On lattice-invariants of complex algebraic surfaces and their applications
On lattice-invariants of complex algebraic surfaces and their applications
Ichiro Shimada
Hiroshima University
December 28, 2009, Busan
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On lattice-invariants of complex algebraic surfaces and their applications
We work overC.
Abstract
We study some lattice-theoretic topological invariants of complex algebraic surfaces inP3, and
present an application to the construction of examples ofweak (arithmetic) Zariski pairs of surfaces with only RDPs inP3. This is a joint work with A. Katanaga and M. Oka.
On lattice-invariants of complex algebraic surfaces and their applications Zariski pairs
LetS andS0 be reduced (possibly reducible) hypersurfaces inPn. Definition
(1) We say thatS andS0 are of the same configuration typeand write
S ∼cfgS0
if there are tubular neighborhoodsT ⊂Pn ofS andT0 ⊂Pn ofS0, and a homeomorphism (T,S) →∼ (T0,S0) that preserves the degrees of the irreducible components ofS andS0.
(2) We say thatS andS0 have the same embedding topology and write
S ∼topS0
if there is a homeomorphism between (Pn,S) and (Pn,S0).
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On lattice-invariants of complex algebraic surfaces and their applications Zariski pairs
If two surfacesS andS0 in P3 with only RDPs are of the same configuration type, then
degS = degS0, and
theADE-typeRS ofSingS is equal to the ADE-type RS0 of SingS0.
Definition
We say that two surfacesS andS0 in P3 with only RDPs areof the weakly same configuration typeand write
S ∼wcfgS0 if degS = degS0 andRS =RS0.
On lattice-invariants of complex algebraic surfaces and their applications Zariski pairs
It is obvious thatS ∼top S0 impliesS ∼cfgS0 andS ∼wcfgS0. Definition
The pair [S,S0] of reduced hypersurfaces in Pn is called aZariski pairifS ∼cfgS0 butS6∼topS0.
Definition
The pair [S,S0] of surfacesS andS0 inP3 with only RDPs is called aweak Zariski pairifS ∼wcfgS0 butS6∼topS0.
Many examples of Zariskim-ples of plane curves (n= 2) have been constructed.
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On lattice-invariants of complex algebraic surfaces and their applications Zariski pairs
The first example was discovered by Zariski in 1930’s.
Example
LetB ⊂P2 be a plane curve of degree 6 defined by f3+g2 = 0, degf = 2, degg = 3, general.
ThenB is irreducible and has six cusps as its only singularities.
The six cusps are lying on the conicf = 0, and we have π1(P2\B)∼=Z/(2)∗Z/(3).
Del Pezzo had observed that there is a plane sexticB0 with only six cusps that arenotlying on a conic. Zariski exhibited suchB0 and showed that
π1(P2\B0)∼=Z/(2)×Z/(3).
On lattice-invariants of complex algebraic surfaces and their applications Zariski pairs
The main problem in these studies is how to distinguish the embedding topologies of plane curvesB ⊂P2 of the same configuration type.
The major tool is the fundamental groupsπ1(P2\B) or its variations like Alexander polynomials.
Aim: Construct Zariski pairs [S,S0] of surfaces in P3 with only isolated singularities.
In the construction, we cannot useπ1(P3\S):
By Zariski’s hyperplane section theorem, we have π1(P3\S)∼=π1(P3\S0)∼=Z/(degS).
We need new topological invariants.
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On lattice-invariants of complex algebraic surfaces and their applications Arithmetic Zariski pairs
LetAut(C) be the automorphism group of C.
For a schemeV →SpecC and an elementσ ∈Aut(C), we define Vσ →SpecCby the following Cartesian diagram:
Vσ −→ V
↓ ¤ ↓
SpecC −→
σ∗ SpecC.
Two schemesV and V0 overCare said to be conjugate ifV0 is isomorphic overCto Vσ overC for someσ∈Aut(C).
On lattice-invariants of complex algebraic surfaces and their applications Arithmetic Zariski pairs
Conjugate complex varieties can never be distinguished by any algebraic methods (they are isomorphic overQ),
but they can be non-homeomorphic in the classical complex topology.
The first example was given by Serre in 1964.
Other examples have been constructed by:
Abelson (1974),
Grothendieck’s dessins d’enfants (1984), Bartolo, Ruber, and Agustin (2004), Easton and Vakil (2007),
F. Charles (2009).
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On lattice-invariants of complex algebraic surfaces and their applications Arithmetic Zariski pairs
Example (S.- and Arima)
Consider two smooth irreducible surfacesS± in C3 defined by w2(G(x,y)±√
5·H(x,y)) = 1, where
G(x,y) := −9x4−14x3y+ 58x3−48x2y2−64x2y +10x2+ 108xy3−20xy2−44y5+ 10y4, H(x,y) := 5x4+ 10x3y−30x3+ 30x2y2+
+20x2y−40xy3+ 20y5. ThenS+ and S− are not homeomorphic.
On lattice-invariants of complex algebraic surfaces and their applications Arithmetic Zariski pairs
Definition
A Zariski pair [S,S0] of hypersurfaces in Pn is called an arithmetic Zariski pairif S andS0 are conjugate.
Definition
A weak Zariski pair [S,S0] of surfaces in P3 with only RDPs is called aweak arithmetic Zariski pairif S andS0 are conjugate.
Aim: Construct arithmetic Zariski pairs [S,S0] of surfaces in P3 with only isolated singularities.
The first example of arithmetic Zariski pair was given by Bartolo, Ruber, and Agustin (2004) for plane curves.
Their tool was thebraid monodromy, and cannot be used for surfaces with only isolated singularities.
We need new topological invariants.
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On lattice-invariants of complex algebraic surfaces and their applications Topological invariantst(S) andT(S)
Definition
Aquasi-lattice is a finitely generatedZ-module Lwith a symmetric bilinear form
L×L→Z.
For a quasi-latticeL, we put
kerL:={x ∈L | (x,y) = 0 for all y ∈L}=L⊥. Note thatkerLcontains the torsion part of L.
Definition
A quasi-latticeLis called alattice if the symmetric bilinear form is non-degenerate (that is,kerL= 0).
For a quasi-latticeL, the free Z-moduleL/kerLis a lattice.
On lattice-invariants of complex algebraic surfaces and their applications Topological invariantst(S) andT(S)
LetS ⊂P3 be a surface with only RDPs.
We will define two topological invariantst(S) andT(S) of (P3,S), which allow us to constructweak (arithmetic) Zariski pairs.
Definition We put
t(S) := the torsion part ofH3(P3\S,Z)
= the torsion part ofH3(P3,S,Z)
= the torsion part ofH2(S,Z).
It is obvious thatS ∼top S0 impliest(S)∼=t(S0).
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On lattice-invariants of complex algebraic surfaces and their applications Topological invariantst(S) andT(S)
We consider the smooth open surface S◦ :=S\SingS and the intersection pairing
H2(S◦,Z)×H2(S◦,Z)→Z.
We then put
V(S◦) :=Ker(H2(S◦)→H2(P3)).
Definition
We define the invariantT(S) by
T(S) :=V(S◦)/kerV(S◦), which is a lattice.
It is obvious thatS ∼top S0 impliesT(S)∼=T(S0).
On lattice-invariants of complex algebraic surfaces and their applications Topological invariantst(S) andT(S)
Calculation of the invariants t(S) andT(S) LetRS denote theADE-type of Sing(S).
Consider the minimal resolution ρ:X →S
ofS. We regardH2(X,Z) as a lattice by the cup-product. Let h∈H2(X)
be the class of the pull-back of a plane section ofS.
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On lattice-invariants of complex algebraic surfaces and their applications Topological invariantst(S) andT(S)
LetEρbe the set of exceptional curves E ⊂X of ρ.
EachE ∈ Eρis a smooth rational curve with E2=−2, and the dual graph of them is a Dynkin diagram of typeRS. We consider the submodule
hEρi ⊂H2(X)
generated by the classes of the curvesE ∈ Eρ. Then hEρi is a sublattice ofH2(X) isomorphic to the negative-definite root lattice ofADE-typeRS. Let
hEρi:= (hEρi ⊗Q)∩H2(X) be the primitive closure ofhEρi in H2(X).
On lattice-invariants of complex algebraic surfaces and their applications Topological invariantst(S) andT(S)
Looking at the topology of the minimal resolutionρ, we obtain the following:
Theorem The invariant
t(S) = the torsion part ofH2(S,Z) is isomorphic tohEρi/hEρi.
Theorem The lattice
T(S) :=V(S◦)/kerV(S◦),
whereS◦ :=S\SingS andV(S◦) :=Ker(H2(S◦)→H2(P3)) is isomorphic to the orthogonal complement ofhEρi ⊕ hhi inH2(X).
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On lattice-invariants of complex algebraic surfaces and their applications Topological invariantst(S) andT(S)
Therefore, if we know the data (hEρi,h), then we can calculatet(S) andT(S).
When degS = 4,X is aK3 surface, andH2(X) is isomorphic to theK3 lattice
L:= (−E8)2⊕
µ 0 1 1 0
¶3 .
Definition
Aquartic lattice datais a pair (Λ,v)
of a negative-definite root sublattice Λ of theK3 latticeLand a vectorv ∈Lwith v2 = 4.
On lattice-invariants of complex algebraic surfaces and their applications Topological invariantst(S) andT(S)
Definition
A quartic lattice data (Λ,v) is realizableif there is a quartic surfaceS ⊂P3 with only RDPs and an isomorphism
φ:H2(X)∼=L of lattices such thatφ(hEρi) = Λ and φ(h) =v.
If suchS exists, then RS is equal to the ADE-type of the root sublattice Λ.
By the Torelli theorem forK3 surfaces, we have the complete list of realizable lattice data.
This task was done by J. G. Yang with an aid of computer.
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On lattice-invariants of complex algebraic surfaces and their applications Examples of weak Zariski pairs
Example
There is a weak Zariski pair [S0,S1] of quartic surfaces such that each Si has 8 nodes as its only singularities, and
t(S0) = 0, whilet(S1)∼=Z/2Z.
This pair was already observed by Coble in 1930’s:
S0 is called azygetic, whileS1 is called syzygetic.
Their difference is also expressed by h0(P3,IQ(2)) =
(2 ifQ =SingS0, 3 ifQ =SingS1, whereIQ ⊂ OP3 is the ideal sheaf ofQ ⊂P3.
A syzygetic memberS1 is defined by an equation of the form PaijAiAj = 0,whereA0,A1,A2 are quadratics.
On lattice-invariants of complex algebraic surfaces and their applications Examples of weak Zariski pairs
Example
There is a weak Zariskiquartet[S0,S1,S2,S3] of quartic surfaces with RDPs of type
2A1+ 2A2+ 2A5 as their only singularities such that
t(S0) = 0, t(S1)∼=Z/2Z, t(S2)∼=Z/3Z, t(S3)∼=Z/6Z.
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On lattice-invariants of complex algebraic surfaces and their applications SingularK3 surfaces
Definition
AK3 surface X is calledsingular if the Picard number of X is 20.
LetX be a singularK3 surface. Then the transcendental lattice T(X) :=NS(X)⊥ inH2(X,Z)
is a positive-definite even lattice of rank 2.
The Hodge decomposition
T(X)⊗C=H2,0(X)⊕H0,2(X) induces an orientation onT(X). We denote by
T˜(X) the oriented transcendental lattice ofX. By Torelli theorem, we have
T˜(X)∼= ˜T(X0) =⇒ X ∼=X0.
On lattice-invariants of complex algebraic surfaces and their applications SingularK3 surfaces
Construction by Shioda and Inose
Every singularK3 surfaceX is obtained as a certain double cover of the Kummer surface
Km(E ×E0),
whereE andE0 are elliptic curves withCM by some orders of Q(p
−|disc(T(X))|).
Theorem (Shioda and Inose)
(1) For any positive-definite oriented even lattice ˜T of rank 2, there exists a singularK3 surface X such that ˜T(X)∼= ˜T. (2) Every singularK3 surface is defined over a number field.
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On lattice-invariants of complex algebraic surfaces and their applications SingularK3 surfaces
A latticeL is naturally embedded into the dual lattice L∨:=Hom(L,Z).
Thediscriminant group ofL is the finite abelian group DL:=L∨/L.
TheZ-valued symmetric bilinear form on Lextends to L∨×L∨→Q.
A latticeL is said to be even ifx2 ∈2Z for allx ∈L. If Lis even, then we have a quadratic form
qL :DL→Q/Z, ¯x7→x2 mod 2Z, which is called thediscriminant formof L.
On lattice-invariants of complex algebraic surfaces and their applications SingularK3 surfaces
We have the following:
Proposition
LetLand L0 be even lattices of the same rank.
IfLandL0 have isomorphic discriminant forms and
the same signature, thenLandL0 belong to the same genus.
SinceH2(X) is unimodular and both ofT(X) andNS(X) are primitive inH2(X), we have the following:
Proposition
(DT(X),qT(X))∼= (DNS(X),−qNS(X)).
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On lattice-invariants of complex algebraic surfaces and their applications SingularK3 surfaces
LetX andX0 be singularK3 surfaces.
It is obvious that, ifX andX0 are conjugate, thenNS(X) and NS(X0) are isomorphic. Therefore we have the following:
Corollary
IfX andX0 are conjugate, thenT(X) andT(X0) are in the same genus.
The class field theory of imaginary quadratic fields tells us how the Galois group acts on thej-invariants of elliptic curves with CM.
Using this, S.- and Sch¨utt (2007) proved the following converse:
Theorem
IfT(X) andT(X0) are in the same genus, thenX andX0 are conjugate.
On lattice-invariants of complex algebraic surfaces and their applications An example of weak arithmetic Zariski pairs
Definition
A quartic surfaceS ⊂P3 is maximizingif it has only RDPs and its total Milnor number is 19.
IfS is a maximizing quartic, thenX is a singular K3 surface, and we have
T(S)∼=T(X).
Hence, if maximizing quarticsS andS0 are of the weakly same configuration type, andT(X) andT(X0) are not isomorphic but in the same genus, then [S,S0] is a weak arithmetic Zariski pair.
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On lattice-invariants of complex algebraic surfaces and their applications An example of weak arithmetic Zariski pairs
Example
There is a weak arithmetic Zariski pair [S,S0] of maximizing quartic surfaces such that
each of them has RDPs of typeA1+A18 as its only singularities, and
the minimal resolutions X of S andX0 of S0 have the transcendental lattices
· 4 0 0 38
¸ and
· 6 2 2 26
¸ , which are in the same genus but are not isomorphic.
Problem: Find the explicit defining equations ofS and S0.
On lattice-invariants of complex algebraic surfaces and their applications An example of weak arithmetic Zariski pairs
Thank you!
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