Non-homeomorphic conjugate complex varieties Toyama, 2007 August 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 . • A lattice Λ is said to be even if ( v, v ) ∈ 2 Z for any v ∈ Λ. 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. (Replace “polynomials” by “homogeneous polynomials”.) 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 (over C ) 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
§ 2. Topology of conjugate varieties Conjugate varieties cannot be distinguished by any algebraic methods. In particular, they are homeomorphic in Zariski topology. How about in the complex topology? Example (Serre (1964)). There exist conjugate non-singular projective varieties X and X σ such that their fundamental groups are not isomorphic: π 1 ( X ) �∼ = π 1 ( X σ ) . In particular, they are not homotopically equivalent. 4
Other examples of non-homeomorphic conjugate varieties. • Abelson: Topologically distinct conjugate varieties with fi- nite fundamental group. Topology 13 (1974). • Artal Bartolo, Carmona Ruber, Cogolludo Agust´ ın: Effec- tive invariants of braid monodromy. Trans. Amer. Math. Soc. 359 (2007). • S.-: On arithmetic Zariski pairs in degree 6. arXiv:math/0611596, to appear in Adv. Geom. • S.-: Non-homeomorphic conjugate complex varieties. arXiv:math/0701115 • Easton, Vakil: Absolute Galois acts faithfully on the com- ponents of the moduli space of surfaces: A Belyi-type the- orem in higher dimension. arXiv:0704.3231 • Bauer, Catanese, Grunewald: The absolute Galois group acts faithfully on the connected components of the moduli space of surfaces of general type. arXiv:0706.1466 • F. Charles: Conjugate varieties with distinct real cohomol- ogy algebras. arXiv:0706.3674 5
Main result . We introduce a new topological invariant ( B U , β U ) of open algebraic varieties U , which allows us to distinguish conjugate varieties topologically in some cases. Combining this topological invariant with the arithmetic theory of abelian surfaces and K 3 surfaces, we obtain examples of non- homeomorphic conjugate varieties. Our examples are as follows: • Zariski open subsets of abelian surfaces. • Zariski open subsets of K 3 surfaces. • Arithmetic Zariski pairs in degree 6. 6
§ 3. Arithmetic Zariski pairs Definition. A pair [ C, C ′ ] of complex projective plane curves is said to be a Zariski pair if the following hold: (i) There exist tubular neighborhoods T ⊂ P 2 of C and T ′ ⊂ P 2 of C ′ such that ( T , C ) and ( T ′ , C ′ ) are diffeomorphic. (ii) ( P 2 , C ) and ( P 2 , C ′ ) are not homeomorphic. Example. The first example of Zariski pair was discovered by Zariski in 1930’s, and studied by Oka. They presented a Zariski pair [ C, C ′ ] of plane curves of degree 6, each of which has six ordi- nary cusps as its only singularities. The fact ( P 2 , C ) and ( P 2 , C ′ ) are not homeomorphic follows from π 1 ( P 2 \ C ) ∼ π 1 ( P 2 \ C ′ ) ∼ = ( Z / 2 Z ) ∗ ( Z / 3 Z ) and = Z / 6 Z . 7
Definition. A Zariski pair [ C, C ′ ] is said to be an arithmetic Zariski pair if the following hold. Suppose that C = { Φ = 0 } . Then there exists an embedding → C such that C ′ is isomorphic (as a plane curve) to σ : F C ֒ C σ := { Φ σ = 0 } P 2 . ⊂ Remark. The Zariski pair of Zariski and Oka is not an arithmetic Zariski pair, because the pro-finite completion of π 1 ( P 2 \ C ) ∼ π 1 ( P 2 \ C ′ ) ∼ = ( Z / 2 Z ) ∗ ( Z / 3 Z ) and = Z / 6 Z are not isomorphic; there exists a surjective homomorphism from π 1 ( P 2 \ C ) to the symmetric group S 3 on three letters, while there are no such homomorphism from π 1 ( P 2 \ C ′ ). Remark. The first example of an arithmetic Zariski pair was discovered by Artal, Carmona, Cogolludo (2007) in degree 12. They used the invariant of braid monodromies in order to dis- tinguish ( P 2 , C ) and ( P 2 , C ′ ) topologically. 8
Example (Artal, Carmona, Cogolludo (2002)). 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 such that the plane curve C = { Φ = 0 } has three simple singular points of type A 16 + A 2 + A 1 as its only singularities. Consider the conjugate plane curves C α = { Φ σ α = 0 } , α = 0 } C β = { Φ σ β = 0 } . α = { Φ σ ¯ C ¯ and They show that, if C ′ is a plane curve possessing A 16 + A 2 + A 1 as its only singularities, then C ′ is projectively isomorphic to C α , C ¯ α or C β . Since simple singularities have no moduli, there are tubular neighborhoods 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 ). 9
Using the new topological invariant, we can show that there are no homeomorphisms between ( P 2 , C α ) and ( P 2 , C β ). Let Y C → P 2 be the double covering branching exactly along the curve C : Φ = 0, and U ⊂ Y C the pull-back of P 2 \ C . Then U is a variety defined over K . Consider the conjugate open varieties U α and U β corresponding to the embeddings σ α and σ β . Then the topological invariants ( B U α , β U α ) and ( B U β , β U β ) differ. Hence [ C α , C β ] is an arithmetic Zariski pair in degree 6. 10
§ 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 . 11
We study the invariant ( B U , β U ) for an open algebraic variety 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 ) ). 12
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