Poincaré series of algebraic links and lattice homology

0.2 MB·English

by  Eugene Gorsky

POINCARE´ SERIES OF ALGEBRAIC LINKS AND LATTICE HOMOLOGY
EUGENEGORSKYANDANDRA´SNE´METHI
ABSTRACT. We construct a version of the lattice homology for plane curve singularities us-
ingthenormalizationoftheircomponents. Weprovethatthe Poincare´ seriesoftheassociated
3 graded homologiescan be identified by an algebraic procedurewith the motivic Poincare´ se-
1 ries. Hence, for a plane curve singularity the following objects carry the same information:
0 the multi-variableAlexanderpolynomial, the multi-variableHilbertseries associated with the
2
normalization,themotivicPoincare´ series,andthePoincare´seriesofthenewlyintroducedlat-

n ticehomology. WealsoconjecturearelationofthislatticehomologywiththeHeegaard–Floer
a
homologyofthecorrespondinglink.
J

1
3 1. INTRODUCTION

] LetC = ∪r C beareducedplanecurvesingularityattheoriginin C2,whereC aretheir-
G i=1 i i
reduciblecomponents. Letγ : (C,0) → (C ,0)besomeuniformizationsofthesecomponents.
A i i
Wedefine r integer-valuedfunctionson theC–algebra O = O by
. C2,0
h
t v (f) = Ord(f (γ (t))),
a i i
m and aZr-indexedfiltration
[
J(v) = {f ∈ O|vi(f) ≥ vi forall i}.
1
v Campillo,DelgadoandGusein-Zadeconsideredin[4]thePoincare´ seriesofthefiltration J(v)
6
defined as theintegralwithrespect totheEulercharacteristicovertheprojectivizationof O:
3
6
7 (1.0.1) P(t ,...,t ) = tv1 ·...·tvrdχ.
. 1 r 1 r
1 ZPO
0 ThePoincare´ series P can berelated withseveraltopologicaland analyticalobjects.
3
For example, Campillo, Delgado and Gusein-Zade have shown in [4] that P basically is
1
: the multi-variable Alexander polynomial of the link of C (the intersection of C with a small
v
i three-dimensionalspherecentered attheorigin). Fortheprecise statementseeTheorem 2.1.1.
X
A more analytic invariant is the Hilbert series H associated with the filtration {J(v)} , it
v
r
a hascoefficientsh(v) = dimO/J(v),seeDefinition2.2.1. ItisknownthatP andH determine
each other, see[3, 11], insection2 wereprovethisfact.
Finally,weconsiderthemotivicversionofthePoincare´ series as well([5, 7, 12]):
v v
(1.0.2) P (t ,...,t ;q) = t 1 ·...·t rdµ.
g 1 r 1 r
ZPO
HereµdenotesthemotivicmeasureonPO[5,7]. SinceP canbederivedfromthecoefficients
g
ofH aswell,seesubsection2.4,P carriesthesameinformationas H andP. Nevertheless,all
g
these invariants capture different aspects and highlight different geometrical structures of the
local planecurvesingularities.
This note introduces another object, the lattice cohomology of the singularity. In [17] the
lattice cohomology of a normal surface singularity was introduced via the lattice provided by
itsresolutiongraph(orplumbinggraphofthelink). Theinvariantcreatedabridgebetweenthe
1
2 EUGENEGORSKYANDANDRA´SNE´METHI
analyticinvariantsof thesingularitywithseveral topologicalones ofthelink,namely Seiberg-
Witten and Heegaard-Floertheories. The goal of the present construction is similar; neverthe-
less here we rely on a different lattice: the needed weight function is provided by the normal-
ization, by the coefficients h(v). (In order to eliminate any further confusion, we will call the
present invariant“latticehomologyvianormalization”).
In short, the definition runs as follows. The lattice complex L− is generated over Z[U] by
elementarycubes (cid:3)ofalldimensionsinRr,withverticesinthelatticeZr. Forsuchacubewe
defineh((cid:3)) = maxx∈(cid:3)∩Zr h(x). Thedifferentialisdefined as
∂ ((cid:3)) = ε Uh((cid:3))−h((cid:3)i)(cid:3) ,
U i i
i
X
where (cid:3) aretheorientedboundary cubesof (cid:3), andε are thecorrespondingsigns.
i i
The complex L− is naturally Zr-filtered: the subcomplex L−(v) is generated by the cubes
contained in the positivequadrant originatingat v. Ourmain theorem describes the homology
ofthesubcomplexesL−(v)and theassociatedgraded complexesgr L− forall v.
v
Theorem 1.0.3. (1) The homology of L−(v) is given by H∗(L−(v)) = Z[U][−2h(v)].
Hence, thePoincare´ series forthishomologyequals
t−2h(v)
P (t) = .
L−(v) 1−t−2
(2) ThePoincare´ polynomialofthehomologyofgr L− isgiven bytheformula
v
P (t) = (−t)−h(v)H (−t−1),
gr L− v
v
where H (q)is thecoefficient inthemotivicPoincare´ series:
v
P (t ,...,t ;q) = H (q)tv1 ···t vr.
g 1 r v 1 r
v
X
WeprovethepartsofthistheoremasTheorem4.2.2andTheorem4.3.2. Themainingredient
isthestructureoftheOrlik-Solomonalgebrasassociatedwithcentralhyperplanearrangements.
Corollary1.0.4. Thepolynomial(−1)h(v)H (−t) hasonlynon-negativecoefficients.
v
Corollary1.0.5. ThegeneratingfunctionfortheEulercharacteristicsofgr L−,
v
P (−1)·tv1 ···tvr,
grvL− 1 r
v
X
equalsP(t ,...,t ). Hence, itcan beexpressed bythemulti-variableAlexander polynomial.
1 r
Motivatedby Theorem1.0.3and Corollary 1.0.5, weformulatethefollowing
Conjecture1.0.6. ThegradedhomologyofgrL− isisomorphictotheHeegaard-Floerhomol-
ogyHF− ofthelinkof C ([21,22, 23, 24], seealso[26]).
We compare various structural properties of both homology theories and show that the con-
jectureholds for r = 1. We also computethehomologyof grL− in otherexamplesand match
themto theHeegaard-Floerhomology.
POINCARE´ SERIESOFALGEBRAICLINKSANDLATTICEHOMOLOGY 3
ACKNOWLEDGEMENTS
The authors are grateful to S. Gusein-Zade, J. Moyano-Fernandez, P. Ozsva´th and J. Ras-
mussen for the useful discussions. We also thank the Oberwolfach Mathematical Institute,
where part of this work was done, for the hospitality. The research of E. G. is partially sup-
ported by the grants RFBR-10-01-678, NSh-8462.2010.1and the Simons foundation. A. N. is
partiallysupportedbyOTKA Grants81203and 100796.
2. HILBERT FUNCTION
2.1. The Poincare´ seriesofmulti-index filtrations. Let usfix alocalplanecurvesingularity
withr irreduciblecomponentsasintheintroduction. Set K = {1,...,r}. Lete denotethei-
0 i
thcoordinatevectorin Zr. ForasubsetK ⊂ K wewritee = e ande = e = e .
0 K i∈K i K0 i
Foravectorv weset v = v e .
K i∈K i i P P
Furthermore, let P be the series defined in (1.0.1), and ∆(t ,...,t ) be the multi-variable
1 r
P
Alexanderpolynomialofthelinkof C. Then thefollowingholds.
Theorem 2.1.1 ([4]). If r = 1,then P(t)(1−t) = ∆(t), whileif r > 1, then
(2.1.2) P(t ,...,t ) = ∆(t ,...,t ).
1 r 1 r
2.2. The Hilbertseries ofmulti-index filtrations. Weset apartial orderon Zr by
u (cid:22) v ⇐⇒ u ≤ v forall i.
i i
We define r integer-valued functions on O = O by v (f) = Ord (f (γ (t))), and a
C2,0 i 0 i
Zr-indexedfiltration
J(v) = {f ∈ O | v(f) (cid:23) v}.
NotethattheidealsJ(v)arealsodefinedfornegativevaluesofv. Thisfiltrationisdecreasing
inthesensethatifu (cid:22) v, thenJ(u) ⊃ J(v).
Definition 2.2.1. Let h(v) = dimO/J(v). Wedefine theHilbert series ofa multi-indexfiltra-
tionJ as theseries
(2.2.2) H(t ,...,t ) = h(v)·tv1 ···tvr.
1 r 1 r
v
X
Furthermore, we define the set S := {v ∈ Zr : there existsf ∈ O withv(f) = v} as well.
It iscalled thesemigroupoff. Fromthedefinitiononegetsthefollowingelementaryfacts.
Lemma 2.2.3. S = {v ∈ Zr : h(v +e ) > h(v) forevery i = 1,...,r}. Furthermore, fix
≥0 i
any0 (cid:22) v ande . Thenh(v+e ) = h(v)+1ifthereisanelementu ∈ S suchthatu = v and
i i i i
u ≥ v for j 6= i. Otherwiseh(v +e ) = h(v).In particular,H andS determineeach other.
j j i
Proof. If h(v + e ) > h(v) for all i, then there exist functions f such that v (f ) = v and
i i i i i
v (f ) ≥ v for j 6= i. Therefore v( r λ f ) = v for generic coefficients λ . For the second
j i j i=1 i i i
partnotethath(v+e )−h(v) = dimJ(v)/J(v+e ). Thisquotientspaceistrivialifthereisno
i i
functionf suchthatv (f) = v andPv (f) ≥ v forj 6= i. Otherwiseitisone-dimensional. (cid:3)
i i j j
Thefollowinglemmais avariationoftheanalogousstatementfrom[3].
Lemma 2.2.4. Theseries H andP arerelatedby
(2.2.5) P(t ,...,t ) = −H(t ,...,t )· (1−t−1).
1 r 1 s i
i
Y
4 EUGENEGORSKYANDANDRA´SNE´METHI
Atthelevel ofcoefficientsthisreads asfollows. Definetheintegersπ(v) bytheequation
(2.2.6) P(t ,...,t ) = tv1 ...tvr ·π(v).
1 r 1 r
v
X
Then
(2.2.7) π(v) = (−1)|K|−1h(v +e ).
K
KX⊂K0
We will invert the equation (2.2.7), namely, we express the integers h(v) from the Poincare´
series P . Firstnotethat
C
(2.2.8) h(v) = h(max(v,0)),
where max(v,0) = v if v > 0 and = 0 otherwise. Hence it is enough to determine h(v) for
i i i
0 (cid:22) v only. Theseries h(v)tv willbedenotedby H(t)| .
0(cid:22)v 0(cid:22)v
Next, for a subset K = {i ,...,i } ⊂ K , K 6= ∅, consider a curve C = ∪ C . As
1 |K| 0 K i∈K i
P
above, this germ defines a |K|-index filtration of O , hence it provides the Hilbert series
C2,0
H ofC invariables {t } :
CK K i i∈K
H (t ,...,t ) = tvi1 ...tvi|K| ·hK(v).
CK i1 i|K| i1 i|K|
v
X
By construction,for K ⊂ K onehasH (t ,...,t ) = H (t ,...,t )| ; or
0 CK i1 i|K| C 1 r ti=0i6∈K
(2.2.9) ifv = 0forall i ∈/ K, then hK(v) = h(v).
i
Analogously,wehavethePoincare´ series of C :
K
PK(t ,...,t ) = P (t ,...,t ) = tvi1 ...tvi|K| ·πK(v).
1 r CK i1 i|K| i1 i|K|
v
X
Bydefinition,forK = ∅wetakeπ∅(v) = 0.Notethatfromh(v)onecanrecoveranyPK: first
computinghK by(2.2.9) andthen recovering PK from HK using(2.2.7).
Althoughby[29]P determinestheembeddedtopologicaltypeofC,hencealltheseriesPK
forall subsets K ⊂ K as well, theanalogueof(2.2.9)is nottrueforthepair πK(v)and π(v).
0
Indeed, the“restrictingrelation”([28])is oftype
1
(2.2.10) PK0\{1}(t ,...,t ) = P(t ,...,t )| · ,
2 r 1 r t1=1 (1−t(C1,C2))···(1−t(C1,Cr))
2 r
where (C ,C ) denotestheintersectionmultiplicityattheoriginofthecomponentsC andC ,
i j i j
i 6= j. Thisalsoshowsthatgiventheseintersectionmultiplicitiesonecan recoverbyinduction
PK from P easily.
The following theorem was proved in [11] as Corollary 4.3. For the completeness of the
expositionwepresentaproofofit(which isslightlydifferent from[11]).
Theorem 2.2.11. Thefollowingequationholds:
1
(2.2.12) H(t ,...,t )| = (−1)|K|−1 t ·PK(t ,...,t ).
1 r 0(cid:22)v r i 1 r
(1−t )
i=1 i KX⊂K0 (cid:16)iY∈K (cid:17)
Proof. The identity(2.2.12) forthQe generating series is equivalentto thefollowingidentityfor
theircoefficients:
(2.2.13) h(v) = (−1)|K|−1 πK(u).
KX⊂K0 0(cid:22)u(cid:22)XvK−eK
POINCARE´ SERIESOFALGEBRAICLINKSANDLATTICEHOMOLOGY 5
Wewillprovetheidentity(2.2.13)byatwo-stepinduction: thefirstinductionisbythenumber
ofcomponentsr, and thesecond one(forfixed r)is overthenorm |v| = v .
i
Ifr = 1,then(2.2.7)impliesthatπ(v) = h(v+1)−h(v).Therefore π(u) = h(v)
P0≤u≤v−1
sinceh(0) = 0.
P
Letusprove(2.2.13)forthecasewhenatleastoneofcoordinatesv vanish. Wecanassume
i
thatv = 0. By theinductionassumptionweget
r
h(v) = h{1,...,r−1}(v ,...,v ) = (−1)|K|−1 πK(u).
1 r−1
K⊂{X1,...r−1} 0(cid:22)u(cid:22)XvK−eK
On the other hand, in (2.2.13) for all K ⊂ K with r ∈ K we get the vacuous restriction
0
0 ≤ u ≤ −1, henceweget nontrivialcontributiononlyfromterms with K ⊂ {1,...r −1}.
r
Suppose now that v has no vanishing coordinates and we proved (2.2.13) for v −e for all
K
non-emptysubsetsK ⊂ K . Wecan rewrite(2.2.7)as alinearequationon h(v):
0
π(v −e) = (−1)r−|K|−1h(v −e ).
K
KX⊂K0
By theinductionassumptionforK 6= ∅ wehave
h(v −e ) = (−1)|M|−1 πM(u),
K
MX⊂K0 0(cid:22)u(cid:22)(vMX−eK∩M−eM)
and weshouldestablishthesameidentityforK = ∅. Thereforewehavetoprovethat
(2.2.14) π(v−e) = (−1)r−|K|+|M| πM(u).
KX⊂K0MX⊂K0 0(cid:22)u(cid:22)(vMX−eK∩M−eM)
Let us fix M and u (cid:22) v − e and sum the expression (−1)|K| over all sets K ⊂ K such that
0
u ≤ v −2 for i ∈ K ∩M. This sum does not vanish iff M = K and u = v −1 for all i.
i i 0 i i
Thisproves(2.2.14). (cid:3)
Corollary 2.2.15. The restricted Hilbert function H(t)| of a multi-component curve is a
0(cid:22)v
rationalfunctionwith denominator r (1−t )2.
i=1 i
Example2.2.16. ConsiderthesinguQlarityA . ItsPoincare´ seriesis1+t t +···+(t t )n−1,
2n−1 1 2 1 2
and thePoincare´ series ofbothitscomponentsequals 1/(1−t). TheHilbertseries is
1 t t
H(t ,t )| = 1 + 2 −t t (1+...+(t t )n−1) .
1 2 0(cid:22)v 1 2 1 2
(1−t )(1−t ) 1−t 1−t
1 2 (cid:18) 1 2 (cid:19)
Therefore, fornon-negativeintegers (v ,v ) onehas
1 2
max(v ,v ), if min(v ,v ) < n
1 2 1 2
h(v) =
v +v −n, otherwise.
( 1 2
Figure1illustratesthisformulafortheHilbertfunctionforA singularity. Thepointscorre-
3
spondingto thesemigroup S aremarked inbold.
Example2.2.17. Considerthesingularity D ,that is,equationy ·(x2 −y3) = 0. Then
5
1 1−t +t2
P(t ,t ) = 1+t t3,P (t ) = ,P (t ) = 2 2.
1 2 1 2 1 1 1−t 2 2 1−t
1 2
6 EUGENEGORSKYANDANDRA´SNE´METHI
4 4 4 5 6
3 3 3 4 5
2 2 2 3 4
1 1 2 3 4
0 1 2 3 4
FIGURE 1. ValuesoftheHilbertfunctionfor A
3
4 4 4 5 6 7
3 3 3 4 5 6
2 2 3 4 5 6
1 1 2 3 4 5
1 1 2 3 4 5
0 1 2 3 4 5
FIGURE 2. ValuesoftheHilbertfunctionfor D
5
Onecan check that h(v ,v )fornon-negativev and v is (seeFigure2 too):
1 2 1 2
v , ifv < 3,v > 0
1 2 1
v +1, ifv = 3,v > 0
 1 2 1
h(v ,v ) = v −1, ifv < 2,v ≥ 2
1 2  2 1 2

v +v −3, ifv ≥ 2,v ≥ 4
1 2 1 2
0,1,1, ifv = 0 and v = 0,1,2.
 1 2





2.3. Some properties of the Hilbert function. For any i ∈ K let µ and δ (respectively
0 i i
µ(C)andδ(C))betheMilnornumberandthedeltainvariantofC (respectivelyofC)([1,10]).
i
Then, cf. [10], µ = 2δ , andµ(C)+r −1 = 2δ(C). Definel = (l ,...,l ) by
i i 1 r
l = µ + (C ,C ).
i i j i
j6=i
X
It isknownthattheAlexanderpolynomialissymmetricinthe followingsense
∆(t−1,...,t−1) = t1−li ·∆(t ,...,t ) for r > 1,
1 r i 1 r
(cid:16)Y (cid:17)
∆(t−1) = t−µ(C)∆(t) for r = 1.
Thiscan becompared withthefollowing.
POINCARE´ SERIESOFALGEBRAICLINKSANDLATTICEHOMOLOGY 7
Lemma 2.3.1. ([12, 6])TheHilbertfunctionsatisfiesthefollowingsymmetryproperty:
(2.3.2) h(l −v)−h(v) = δ(C)−|v|,
where |v| = r v .
i=1 i
Remark 2.3P.3. Consider v (cid:23) l. It follows from (2.3.2) that h(v) = |v| − δ(C). This can be
verified by the identity (2.2.13) as well. Indeed, for |K| = 1 we have πK(u) =
0≤ui≤vi−1
v −δ(C ),for|K| = 2wehave π{i,j}(u ,u ) = P{i,j}(1,1) = (C ,C ),whilePK(e ) = 0
i i i j i jP K
for|K| > 2. Hence h(v) = (v −δ(C ))− (C ,C ) = |v|−δ(C).
i i i i,j i j
P
Theidentityh(v) = |v|−δ(C)andLemma2.2.3givethatv ∈ S wheneverv (cid:23) l (hence, in
P P
fact, l is theconductorof S). Thisfact combinedagain with2.2.3gives
Corollary2.3.4. Foranybasicvectore andn ≥ l onehash(v+(n+1)e )−h(v+ne ) = 1.
i i i i
Thenextpropertyis notusedinthepresentnote, neverthelessweadditsinceitcontainsthe
keyobservationwhichwillbeusedin aforthcomingarticlewithapplicationsindeformations.
Proposition 2.3.5. (a) Let f′ be a deformation of f, where f and f′ are irreducible. Then
h (v) ≤ h (v)for everyv.
f f′
(b) Let a (possibly reducible) curve C′ be a deformation of an irreducible curve C. Then
h (v) ≥ h (|v|)forevery v.
C′ C
Proof. (a) Forany function g theintersection multiplicityof g with f is greaterorequal to the
intersectionmultiplicityof g withf′. Therefore J (v) ⊂ J (v)and
f′ f
h (v) = codimJ (v) ≤ codimJ (v) = h (v).
f f f′ f′
(b)ConsiderafunctiongfromJ (v). ItsordersonthecomponentsofC′aregreaterorequalto
C′
thecorrespondingcomponentsofv,hencetheintersectionmultiplicityofgwithC′isgreateror
equalto|v|. Hencetheintersectionmultiplicityofg withC isgreaterorequalto |v|. Therefore
J (v) ⊂ J (|v|)and h (|v|) = codimJ (|v|) ≤ codimJ (v) = h (v). (cid:3)
C′ C C C C′ C′
2.4. MotivicPoincare´ series. Following[5], wedefinethemotivicPoincare´ series ofaplane
curvesingularity C = ∪r C bytheformula
i=1 i
1
P (t ,...,t ;q) = tv1 ···tvr (−1)|K|qh(v+eK).
g 1 r 1−q 1 r
vX∈Zr KX⊂K0
It follows from the results of [5] that this definition agrees with the integral (1.0.2). In [7],
and independently in [12], the following properties are proved: P (t ,...,t ;q) is a rational
g 1 r
functionwithdenominator r (1−t q),hence
i=1 i
r
Q
P (t ,...,t ;q) := P (t ,...,t ;q)· (1−t q)
g 1 r g 1 r i
i=1
Y
isapolynomial. Moreover, P satisfies thefunctionalequation
g
1 1
P ( ,..., ;q) = q−δ(C) t−li ·P (t ,...,t ;q).
g qt qt i g 1 r
1 r
i
Y
In [12] this equation was deduced from (2.3.2). Moreover, one can check that for q = 1
one has P (t ,...,t ;q = 1) = P(t ,...,t ). An explicit algorithm of the computation of
g 1 r 1 r
P (t ,...,t ;q) intermsoftheembeddedresolutiontreeof C isprovidedin[7].
g 1 r
8 EUGENEGORSKYANDANDRA´SNE´METHI
2.5. Conclusion. By the above discussions, the following objects associated with a plane
curvesingularitycarry thesameamountofinformation: ∆, S, H,P andP .
g
3. CENTRAL HYPERPLANE ARRANGEMENTS
3.1. Matroids and rank functions.
Definition 3.1.1. (A) ([27]) Let K be a finite set. A function ρ, assigning a non-negative
0
integertoany subset K ⊂ K , iscalled arankfunction,if
0
(1) 0 ≤ ρ(K) ≤ |K|(where |K|denotesthecardinalityof K).
(2) IfK ⊂ K thenρ(K ) ≤ ρ(K ).
1 2 1 2
(3) Foreverypairofsubsets K and K onehasthefollowinginequality:
1 2
ρ(K ∩K )+ρ(K ∪K ) ≤ ρ(K )+ρ(K ).
1 2 1 2 1 2
(B)Amatroidisa finiteset witharank functionon it.
(C)Thecharacteristicpolynomialofamatroid M = (K ,ρ) isdefined as
0
χ (t) = (−1)|K|tρ(K0)−ρ(K).
M
KX⊂K0
Remark 3.1.2. Some authors define the characteristic polynomial using the Mo¨bius function
ofamatroid. Thisdefinitionisequivalenttothepresent one,seee.g. [27, Theorem2.4].
Leth(v)denotetheHilbertfunctionofaplanecurvesingularity. LetusfixK = {1,...,r}
0
and forevery v considerthefollowingfunctiononsubsetsof K :
0
ρ (K) := h(v +e )−h(v) = dimJ(v)/J(v +e ).
v K K
Proposition3.1.3. Foreveryv thefunctionρ is arankfunctiononK .
v 0
Proof. Property(1)followsfromLemma2.2.3. Next,forK ⊂ K onehasρ (K )−ρ (K ) =
1 2 v 2 v 1
dimJ(v + e )/J(v + e ) ≥ 0. To prove the third property notice that J(v + e ) =
K1 K2 K1∪K2
J(v +e )∩J(v +e ), and J(v +e )+J(v+e ) ⊂ J(v +e ), hence
K1 K2 K1 K2 K1∩K2
ρ (K )+ρ (K )−ρ (K ∪K ) = dimJ(v)/(J(v +e )+J(v +e )) ≥ ρ (K ∩K ).
v 1 v 2 v 1 2 K1 K2 v 1 2
(cid:3)
We will call ρ the rank function for a the “local matroid M ”. In the space J(v) we have r
v v
subspaces J(v +e ) of codimension 0 or 1. If v ∈ S, then the set of functions with valuation
i
v can be represented as a complement of a hyperplane arrangement (cf. [12]). If v 6∈ S, then
J(v) = J(v +e ) forsomei , cf. Lemma2.2.3. Moreover,onehasthefollowinglemma.
i0 0
Lemma 3.1.4. Assume that J(v) = J(v + e ) for some i ∈ K . Then J(v + e ) = J(v +
i0 0 0 K
e +e ) foranyK ⊂ E with K 6∋ i . Hence
K i0 0
χMv(t) = (−1)|K| ·th(v+eK0)−h(v+eK) = 0.
KX⊂K0
Proof. Use J(v + e + e ) = J(v + e ) ∩ J(v + e ) for the first statement, and pairwise
K i0 K i0
cancellationforthesecond one. (cid:3)
POINCARE´ SERIESOFALGEBRAICLINKSANDLATTICEHOMOLOGY 9
3.2. Central hyperplane arrangements andOrlik-Solomonalgebras.
Letusrecallsomefactsaboutcentralhyperplanearrangements. LetV beavectorspaceand
let H = {H ,...,H } be acollection oflinearhyperplanes in V. Fora set K = {α ,...,α }
1 r 1 k
wedefinetherank function
ρ(K) = codim(H ∩...∩H ).
α1 αk
SimilarlytoProposition3.1.3,onecancheckthatρisarankfunctionforacertainmatroid. Let
usdenoteby χ (t) itscharacteristicpolynomial.
H
Lemma3.2.1. TheclassintheGrothendieckringofvarietiesofthecomplementof ∪ H inV
i i
equals
[V \∪r H ] = LdimV−ρ(K0)χ (L),
i=1 i H
where Ldenotestheclassof theaffineline.
Proof. Followsfrom theinclusion-exclusionformula:
[V \∪r H ] = (−1)|K| [∩ H ] = (−1)|K| LdimV−ρ(K). (cid:3)
i=1 i α∈K α
KX⊂K0 KX⊂K0
To the arrangement H one can associate the corresponding Orlik-Solomon algebra. Con-
sider the anticommutative algebra E generated by the variables z ,...,z corresponding to
1 r
hyperplanes. For any set K ⊂ K we consider the monomial z = z ∧ ... ∧ z , where
0 K α1 αk
K = {α ,...,α }.Wecan equipE withthenatural differential ∂ sendingz to 1,namely
1 k i
k
∂(z ) = (−1)i−1z .
K K\{αi}
i=1
X
Definition 3.2.2. We call the set K dependent, if the equations of the corresponding hyper-
planesare linearlydependent. Otherwise K iscalled independent.
TheOrlik-SolomonidealI istheidealinE generatedbytheelements∂z foralldependent
K
setsK. TheOrlik-Solomonalgebrais thequotient A = E/I.
Theorem 3.2.3. ([18, Theorem 5.2]) The integral cohomology ring of the complement V \
∪r H is isomorphic to the Orlik-Solomon algebra E/I. It has no torsion, and its Poincare´
i=1 i
polynomialis givenbytheformula
P(H,t) = (−t)r(E) ·χ (−t−1) = (−1)|K|(−t)ρ(K).
H
KX⊂K0
As a corollary, we conclude that the homology of V \ ∪r H is defined by its class in the
i=1 i
Grothendieckring. Thesameistrueforitsprojectivization(seebelow).
Wewillneed thefollowingdeformationsofthedifferential on E.
Definition 3.2.4. Let usdefine thefollowingoperator:
k
∂ : E[U] → E[U], ∂ (z ) = (−1)i−1Uρ(K)−ρ(K\{αi})z ,
U U K K\αi
i=1
X
where U isaformal parameterand K = {α ,...,α }.
1 k
Notethatρ(K)−ρ(K\{α }) ∈ {0,1},hence∂ decomposesintoasumoftwocomponents
i U
(3.2.5) ∂ = ∂ +U∂ , with ∂ +∂ = ∂.
U 0 1 0 1
10 EUGENEGORSKYANDANDRA´SNE´METHI
Lemma 3.2.6. The operator ∂ is a differential on E[U], that is, ∂2 = 0. In particular, the
U U
followingidentitieshold:
∂2 = ∂2 = 0, ∂ ∂ +∂ ∂ = 0.
0 1 0 1 1 0
Proof. Straightforward. (cid:3)
Let J and J⊥ denote the subspaces of E spanned by the elements z for all dependent,
K
respectivelyindependentsubsets K. Clearly E = J ⊕J⊥.
Lemma 3.2.7. Thefollowingstatementshold:
(a) ([18], Lemma 2.7) I = J +∂J.
(b) ∂ J⊥ = 0, henceIm∂ = ∂ J.
0 0 0
(c) ∂ J ⊂ J, henceI = J +∂J = J +∂ J.
1 0
(d) Ker∂ = J⊥ +Im∂ .
0 0
(e) There existsubspacesA ⊂ J,B ⊂ J⊥ suchthatIm∂ = A⊕B.
0
Proof. The claims (b) and (c) are clear. Let us prove(d). The inclusionJ⊥ +Im∂ ⊂ Ker∂
0 0
is also clear, hence we need to prove that if ∂ (φ) = 0 then there exists φ ∈ J⊥ such that
0
φ−φ ∈ Im(∂ ).
0
Let us call z essential in a monomial z ∧ z , if ρ({i} ⊔ K) = ρ(K) +e1, and redundant
i i K
otherewise. Letusdecomposeφ = z ∧φ +z ∧φ +φ ,wherez isessentialineverymonomial
1 1 1 2 3 1
ofz ∧φ ,redundant ineverymonomialof z ∧φ , andφ containsno z .
1 1 1 2 3 1
Then
∂ (φ) = z ∧ψ +φ +∂ (φ )
0 1 2 0 3
forsomeψ, and neitherφ nor∂ (φ ) containz . Hence, if∂ (φ) = 0then φ = −∂ (φ ).
2 0 3 1 0 2 0 3
Since z is redundant in every monomial in z ∧ ∂ (φ ), it is redundant in every monomial
1 1 0 3
inz ∧φ . Therefore
1 3
∂ (z ∧φ ) = φ −z ∧∂ (φ )+z ∧η,
0 1 3 3 1 0 3 1
where z is essential in every monomial of z ∧ η. Indeed, if α ∈ K, z is redundant in
1 1 j αj
K ∪{1}and essentialinK, then z is essentialinK ∪{1}\{α }.
1 j
Weconcludethat
φ−∂ (z ∧φ ) = z ∧(φ −η)
0 1 3 1 1
andz isessentialineverymonomialintherighthandside. Now∂ (φ) = −z ∧∂ (φ −η) = 0,
1 0 1 0 1
hence ∂ (φ − η) = 0. Then we can repeat the procedure inductively replacing φ by φ − η,
0 1 1
and z by z , etc. At the end we reduce φ modulo Im(∂ ) to an element of E where all z are
1 2 0 i
essential;suchan element belongsto J⊥.
Next, we prove (e). Recall that Im∂ = ∂ J and K is dependent iff ρ(K) < |K|. If
0 0
the monomial z appears in ∂ (z ) then ρ(K) = ρ(K′) and |K′| = |K| − 1. Therefore
K′ 0 K
∂ (z ) ∈ J⊥ ifρ(K) = |K|−1,and ∂ (z ) ∈ J otherwise. (cid:3)
0 K 0 K
Wewillneed thefollowingtwo resultsintheconstructionof latticehomology.
Theorem 3.2.8. The Orlik-Solomon algebra is isomorphic to the homology of the differential
∂ :
0
H (E,∂ ) = E/I ≃ H∗(V \∪r H ).
∗ 0 i=1 i
Proof. By Lemma3.2.7onehastheisomorphisms
Ker∂ = J⊥ +Im∂ , Im∂ = ∂ J,
0 0 0 0
hence
H (E,∂ ) = (J⊥ +∂ J)/∂ J ≃ J⊥/(∂ J ∩J⊥) ≃ E/(J +∂ J).
∗ 0 0 0 0 0