
In thesequel we fix a topological type, that is, a resolution graph. The topological invariants are read from Γ, or equivalently, from L. The most elementary one is the
‘Riemann–Roch’expressionχ:L→Qgivenbyχ(l):=−(l,l−ZK)/2,whereZK ∈L istheanticanonicalcycledefinedcombinatoriallybytheadjunctionformulae,cf.2.1.
Thelist ofanalytic invariants, associated withageneric analytic type(withrespect to the fixed graph), which are described in the present article topologically are the following:h1(OZ),h1(OZ(l)) (withcertainrestrictionontheChernclassl),—thislast oneappliedforZ0 providesh1(OX) andh1(OX(l))too—,thecohomologicalcycle ofnaturallinebundles,themultivariableHilbertandPoincaréseriesassociatedwiththe divisorialfiltration, theanalytic semigroup, the maximal idealcycle. See [4,5,19,23,25, 28,31,33] for thedefinitionsand relationshipsbetweenthem.Here somedefinitionswill berecalledinsection6.
Surprisingly, in all the topological characterization we need to use merely χ, how- ever, it is really remarkable the level of complexity and subtlety of the combinatorial expressions/invariantscarriedbythis‘simple’(?)quadraticfunction.Definitely,thiscan
happendue tothefactthatweworkoverthelatticesLandL,andthepositionof the latticepointswithrespecttothelevelsetsofχplaythekeyrole.Itisarealchallengenow tointerprettheseexpressionsintermsoflatticecohomology[26,27] orothertopological 3–manifold invariants.
Theorem A.Fix aresolution graph and assume that the analytic type of X is generic.
Then thefollowingidentitieshold:
(a) Forany effectivecycleZ∈L>0 with |Z| connected h1(OZ) = 1− min
0<l≤Z,l∈L{χ(l)}. (b) If l=
v∈VlvEv ∈L satisfieslv <0forany Ev inthesupportof Z then h1(Z,OZ(l)) =χ(−l)− min
0≤l≤Z,l∈L{χ(−l+l)}.
(For acharacterizationvalidformoregeneral Chern classesl seesection 6.) (c) If pg(X,o)=h1(X, OX) isthegeometricgenus of(X,o)then
pg(X, o) = 1− min
l∈L>0{χ(l)}=−min
l∈L{χ(l)}+
1 if (X, o) is not rational, 0 else.
(d) Moregenerally,forany l∈L h1(X, OX(l)) =χ(−l)−min
l∈L≥0{χ(−l+l)}+
1 if l ∈L≥0 and(X, o)is not rational, 0 else.
(e) LetH(t)=
l∈Lh(l)tl be themultivariable equivariant Hilbert series associated with thedivisorial filtration.Writel asrh+l0 forsomel0∈L andrh∈L theunique representative of h = [l] in the semi-open cube of L. Then h(rh) = 0 for l0 = 0.
Furthermore, forl0>0andh= 0 h(l) = min
l∈L≥0{χ(l+l)} − min
l∈L≥0{χ(rh+l)}. Forh= 0 andl=l0>0
h(l0) = min
l∈L≥0{χ(l0+l)} − min
l∈L≥0{χ(l)}+
1 if(X, o)is not rational, 0 else.
(f)WritethemultivariableequivariantPoincaréseriesP(t)=−H(t)·
v∈V(1−t−1v )as
l∈Sp(l)tl.Itissupportedin theLipman(antinef)cone,inparticular inL≥0.Then p(0)= 1 andforl>0onehas
p(l) =
I⊂V
(−1)|I|+1 min
l∈L≥0
χ(l+l+EI).
(g)ConsidertheanalyticsemigroupSan :={l ∈L : OX(−l)has no fixed components}. Then
San ={l : χ(l)< χ(l+l)for any l∈L>0} ∪ {0}.
(h) Assume that Γ is a non–rational graph and set M = {Z ∈ L>0 : χ(Z) = minl∈Lχ(l)}.
Thenthe uniqueminimal element ofM is thecohomological cycle, while theunique maximalelement of Misthemaximal idealcycleof X.
1.4. The Abel map.The main tool of the present note is the Abel map constructed andstudiedin[21]. Thoughin[21] wealso listedseveral applications,thepresentnote showsits power,itsapplicabilityinareally difficultproblem,withapriori unexpected answerswhich become totallynatural and motivated from the perspective of this new approach.
Letus recall shortly this object(for detailssee [21] or §2and 3.4 here). Let(X,o), X → X, L and L as above. Then for any effective cycle Z supported on E and for any (possible) Chern class l ∈ L we consider the space ECal(Z) of effective Cartier divisorsDsupportedonZ,whoseassociatedlinebundlesOZ(D) havefirstChernclass l. Furthermore, we also consider the Abel map cl(Z) : ECal(Z) → Picl(Z), D → OZ(D).
Using the Abel map, in [21, Th. 5.3.1] we have shown that for any analytic singu- larity and resolution with fixed resolution graph, and for any L ∈ Picl(Z), one has h1(Z,L)≥χ(−l)−min0≤l≤Z, l∈Lχ(−l+l),andequalityholdsforagenericlinebundle Lgen ∈Picl(Z). Inparticular, forany analytic type,Lgen ∈Picl(Z) can beexpressed combinatorially.Now, the expectation and ourguiding principle is thefollowing: fora genericanalyticstructurethenaturallinebundleOZ(l) shouldhavethesameh1asthe genericlinebundleLgen∈Picl(Z) (associatedwithanyanalyticstructure).Thisisthe keytechnicalstatementofthenote (fornotationssee5.1).
TheoremB.AssumethatX isgeneric.Undersome(necessary)negativityrestrictionon theChernclassl (seeTheorem 5.1.1andRemark 6.1.1(b)) thefollowingfactshold.
(I)The followingfacts areequivalent:
(a)OZ(l)∈im(c˜l),where OZ(l)isthenatural linebundlewithChern classl; (b) Lgen ∈ im(c˜l), where Lgen is a generic line bundle in Pic˜l(Z) (that is, c˜l is dominant);
(c) OZ(l) ∈ im(c˜l), and for any D ∈ (c˜l)−1(OZ(l)) the tangent map TDc˜l : TDECa˜l(Z)→TOZ(l)Pic˜l(Z)issurjective.
(II)hi(Z,OZ(l))=hi(Z,Lgen)fori= 0,1andforagenericlinebundleLgen∈Pic˜l(Z).
Theproofislongandtechnical,itfillsinallsection5(the‘hard’partis(a)⇒(c)).It usestheexplicitdescriptionoftangentmapofcl intermsofLauferduality(integration of forms along divisors,cf. 2.2). Inthis section certainfamiliarity with [21] mighthelp thereading.
By this result, if X has generic analytic structure, then the cohomology of natural line bundles canbe expressed by the very same topological formula as Lgen with the sameChernclass.Then alltheformulaeofTheoremA abovefollow directly.
Inthenextparagraphwesayafewwordsabout‘genericanalytictype’.
1.5. Theworkingdefinitionofthegenericanalytictype. Usuallywhenwehaveaparam- eterspaceforafamilyofgeometricobjects,the‘genericobject’mightdependessentially on the fact thatwhat kind of geometrical problem we wish to solve, or, whatkind of anomalieswewishtoavoid.Accordingly,wedetermineadiscriminantspaceofthenon–
wishedobjects,andgenericmeansitscomplement.Inthepresentarticleallthediscrete analyticinvariantswetreatarebasicallyguidedbythecohomologygroupsofthenatural line bundles (fortheir definitionsee[24], [30] or 3.4 here,theyassociate inacanonical way aline bundleto any givenChern class).Hence, thediscriminant spaces(sittingin the base spaceof complete deformationspaces ofLaufer [14]) aredefinedas the‘jump loci’ of the cohomology groups of the natural line bundles. In section 3 we recall the needed results ofLaufer regarding complete deformationsof some X, and we build on this ourworkingdefinitionofgeneralanalytictype.
Note thatthe naturalline bundles arewell–defined onlyif thelinkis arational ho- mology sphere. Furthermore, this assumption appearedin thetheory of Abel maps as well.Hence,inthearticlewealsoimpose thistopologicalrestriction.
2. Preliminariesandnotations
2.1. Notationsregardinga goodresolution.[23,24,28,11,21] Let(X,o) bethegermofa complexanalyticnormalsurfacesingularity,andletusfixagoodresolution φ:X →X of (X,o). LetE betheexceptionalcurve φ−1(0) and∪v∈VEv be itsirreducible decom- position. DefineEI :=
v∈IEv forany subsetI⊂ V.
We willassumethateachEv isrational,and thedual graphis atree.This happens exactlywhenthelinkM of(X,o) isarationalhomologysphere.
TheZ–moduleL:=H2(X, Z) isalatticeendowedwiththenaturalnegativedefinite intersectionform(,).Itisfreelygeneratedbytheclassesof{Ev}v∈V.Theduallatticeis L= HomZ(L,Z)={l ∈L⊗Q : (l,L)∈Z}.It isgeneratedbythe(anti)dual classes {Ev∗}v∈V defined by (Ev∗,Ew) = −δvw (where δvw stands for the Kronecker symbol).
It is also identified withH2(X, Z).The anticanonical cycleZK ∈L is definedvia the adjunctionidentities(ZK,Ev)=Ev2+ 2 for allv.
AlltheEv–coordinatesofanyE∗uarestrictpositive.Wedefinethe(rational)Lipman cone asS:={l∈L : (l,Ev)≤0for allv}.Asamonoiditis generatedoverZ≥0 by {Ev∗}v.
ThelatticeLembeds intoL withL/LH1(M,Z).Thequotient L/Lisabridged byH. Each class h∈H =L/Lhas auniquerepresentative rh ∈L inthesemi-open cube{
vrvEv∈L : rv ∈Q∩[0,1)}, suchthatitsclass[rh] ish.
Thereisanatural(partial)orderingofL andL:wewritel1≥l2ifl1−l2 =
vrvEv
withallrv≥0.Weset L≥0={l∈L : l≥0}andL>0=L≥0\ {0}. Thesupportofacyclel=
nvEv isdefinedas|l|=∪nv=0Ev.
2.2. TheAbel map.[21] Let Pic(X)=H1(X, O∗X) be thegroupofisomorphismclasses ofholomorphic linebundles onX.Thefirst Chernmap c1: Pic(X) →L issurjective;
writePicl(X) =c−11(l).Since H1(M,Q)= 0,bytheexponentialexactsequenceonX onehasPic0(X) H1(X, OX)Cpg,where pg isthegeometricgenus.
Similarly,ifZ is aneffectivenon–zero integralcyclesupportedbyE,then Pic(Z)= H1(Z,OZ∗) denotesthegroupofisomorphismclassesofinvertible sheavesonZ.Again, itappearsintheexactsequence0→Pic0(Z)→Pic(Z)−→c1 L(|Z|)→0,wherePic0(Z) is identified with H1(Z,OZ) by the exponential exact sequence. Here L(|Z|) denotes the sublattice of L generated by the base element Ev ⊂ |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 supportsare zero–dimensional in E. Taking the class of aCartier divisor providestheAbel mapc: ECa(Z)→Pic(Z).LetECa˜l(Z) betheset ofeffectiveCartier divisorswithChern class˜l∈L(|Z|),i.e.ECa˜l(Z):=c−1(Pic˜l(Z)).Therestrictionofc isdenotedbyc˜l(Z): ECa˜l(Z)→Pic˜l(Z).
WealsousethenotationECal(Z):= ECaR(l)(Z) andPicl(Z):= PicR(l)(Z) forany l ∈L, where R : L → L(|Z|) is thecohomological restriction, dual to the inclusion L(|Z|) → L. (This means that R(Ev∗) = the (anti)dual of Ev in the lattice L(|Z|) if Ev⊂ |Z|andR(Ev∗)= 0 otherwise.)
A line bundle L ∈ Pic˜l(Z) is in the image im(c˜l) if and only if it has a section withoutfixedcomponents,thatis,ifH0(Z,L)reg =∅,whereH0(Z,L)reg :=H0(Z,L)\
∪vH0(Z −Ev,L(−Ev)). Here the inclusion of H0(Z −Ev,L(−Ev)) into H0(Z,L) is given bythe longcohomological exactsequence associated with 0→ L(−Ev)|Z−Ev → L → L|Ev → 0, and it represents the subspace of sections, whose fixed components containEv.
Bythisdefinition(see(3.1.5)of[21] andthediscussionbeforeit)ECa˜l(Z)=∅ifand onlyif−˜l∈ S(|Z|)\ {0}.It isadvantageousto haveasimilar statementfor˜l= 0 too, hence we redefine ECa0(Z) as {∅}, a set/space with one element (the empty divisor), andc0: ECa0(Z)→Pic0(Z) byc0(∅)=OZ.Then
H0(Z,L)reg=∅ ⇔ L=OZ ⇔ L ∈im(c0) (c1(L) = 0). (2.2.1) Then the ‘extended equivalence’ reads as: ECa˜l(Z) = ∅ if and only if −˜l ∈ S(|Z|).
In such a caseECa˜l(Z) is a smooth complex algebraic variety of dimension (˜l,Z), cf.
[21, Th. 3.1.10]. Furthermore, the Abel map is an algebraic regular map. It can be
described using Laufer’sduality asfollows, cf. [13],[16,p.1281] or[21]. First,bySerre duality,
H1(X, OX)∗Hc1(X, Ω2
X)H0(X\E,Ω2
X)/H0(X, Ω2
X). (2.2.2) An element of H0(X \E,Ω2
X)/H0(X, Ω2
X) can be represented by the class of a form
ω ∈H0(X\E,Ω2
X).Furthermore, anelement[α] of H1(X, OX) canberepresented by a Cechˇ cocyle αij ∈ O(Ui∩Uj), where {Ui}i isan opencover ofE, Ui∩Uj∩Uk =∅, and eachconnectedcomponentoftheintersections Ui∩Uj iseitheracoordinatebidisc B = {|u| < 2 , |v| < 2 } with coordinates (u,v), such that E ∩B ⊂ {uv = 0}, or a punctured coordinate bidisc B = { /2 < |v| < 2 , |u| < 2 } with coordi- nates (u,v), such that E ∩B = {u = 0}. Then, Laufer’s realization of the duality H0(X\E,Ω2
X)/H0(X, Ω2
X)⊗H1(X, OX)→C is [α],[ω]=
B
|u|=,|v|=
αijω. (2.2.3)
Inparticular, ifω hasnopolealongE inB,thentheB–contributionintheabovesum is zero.
This duality, via the isomorphism exp : H1(X, OX) → c−11(0) ⊂ H1(X, O∗X) = Pic(X), can be transported as follows, cf. [21]. (Here we present the case of a pecu- liar divisordue to the factthatthis versionwill be used later.) Considerthefollowing situation. We fix a smooth point p on E (p ∈ Ev), a local bidisc B p with local coordinates (u,v) such that B∩E = {u= 0}, B = {|u|,|v| < }. Weassume that a certain form ω ∈ H0(X \E,Ω2
X) has local equation ω =
i∈Z,j≥0ai,juivjdu∧dv in B.In thesame time, wefix a divisorD on X, whose local equation inB is v, ≥1.
Let Dt be another divisor, which is thesame as D in the complementof B and inB its localequationis (v+t+
k≥1,l≥0tk,lukvl),where allt,tk,l∈C and|t|,|tk,l|1.
Then Dt−D is the divisor D = div(g), where g := ((v+t+
k≥1,l≥0tk,lukvl)/v), supported in B. In particular, O(D) ∈ Pic0(X) ⊂H1(X, O∗X) can be represented by the cocycleg|B∗ ∈ O∗(B∗), where B∗ ={ /2<|v| < ,|u|< }. Therefore, log(g|B∗) is acocycleinB∗ representingitsliftinginto H1(X, OX).Thispaired withω givesfor Dt,[ω]:=exp−1OX(Dt−D), [ω]theexpression
·
|u|=,|v|=
log 1 +t+
k,ltk,lukvl v
·
i∈Z,j≥0
ai,juivjdu∧dv. (2.2.4)
If ω has no pole then Dt,[ω] = 0. As an example, assume that ω has the form (h(u,v)/uo)du∧dv withhregularand h(0,0)= 0,ando≥1,whileg= (v+tuo−1)/v and = 1,then
Dt,[ω]=
|u|=,|v|=
log 1+tuo−1 v
·h
uodu∧dv=c·t+{higher order terms} (c∈C∗).
(2.2.5) IfZ0 thenH0(X\E,Ω2
X)/H0(X, Ω2
X)H0(X, Ω2
X(Z))/H0(X, Ω2
X).Furthermore, ifω1,. . . ,ωpg arerepresentativesofabasisofthisvectorspaceandDtisconsideredasa pathinECa−E∗v(Z), thenDt →(Dt,[ω1],. . . ,Dt,[ωpg]) istherestriction ofthe AbelmaptoDt(associatedwithZ,andshiftedbytheimageofD) (cf.[21]).
3. Resolutionswithgenericanalyticstructure
3.1. Thesetup.Wefixatopologicaltypeofanormalsurfacesingularity.Thismeansthat we fix eitherthe C∞ orienteddiffeomorphism typeof the link,or, equivalently, oneof thedualgraphsofagoodresolution(allofthemareequivalentuptoblowingup/down rational(−1)–vertices).Weassumethatthelinkisarationalhomologysphere,thatis, thegraphisatreeofrationalvertices.
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 wecannot relyonit. Inparticular, the‘generic analyticstructure’,as a‘generic’ point of this moduli space, inthis way is not well–defined. However, in order to run/prove theconcretepropertiesregardinggenericanalyticstructures,insteadofsuchtheoretical definitionitwouldbeevenmuchbettertoconsideradefinitionbasedonalistofstability propertiesundercertainconcretedeformations(whosevaliditycouldbeexpectedforthe
‘generic’analytic structure inthepresenceof aclassifyingspace). Hence,forus inthis note, ageneric analytic structure will be a structure,which will satisfy such stability properties.Inordertodefine themitisconvenienttofixaresolution graphΓ andtreat deformationofanalyticstructuressupportedonresolutionspaceshavingdual graphΓ.
Thetypeof stability we wishto haveis thefollowing.The topological type(or, the graphΓ)determinesalowerboundforthepossiblevaluesofthegeometricgenus(which usually dependsonthe analytictype).Let MIN(Γ) bethe uniqueoptimalbound, that is, MIN(Γ)≤pg(X,o) for any singularity(X,o) which admitsΓ as aresolution graph, and MIN(Γ)=pg(X,o) for some(X,o). Thenone ofthe requirements forthe ‘generic analytic structure’ (Xgen,o) is that pg(Xgen,o) = MIN(Γ). (Inthe body of the paper MIN(Γ) will be determined explicitly.) However, we will need several similar stability requirementsinvolvingotherlinebundlesaswell(besidesthetrivialone,whichprovides pg).Fortheirdefinitionweneedapreparation.
3.2. Laufer’sresults.In this subsectionwe review someresultsof Laufer regarding de- formationsoftheanalyticstructureonaresolutionspaceofanormalsurfacesingularity withfixedresolutiongraph(anddeformationsofnon–reducedanalyticspacessupported onexceptionalcurves)[14].
First,letusfixanormalsurfacesingularity(X,o) andagoodresolutionφ: (X, E)→ (X,o) with reduced exceptionalcurve E =φ−1(o), whose irreducible decomposition is
∪v∈VEv and dual graph Γ. Let Iv be the ideal sheaf of Ev ⊂ X. Then for arbitrary positive integers {rv}v∈V one defines two objects, an analytic one and a topological (combinatorial) one. Atanalytic level, onesets theideal sheafI(r):=
vIvrv and the non–reduces spaceZ(r) withstructuresheafOZ(r):=OX/I(r) supportedonE.
Thetopologicalobjectisagraphdecoratedwithmultiplicities,denotedbyΓ(r).Asa non–decoratedgraphΓ(r) coincideswiththegraphΓ withoutdecorations.Additionally each vertex v has a ‘multiplicity decoration’ rv, and we put also the self–intersection decorationE2vwheneverrv>1.(Hence,thevertexvdoesnotinherittheself–intersection decoration of v if rv = 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–
reducedstructureallthemultiplicities{rv}v∈V,andadditionally,alltheself–intersection numbersEv2 forthosev’swhenrv>1 (see[14,Lemma3.1]).
WesaythatthespaceZ(r) hastopologicaltypeΓ(r).
Clearly,theanalyticstructureof(X,o),henceofXtoo,determineseach1–dimensional non–reducedspaceZ(r).Theconverseisalsotrueinthefollowing sense.
Theorem 3.2.1. [12,Th. 6.20], [14, Prop. 3.8] (a)Consider an abstract 1–dimensional space Z(r), whose topological type Γ(r) can be completed to a negative definite graph Γ (or, lattice L). Then there exists a 2–dimensionalmanifold X in which Z(r) can be embeddedwithsupportE suchthattheintersectionmatrixinheritedfromtheembedding E ⊂X is thenegativedefinite latticeL.In particular (sinceby Grauerttheorem [7] the exceptional locus E in X can be contracted to anormal singularity),any suchZ(r) is alwaysassociated withanormal surfacesingularity(asabove).
(b) Supposethatwehave twosingularities (X,o)and(X,o)withgoodresolutions as abovewiththesameresolutiongraphΓ.DependingsolelyonΓ,theintegers{rv}v maybe chosensolargethatifOZ(r) OZ(r),thenE⊂X andE⊂X havebiholomorphically equivalent neighbourhoodsviaamaptaking E toE.(Foraconcreteestimatehow large r shouldbe seeTheorem 6.20in[12].)
Inparticular,inthedeformationtheoryofX itisenoughtoconsiderthedeformations of non–reducedspacesoftypeZ(r).
Fix a non–reduced 1–dimensional space Z = Z(r) with topological type Γ(r). Fol- lowingLaufer andfortechnicalreasons(partlymotivatedbyfurtherapplicationsinthe forthcomingcontinuationsoftheseriesofmanuscripts)wealsochooseaclosedsubspace Y of Z (whose supportcan be smaller, it canbe even empty). More precisely, (Z,Y) locallyisisomorphicwith(C{x,y}/(xayb),C{x,y}/(xcyd)),wherea≥c≥0,b≥d≥0, a>0.TheidealofY inOZ isdenotedbyIY.
Definition 3.2.2.[14, 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 containingadistinguishedpoint0.
(ii)Overapointq∈QthefiberZqisthesubspaceofZdeterminedbytheidealsheaf λ∗(mq) (where mq isthemaximalidealofq). Z isisomorphicwithZ0,usually theyare identified.
(iii)λisatrivialdeformationofY (thatis,thereisaclosedsubspaceY ⊂ Z andthe restrictionofλtoY isatrivialdeformationofY).
(iv)λislocallytrivial inawaywhichextendsthetrivialdeformationλ|Y.Thismeans thatforantq∈Qandz∈ Z thereexistaneighbourhoodW ofzinZ,aneighbourhood V of z inZq, aneighbourhoodU of qinQ,and anisomorphismφ:W →V ×U such thatλ|W =pr2◦φ(compatiblywiththetrivializationofY from(iii)),where pr2 isthe secondprojection;formoresee[14].
Oneverifiesthatunderdeformations(withconnectedbasespace)thetopologicaltype ofthefibersZq,namelyΓ(r),remainsconstant(see[14, Lemma3.1]).
Definition3.2.3.[14,Def.2.4] Adeformationλ:Z →QofZ,fixingY,iscompleteat0 if,givenanydeformationτ:P →RofZ fixingY,thereisaneighbourhoodRof0 inR andaholomorphicmapf :R→Qsuchthatτ restrictedtoτ−1(R) isthedeformation f∗λ. Furthermore,λiscompleteifitiscompleteateachpointq∈Q.
Lauferprovedthefollowingresults.
Theorem3.2.4.[14,Theorems2.1,2.3,3.4,3.6] LetθZ,Y =HomZ(Ω1Z,IY)bethesheaf of germsofvector fieldson Z,whichvanish on Y,andletλ:Z →Qbe adeformation ofZ,fixingY.
(a)If theKodaira–Spencermap ρ0 :T0Q→H1(Z,θZ,Y) issurjectivethen λiscom- plete at0.
(b)If ρ0 issurjectivethenρq issurjectiveforallqsufficiently nearto0.
(c)Thereexistsadeformationλwith ρ0 bijective.In suchacaseinaneighbourhood U of 0 the deformationis essentially unique, and the fiberabove q is isomorphic toZ foronlyatmost countablymanyq inU.
It is worth to stress thatany two analytic typeson afixed topological type canbe connectedbyapath, whichcanbe coveredbyfinitely manydeformationsoftheabove type(see[14,Th.3.2]).
3.2.5. Functoriality.Let Z be a closed subspace of Z such that IZ ⊂ IY ⊂ OZ. Then there is a natural reduction of pairs (OZ,OY) → (OZ,OY). Hence, any de- formation λ : Z → Q of Z fixing Y reduces to a deformation λ : Z → Q of Z fixingY. Furthermore,ifλiscomplete thenλ is automaticallycomplete as well(since H1(Z,θZ,Y)→H1(Z,θZ,Y) isonto).
3.3. The‘0–genericanalyticstructure’.Wewishtodefinewhenistheanalyticstructureof afiberZq(q∈Q) ofadeformation‘generic’.Weproceedintwosteps.The‘0–genericity’
is the firstone(corresponding to the Chernclass l = 0), which will be definedinthis subsection.
Itisratheradvantageoustosetadefinition,whichiscompatiblewithrespecttoallthe restrictionsOZ → OZ.Inorderto dothis,letus fixthecoefficientsr˜={r˜v}v so large thatforthemTheorem3.2.1isvalid.Inthiswaybasicallywefixaresolution(X, E) and somelargeinfinitesimalneighbourhoodZ(˜r) associatedwithit.Moreover,letusalsofix acomplete deformationλ(˜r):Z(˜r)→QwhosefibershavethetopologicaltypeofΓ(˜r).
Next, we consider all the other coefficient sets r := {rv}v such that 0 ≤ rv ≤ r˜v for all v,not allrv = 0. Such achoice,by restrictionas in3.2.5, automaticallyprovides a deformationλ(r):Z(r)→Q. Thenset
Δ(0, r) :={q∈Q : hi(Z(r)q,OZ(r)q) is not constant in a neighbourhood ofq
for somei}. (3.3.1)
ThenΔ(0,r) isaclosed(reduced)propersubspaceofQ,see[34,35] (onecanusealso anargumentsimilar toLemma3.6.1writtenforl = 0).DefineΔ0(˜r):=∪rv≤˜rvΔ(0,r).
Then Δ0(˜r) isalsoclosedandΔ0(˜r)=Q.
Definition 3.3.2.We say that the fiber Z(˜r)q of λ(˜r) : Z(˜r) → Q is 0–generic if q ∈ Q\Δ0(˜r).
Next, we wish to generalize this definition for all Chern classes l ∈ L, or, for all
‘naturalline bundles’,asgeneralizationsofthetrivialbundlecorrespondingto l= 0.
3.4. Natural linebundles.Letusstartagainwithagoodresolutionφ: (X, E)→(X,o) of a normal surface singularity with rational homology sphere link, and consider the cohomology exactsequenceassociated withtheexponentialexactsequenceofsheaves
0→Pic0(X)−→ Pic(X)−→c1 H2(X, Z)→0. (3.4.1) Here c1(L)∈H2(X, Z)=L is thefirst Chern class ofL. Then, seee.g. [30,24], there exists auniquehomomorphism(split)s:L →Pic(X) ofc1 suchthatc1◦s=idands restrictedtoLisl → OX(l).Thelinebundless(l) arecallednatural linebundlesofX, and aredenotedbyOX(l).Forseveraldefinitionsofthemsee [24].E.g.,L isnaturalif and onlyifoneofitspowerhastheform OX(l) forsomeintegral cyclel∈Lsupported on E. Herewe recallanotherconstructionfrom [30,24], whichwillbe extendedlaterto thedeformationsspaceofsingularities.
Fixsomel∈LandletnbetheorderofitsclassinL/L.Thennlisanintegralcycle;
its reinterpretation as adivisor supportedon E will be denotedby div(nl). Weclaim thatthereexists adivisorD =D(l) in X suchthatonehasalinearequivalence nD∼ div(nl) and c1(OX(D)) = l. Furthermore, D(l) is unique up to linear equivalence, hencel → OX(D(l)) isthewished splitof(3.4.1).Indeed,sincec1 isonto,thereexists
adivisorD1 suchthatc1(OX(D1))=l.HenceOX(nD1−div(nl)) hasthe form (L) for some L ∈ Pic0(X) = H1(X, OX) = Cpg. Define D2 such that OX(D2) = n1L in H1(X, OX).ThenD1−D2works.TheuniquenessfollowsfromthefactthatPic0(X) is torsionfree.
Thefollowingwarningisappropriate.NotethatifX1isaconnectedsmallconvenient neighbourhoodof the union of someof the exceptional divisors (henceX1 also stands astheresolutionofthesingularity obtainedbycontractionofthatunionofexceptional curves)then onecanrepeat thedefinitionof naturalline bundlesat the levelof X1 as well.However,therestrictiontoX1ofanaturallinebundleofX(evenoftypeOX(l) with l integralcycle supportedonE)usually isnotnaturalon X1:OX(l)|X1 =OX1(R(l)) (where R : H2(X, Z) → H2(X1,Z) is the natural cohomological restriction), though theirChernclassescoincide.
Inthesequelwewilldealwiththefamilyof‘restrictednaturallinebundles’obtained byrestrictions ofOX(l).Evenifweneedto descendto a‘lower level’X1 withsmaller exceptional curve, or to any cycle Z with support included in E (but not necessarily E)our‘restricted naturalline bundles’ willbe associated with Chernclasses l ∈L = L(X) viathe restrictions Pic(X) → Pic(X1) or Pic(X) → Pic(Z) of bundles of type OX(l)∈Pic(X). Thisbasically meansthatwe fixatowerofresolution ofsingularities {X1}X1⊂X,or{OZ}|Z|⊂E,determinedbythe‘toplevel’X, andalltherestrictednatural linebundles,even atintermediatelevels,arerestrictionsfrom thetoplevel.
WeusethenotationsOX1(l):=OX(l)|X1 andOZ(l):=OX(l)|Z respectively.
3.5. Theuniversalfamilyofnaturallinebundles.Next,wewishtoextendthedefinition ofthelinebundlesOZ(l) tothetotalspaceofadeformation(atleastlocally,oversmall ballsinthecomplementofΔ0(˜r)).
We fix some Z = Z(˜r) with all ˜rv 0, supportedon E, such thatTheorem 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 inDefinition3.2.2 such thatall the fibershave thesame fixed topological typeΓ(˜r). Weconsider the discriminant Δ0(˜r) ⊂Q, and wefix some q0 ∈ Q\Δ0(˜r), and asmall ball U, q0 ∈U ⊂Q\Δ0(˜r). Above U the topologically trivial family of irreducible exceptional curves form the irreducible divisors {Ev}v, such that Ev above any point q ∈ U is the corresponding irreducible exceptional curve Ev,q of Xq. With the notationsof theprevious paragraph, if nl hasthe form
vnvEv write divλ(nl):=
vnvEv for the corresponding divisorinλ−1(U).Since U iscontractible, one has H2(λ−1(U),Z) = L and H1(λ−1(U),Z) = 0, hence the exponential exact sequenceonλ−1(U) gives
0→Pic0(λ−1(U))−→Pic(λ−1(U))−→c1 L →H2(λ−1(U),Oλ−1(U)). (3.5.1) Lemma3.5.2.H2(λ−1(U),Oλ−1(U))= 0andthefirstChernclassmorphismc1in(3.5.1) isonto.
