pairs of invariants of normal surface singularities

Pairs of Invariants ofNormal Surface Singularities

András Némethi

Rényi Institute of Mathematics, Budapest, Hungary

ICM, Rio de Janeiro, August 4, 2018

András Némethi Pairs of Invariants of Normal Surface Singularities

(X ,o) = analytic germ of a normal surface singularity / CM := X ∩S2N−1

ε = the link of (X ,o) (where (X ,o)⊂ (CN ,o)),oriented 3–manifold

Fix a good resolution X π−→ X ,• exceptional curve: π−1(o) = E = ∪v∈V Ev , Ev irreducible• dual graph: Γ• intersection form (Ev ,Eu)u,v

• L := (Z〈Ev 〉v ,( , )) = H2(X ,Z) negative definite lattice• L′ = H2(X ,Z) = HomZ(L,Z)⊂ L⊗Q dual lattice• discriminant group H := L′/L, H = Pontryagin dual of H• θ : H → H natural isomorphism [l ′] 7→ θ([l ′]) := e2π i(l ′,·)

• The Lipman cone: S ′ := {l ′ ∈ L′ : (l ′,Ev )≤ 0 ∀v}


Assume:

M is a rational homology sphere (H1(M,Z) is finite)

⇔ L′/L = H = H1(M,Z),

⇔ Γ is a tree and Ev ' P1 for all v


Analytic invariants ⇔ local algebra OX ,o , or⇔ complex analytic manifold X

Topological invariants ⇔ topological/smooth invariants of M, or⇔ combinatorial invariants of Γ, or of L

Goal: connect the analytic and topological invariants

Question: how the analytic and topological typesinfluence each other ?


Some invariants of the analytic type:

I Cohomology of holomorphic line bundles on XI Cohomology of natural line bundles on X [Okuma], [N.]I H–equivariant multivariable divisorial multifiltration F and

family of linear subspace arrangements, both indexed by L′

[Campillo, Delgado, Gusein–Zade]I H–equivariant multivariable Hilbert and Poincaré series of F

I Principle cycles (cut out by sections of line bundles orfunctions)

I geometric genus pg = h1(OX ), H–equivariant geometric genus

Problem: determine these invariants,find computational methods (e.g. surgery formulae)

Question: when are they topological ? (computable from Γ or M)


Some invariants of the topological type:

I Seiberg–Witten invariant of the linkI Heegaard–Floer (co)homology of the link [Ozsváth–Szabó], or

Monopole Floer (co)homology of the link[Kronheimer–Mrowka]

I Lattice (co)homology, graded roots [N.]I a topological family of linear subspace arrangements indexed

by L′ [N.]I H–equivariant multivariable zeta–function (series)

Problem: determine these invariants,find computational methods (e.g. surgery formulae)

Question: what are their peculiar/additional properties(valid for singularity links)?


Some definitions of the analytic part:‘Natural’ line bundles (with given Chern class):the cohomological exponential exact sequence splits uniquelyextending the natural L 3 l 7→ O(l) split over integral cycles L

L

��||yy

yy

y

0 // H1(X ,OX ) // Pic(X )c1 // L′ //O

oo_ _ _ 0.

This provides to each rational cycle l ′ ∈ L′ = H2(X ,Z) aline bundle O(l ′) ∈ Pic(X ), whose first Chern class c1 is l ′.


Example of natural line bundle:Let c : (X a,o)→ (X ,o) be the universal abelian cover of (X ,o),

πa : X a→ X a the normalized pullback of the resolution π by c ,

X a πa−→ (X a,o)yc

yc

X π−→ (X ,o)

The action of H on (X a,o) lifts to X a and gives an H-eigensheafdecomposition into natural line bundles

c∗OX a =⊕l ′∈C

O(−l ′), with C = {∑ l ′vEv ∈ L′, 0≤ l ′v < 1}

#C = #H

Notations: ∀h ∈ H set rh ∈ C with [rh] = h ∈ L′/L = H


c∗OX a =⊕h∈H

O(−rh)

pg (X a,o) = ∑hh1(O(−rh))

equivariant geometric genus: {h1(O(−rh))}hH–invariant, geometric genus:

rh = 0, O(0) = OX , h1(OX ) = pg (X ,o)


For fixed π, OX a,o inherits the L′–indexed divisorial filtration

F (l ′) := {f ∈ OX a,o |div(f ◦πa)≥ c∗(l ′)}.

h(l ′) = dimension of the θ([l ′])-eigenspace of OX a,o/F (l ′).

The equivariant divisorial Hilbert series is

H (t) = ∑l ′=∑ lvEv∈L′

h(l ′)t l11 · · · tlss = ∑

l ′∈L′h(l ′)tl

′ ∈ Z[[L′]].

h 7→Hh(t) = ∑[l ′]=h h(l ′)tl′ ↔ H-eigensheaf decomposition.

H (t) = ∑h∈H Hh(t)H0(t) = ∑l∈L h(l)tl ↔ H-invariants,

= the Hilbert series of OX ,o associated with theL–indexed {Ev}v–divisorial filtration.


Equivariant Poincaré series: (‘multi–graded version’)

P(t) =−H (t) ·∏v

(1− t−1v ) = ∑

l ′∈S ′p(l ′)tl

′ ∈ Z[[S ′]]

P(t) = ∑h∈H

Ph(t)

Fact:

P(t) ⇔ H (t)

advantage of P: it is supported on the positive cone S ′

��

��

...................................

S ′


Aside from generalized Ehrhart theory:Fix a series S(t) = ∑l ′∈L′ s(l ′)tl

′, supported on a positive cone,

Decompose as S = ∑h∈H Sh, where Sh(t) = ∑[l ′]=h s(l ′)tl′,

Consider the counting function of the coefficients of Sh

Qh(l ′) = ∑a∈L,a�0

s(l ′+a) ([l ′] = h).

��

��

��

....................................

.............l ′q

Assume that there exist: a real cone K ⊂ L′⊗R, l ′∗ ∈K ,a sublattice L⊂ L of finite index, anda quasipolynomial Qh(l) (l ∈ L) such that

Qh(l + rh) = Qh(l) ∀ l + rh ∈ (l ′∗+K )∩ (L+ rh).

Then we say that Sh(t) admits a quasipolynomial in K , namelyQh(l), and also an (equivariant, multivariable) periodic constantassociated with K , which is defined as

pcK (Sh(t)) := Qh(0).


The quasipolynomial of P:

Theorem: The counting function of P satisfies for l ′ = l + rh ∈ L′

∑a∈L,a 6≥0

p(l ′+a) = h(l ′)

=−h1(X ,O(−l ′)) + χ(l)− (rh, l) +h1(X ,O(−rh)).

Here χ(l) =−(l , l +K )/2, K = KX ∈ L′ is the canonical class.

If l ′ ∈ −K +S ′ then by the vanishing h1(O(−l ′)) = 0

∑a∈L,a�0

p(l ′+a) = χ(l)− (rh, l) +h1(X ,O(−rh)).

This quadratic function is the quasipolynomial of P inK = S ′

R, with periodic constant

pcS ′(Ph(t)) = h1(X ,O(−rh)).


The topological analogue:Set E ∗v ∈ L′ defined by (E ∗v ,Eu) =−δv ,u for all u ∈ V .Remark: L′ = Z〈E ∗v 〉v , and S ′ = Z≥0〈E ∗v 〉v .

The topological multivariable (‘zeta’) series, supported on S ′,is the Taylor expansion Z (t) = ∑l ′ z(l ′)tl

′at the origin of

∏v∈V

(1− tE ∗v )δv−2 (δv = valency of v in Γ)��

��

�

��*........................

E∗v

E∗u

Seiberg–Witten invariants of the link M.Let σcan be the canonical spinc–structure on X identified byc1(σcan) =−K , and let σcan ∈ Spinc(M) be its restriction to M.Spinc(M) is an H–torsor with action denoted by ∗.We denote by swσ (M) ∈Q the Seiberg–Witten invariants of Mindexed by the spinc–structures σ ∈ Spinc(M).


The quasipolynomial of Z :

Theorem [N.] The counting function of Zh(t) in the cone S ′Radmits the (quasi)polynomial

Qh(l) =−(K +2rh +2l)2 + |V |8

− sw−h∗σcan(M),

whose periodic constant is

(∗) pcS ′R(Zh(t)) = Qh(0) =−sw−h∗σcan(M)− (K +2rh)2 + |V |8

.

The right hand side of (∗) with opposite sign is called therh–normalized Seiberg–Witten invariant of M.


The analogy between P(t) and Z (t) culminates in the fact that formany analytic types P = Z , hence P and H have topologicaldescriptions.

Theorem: P(t) = Z (t) holds in the following cases:(a) rational singularities [Campillo—Delgado—Gusein-Zade](b) minimally elliptic singularities [N.](c) splice quotient singularities [N.] (this includes (a), (b) and

the weighted homogeneous case as well).(Splice quotient germs were introduced by Neumann–Wahl, laterredefined by Okuma.)

Remark: P = Z is not always true (e.g. for superisolated germs).Question: What is the limit of the identity P(t) = Z (t)?Answer: P(t) = Z (t) iff (X ,o) is splice quotient. [N.]


Recall: pcS ′(Ph)=h–part of the equivariant geometric genus,

pcS ′(Zh)=rh-normalized Seiberg–Witten invariant.

The identity of the right hand sides was conjectured/studied evenbefore the appearance of P = Z identity: this is theSeiberg–Witten Invariant Conjecture of Nicolaescu-N., asextension of the Casson Invariant Conjecture of Neumann-WahlWe say that (X ,o) satisfy the equivariant SWIC if for any h ∈ H

h1(X ,O(−rh)) =−sw−h∗σcan(M)− (K +2rh)2 + |V |8

This (automatically) extends to arbitrary natural line bundles as

h1(X ,O(l ′)) =−sw[l ′]∗σcan(M)− (K −2l ′)2 + |V |8

We say that (X ,o) satisfies the SWIC (for h = 0) if

pg (X ,o) =−swσcan(M)− K 2 + |V |8

.


Remark: The (equivariant) SWIC might hold even if P = Z fails.Theorem: The equivariant SWIC holds in the following cases:(a) rational singularities [Nicolaescu–N.](b) weighted homogeneous singularities [Nicolaescu–N.](c) splice quotient singularities [N.]

Additionally, the SWIC (for h = 0) folds for(a) suspensions {f (x ,y) + zN = 0} with f irred. [Nicolaescu–N.](b) hypersurface Newton non–degenerate germs [N.–Sigurdsson](c) superisolated singularities with one cusp[Bobadilla–Luengo–Melle-Hernández–N.] and [Borodzik–Livingston]

Since the identity of the SWIC is stable with respect to equisingulardeformations, the SWIC remains valid for such deformations of anyof the above cases. (The same is true for the identity P = Z .)


SWIC might fail, e.g. for some superisolated singularitiesbut still it indicates several information...

Classification problem of projective plane curvesC = {f (x ,y ,z) = 0} ⊂ P2, deg(f ) = dSing(C ) = {p1, ...,pr} has local knots K1, ...,Kr , Ki ⊂ S3.Question: For fixed d what local singularity types can be realized?

Construction of superisolated surface singularitiesf := f +generic > d terms, (X ,o) = {f = 0} ⊂ (C3,0) SI sing.pg = d(d −1)(d −2)/6, M = S3

−d (K1# · · ·#Kr )M QHS3 ⇔ C is rational cuspidalSWIC ⇔ interesting distribution property of the semigroups of localsingularity types Ki ⊂ S3 ⇒ obstructions for classification of curvesReinterpretations, generalizations created a lot of activity....[Bobadilla–Luengo–Melle-Hernández–N.], [Borodzik–Livingston],[Bodnár–Colaria–Golla], [Borodzik–Hom], [Borodzik–Moe], ...


More analogy between P(t) and Z (t): Surgery formulae(independently of Z = P, for any topological/analytic type)

Topological surgery formula (any Γ, any I ⊂ V fixed):Inv(Γ)=normalized Seiberg–Witten invariant associated with M(Γ),

Inv(Γ)− Inv(Γ\{I}) =−pc(Z |tv=1for all v 6∈ I )

[Braun-N.], [László-Nagy-N.] (this is not the surgery given by ‘exacttriangles of topological cohomology theories’, e.g. via HF± theory)Analytic surgery formula (‘almost any’ (X ,o), any I ⊂ V fixed):Inv(X (Γ))=equivariant geometric genus associated with X ,

Inv(X (Γ))− Inv(X (Γ\{I})) = pc(P|tv=1for all v 6∈ I )

[Okuma], [Nagy-N.](analytic theory is ‘rigid’, surgery formulae are surprising, rare....)


Linear subspace arrangements indexed by L′,as ‘lifts’ of the series P(t) and Z (t)

Fix a Chern class l ′ ∈ L′. For any analytic type supported on Γ

∀v : H0(OX (−l ′−Ev ))⊂ H0(OX (−l ′)) ∞−dim’l arrangement

truncation:

A(l ′) := (F (l ′)/F (l ′+E ))θ([l ′]) = H0(OX (−l ′))/H0(OX (−l ′−E ))

and for any v ∈ V a linear subspace in A(l ′)

Av (l ′) := H0(OX (−l ′−Ev )/H0(OX (−l ′−E ))

Aan(l ′) := {Av (l ′)}v in A(l ′) = ‘analytic’ subspace arrangement.[Campillo—Delgado—Gusein-Zade]


Fact: Aan(l ′) embeds into another arrangementSet T (l ′) := H0(OE (−l ′)), and for any v ∈ V set

Tv (l ′) := H0(OE−Ev (−l ′−Ev ))⊂ T (l ′).

Fact. The linear subspace arrangement Atop(l ′) := {Tv (l ′)}v inT (l ′) is topological, it depends only on Γ.

A(l ′) ↪→ T (l ′)

Av (l ′) = A(l ′)∩Tv (l ′) ↪→ TV (l ′)

∪∪ (∗)

Fixed topological type Γ ⇒ Atop(l ′)Any analytic type supported by Γ ⇒ A(l ′)⊂ T (l ′) (with ‘movingdimension and position’) which determines Aan(l ′) by (∗)


Theorem:

P(t) = ∑l ′

χtop(P(A(l ′)\∪vAv (l ′))) · tl ′

Z (t) = ∑l ′

χtop(P(T (l ′)\∪vTv (l ′))) · tl ′

E.g.• (X ,o) rational ⇒ Aan(l ′) = Atop(l ′) ∀ l ′ ⇒ P = Z

• Examples of splice quotient: A(l ′) t Atop(l ′) ∀ l ′ ⇒ P = Z(Recall: P = Z ⇒ SW–characterization of all h1(O(l ′)))

• P 6= Z ⇒ A(l ′) ‘small’ and it has ‘bad’ position in Atop(l ′)


Task: Characterize all linear subspace arrangements, indexed bysome lattice, which might appear as {Atop(l ′)}l ′ .

Task: For any fixed {Atop(l ′)}l ′ , characterize all linear subspacearrangements (up to isotopy, or even the corresponding modulispace in some flag manifolds) which might appear as {Aan(l ′)}l ′ in{Atop(l ′)}l ′ .

Task: Lift all the invariants above to the ‘motivic’ level in theGrothendieck ring. E.g., at topological level, set

Z (L,t) = ∑l ′∈S ′

[P(T (l ′)\∪vTv (l ′))] · tl ′ .

Then (with V =vertices, E=edges of Γ)

Z (L,t) =∏(u,v)∈E (1− tE ∗u − tE ∗v +LtE ∗u+E ∗v )

∏v∈V (1− tE ∗v )(1−LtE ∗v ).


There is a cohomology theory behind the topological part [N.]:

the lattice cohomology H∗(M,σ), σ ∈ Spinc(M)

• L⊗R has a cellular decomposition into cubes.0–cubes = L, 1–cubes = ‘segments’ [l , l +Ev ], etc.• For a characteristic element k ∈ L′, set χk(l) =−(l , l +k)/2,(Note Char = K +2L′ and Char/2L = Spinc(M))• Sn = the union of all q–cubes in {l ∈ L⊗R : χk(l)≤ n}.

Definition [N.]Hp(Γ,k) :=⊕nHp(Sn,Z).

Each Hq (q ≥ 0) is a graded Z[U]–module.U–action = cohomological restrictions given by the inclusions

. . .⊂ Sn−1 ⊂ Sn ⊂ Sn+1 ⊂ . . .


Properties of H∗ [N.]:• it is an invariant of M• it characterizes families of singularities (rational, elliptic,...)• ∃ cohomological long exact sequences (associated with surgeries)• vanishing : # of ‘bad’ vertices ≤ n ⇒ Hq = 0 for q ≥ n• reduction (to the – much smaller – sublattice of ‘bad’ vertices)• conjecture: H∗⇒ HF (true for # of ‘bad’ vertices ≤ 2)• ‘reduced’ H∗ = 0 ⇔ M is an L–space (reduced HF =0)• normalised Euler characteristic of H∗ is the Seiberg–Witten inv• multi-graded Euler characteristic ⇒ Z (t)• ∃ a modified version : path lattice cohomology• ∃ an improvement of H0 : graded roots


Analytic counterpart.... ?

The analytic type ‘makes a choice of a part’ of H∗...

E.g., at pg level, some possible choices made by pg are

• sw(M)= normalised Euler characteristic of H∗ (splice quotient)• normalised rank of reduced H0 (superisolated germs)• normalised Euler characteristic of path lattice cohomology(Newton non–degenerate case)

Problem: List all the choices that the analytic structure can make!(for pg , for P(t), for .....)


Thank you!


pairs of invariants of normal surface singularities

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