Non-homeomorphic conjugate complex varieties Sopot, 2007 Ichiro - PDF document

Non-homeomorphic conjugate complex varieties Sopot, 2007 Ichiro Shimada (Hokkaido University, Sapporo, JAPAN) • We work over the complex number field C . • The coefficients of the (co-)homology groups are in Z . • By a lattice, we mean a finitely generated free Z -module Λ equipped with a non-degenerate symmetric bilinear form Λ × Λ → Z . 1

§ 1. Conjugate varieties An affine algebraic variety X ⊂ C N is defined by a finite num- ber of polynomial equations: X : f 1 ( x 1 , . . . , x N ) = · · · = f m ( x 1 , . . . , x N ) = 0 . Let c j,I ∈ C be the coefficients of the polynomial f j : � x I = x i 1 1 · · · x i N c j,I x I , f j ( x 1 , . . . , x N ) = where N . I We then denote by F X := Q ( . . . , c j,I , . . . ) ⊂ C the minimal sub-field of C containing all the coefficients of the defining equations of X . There are many other embeddings σ : F X ֒ → C of the field F X into C . Example. √ (1) If F X = Q ( 2 , t ), where t ∈ C is transcendental over Q , then the set of embeddings F X ֒ → C is equal to √ √ { 2 , − 2 } × { transcendental complex numbers } . (2) If all c j,I are algebraic over Q , then the set of embeddings is finite, and the Galois group of the Galois closure of the al- gebraic extension F X / Q acts on the set transitively. 2

For an embedding σ : F X ֒ → C , we put � f σ c σ j,I x I , j ( x 1 , . . . , x N ) := I and denote by X σ ⊂ C N the affine algebraic variety defined by f σ 1 = · · · = f σ m = 0 . We can define X σ for a projective or quasi-projective variety X ⊂ P N in the same way. Definition. We say that two algebraic varieties X and Y are said to be conjugate if there exists an embedding σ : F X ֒ → C such that Y is isomorphic to X σ . In the language of schemes, two varieties X and Y over Spec C are conjugate if there exists a diagram Y − → X ↓ ↓ � σ ∗ Spec C − → Spec C . of the fiber product for some morphism σ ∗ : Spec C → Spec C . It is obvious that being conjugate is an equivalence relation. 3

√ √ y 2 = x 3 + 6 2 x + 2 . √ √ y 2 = x 3 − 6 2 x − 2 . 4

§ 2. Topology of conjugate varieties Conjugate algebraic varieties cannot be distinguished by any algebraic methods. In particular, they are homeomorphic in Zariski topology. How about their complex topology? The following is due to Serre, Grothendieck, Artin, . . . . Theorem. Let X and Y be conjugate non-singular projective varieties. (1) They have the same betti numbers: B i ( X ) = B i ( Y ) for i = 0 , . . . , 2 dim X. (2) The profinite completions of their fundamental groups are 1 ( X ) ∼ isomorphic: π ∧ = π ∧ 1 ( Y ). The following example is due to Serre (1964). Example. There exist conjugate non-singular projective varieties X and Y such that their fundamental groups are not isomorphic: π 1 ( X ) �∼ = π 1 ( Y ). 5

Other examples of non-homeomorphic conjugate varieties: • Abelson (1974). • E. Artal, J. Carmona, and J.-I. Cogolludo. (2003-). • Bauer, Catanese, Grunewald. (2005-). . . . Grothendieck’s “dessins d’enfant”. Let f : C → P 1 be a finite covering of a projective line branch- ing only at the three points 0 , 1 , ∞ ∈ P 1 . We have defining equations of f with coefficients in Q ⊂ C . For σ ∈ Gal( Q / Q ), consider the conjugate covering f σ : C σ → P 1 . Then f and f σ have, in general, different topology. 6

§ 3. Main result We introduce a new topological invariant of open algebraic va- rieties, which allows us to distinguish conjugate varieties topo- logically in some cases. Combining this topological invariant with the following results, we obtain several explicit examples of non-homeomorphic con- jugate varieties. • Arithmetic theory of abelian and K 3 surfaces due to S.- and Sch¨ utt. • Degtyarev’s theorem on the connected components of plane curves of degree 6 with only simple singularities. • Artal, Carmona and Cogolludo’s calculation of defining equations of plane curves of degree 6 with prescribed sim- ple singularities. Our examples consist of the following: • Zariski open subsets of abelian surfaces. • Zariski open subsets of K 3 surfaces. • Singular plane curves C of degree 6 with only simple sin- gularities and of Milnor number 19. (In this example, the homeomorphism types of the pairs ( P 2 , C ) are distinct.) 7

Example. We consider the following cubic extension of Q : ϕ = 17 t 3 − 18 t 2 − 228 t + 556 . K := Q [ t ] / ( ϕ ) , where The roots of ϕ = 0 are α, ¯ α, β , where √ α = 2 . 590 · · · + 1 . 108 · · · − 1 , β = − 4 . 121 · · · . There are three corresponding embeddings σ α : K ֒ → C , σ ¯ α : K ֒ → C and σ β : K ֒ → C . There exists a homogeneous polynomial Φ( x 0 , x 1 , x 2 ) ∈ K [ x 0 , x 1 , x 2 ] of degree 6 with coefficients in K with the following properties. We consider the conjugate plane curves C α = { Φ σ α = 0 } C β = { Φ σ β = 0 } . and Then each of them has three simple singular points of type A 16 + A 2 + A 1 as its only singularities. In particular, there exist tubular neigh- borhoods T α ⊂ P 2 of C α ⊂ P 2 and T β ⊂ P 2 of C β ⊂ P 2 such that ( T α , C α ) is diffeomorphic to ( T β , C β ). However, there are no homeomorphisms between the pairs ( P 2 , C α ) and ( P 2 , C β ). Namely, C α and C β form an arithmetic Zariski pair . Let X → P 2 be the double covering of the plane branching ex- actly along the curve C : Φ = 0, and U ⊂ X the pull-back of P 2 \ C . Then U is a variety defined over K . Consider the conju- gate open varieties U α and U β corresponding the embeddings σ α and σ β . Then U α and U β are not homeomorphic. 8

§ 4. The topological invariant Let U be an oriented topological manifold of dimension 4 n . Let ι U : H 2 n ( U ) × H 2 n ( U ) → Z be the intersection pairing. Definition. We put � J ∞ ( U ) := Im( H 2 n ( U \ K ) → H 2 n ( U )) , K where K runs through the set of all compact subsets of U . We then put � B U := ( � B U := H 2 n ( U ) /J ∞ ( U ) and B U ) / torsion . Since any topological cycle is compact, the intersection pairing ι U induces a symmetric bilinear form β U : B U × B U → Z . It is obvious that, if U and U ′ are homeomorphic, then there exists an isomorphism ( B U , β U ) ∼ = ( B U ′ , β U ′ ) , and hence the isomorphism class of ( B U , β U ) is a topological invariant of U . 9

We study the invariant ( B U , β U ) for the space U := X \ Y, where X is a non-singular projective variety of complex di- mension 2 n , and Y is a union of irreducible (possibly singular) subvarieties Y 1 . . . , Y N of complex dimension n : Y = Y 1 ∪ · · · ∪ Y N . We denote by � Σ ( X,Y ) := � [ Y 1 ] , . . . , [ Y N ] � ⊂ H 2 n ( X ) the submodule of H 2 n ( X ) generated by the homology classes [ Y i ] ∈ H 2 n ( X ), and put Σ ( X,Y ) := ( � Σ ( X,Y ) ) / torsion . We then put Λ ( X,Y ) := { x ∈ H 2 n ( X ) | ι X ( x, y ) = 0 for any y ∈ � � Σ ( X,Y ) } , Λ ( X,Y ) := ( � Λ ( X,Y ) ) / torsion . Finally, we denote by σ ( X,Y ) : Σ ( X,Y ) × Σ ( X,Y ) → Z and λ ( X,Y ) : Λ ( X,Y ) × Λ ( X,Y ) → Z the symmetric bilinear forms induced from the intersection pairing ι X : H 2 n ( X ) × H 2 n ( X ) → Z . Theorem. Let X , Y and U be as above. Suppose that σ ( X,Y ) is non- degenerate. Then ( B U , β U ) is isomorphic to (Λ ( X,Y ) , λ ( X,Y ) ). 10

Sketch of the proof. Since X is non-singular and complete, the intersection pair- ing ι X on H 2 n ( X ) / torsion is non-degenerate. Hence the as- sumption that σ ( X,Y ) is non-degenerate implies that λ ( X,Y ) is non-degenerate. We consider the homomorphism j U : H 2 n ( U ) → H 2 n ( X ) induced by the inclusion. It is obvious that the image of j U is contained in � Λ ( X,Y ) . We first show that Im( j U ) = � Λ ( X,Y ) . Let [ W ] ∈ � Λ ( X,Y ) be represented by a real 2 n -dimensional topo- logical cycle W . We can assume that W ∩ Y consists of a finite number of points in Y \ Sing( Y ), and that the intersection of W with Y is transverse at each intersection point. Let P i, 1 , . . . , P i,k ( i ) (resp. Q i, 1 , . . . , Q i,l ( i ) ) be the intersection points of W and Y i with local intersection number 1 (resp. − 1). Since ι X ([ W ] , [ Y i ]) = 0, we have k ( i ) = l ( i ) . Modifying W by adding the tube ∂ ( D 2 n × I ) for each pair ( P i,j , Q i,j ), we obtain a topological cycle W ′ that is homologous to W in X and is disjoint from Y . Hence [ W ] = [ W ′ ] is represented by W ′ ⊂ U . Thus Im( j U ) = � Λ ( X,Y ) holds. 11

Figure 12

Using Mayer-Vietris sequence, we can prove Ker( j U ) ⊆ J ∞ ( U ) from the assumption that λ ( X,Y ) is non-degenerate. By the commutative diagram j U � 0 − → Ker( j U ) − → H 2 n ( U ) − → Λ ( X,Y ) − → 0 v ˜ → ֒ → = → � 0 − → J ∞ ( U ) − → H 2 n ( U ) − → B U − → 0 , we obtain the isomorphism (Λ ( X,Y ) , λ ( X,Y ) ) ∼ = ( B U , β U ). 13