ON THE INTEGRAL HODGE CONJECTURE FOR REAL THREEFOLDS. 1. Motivation


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1 ON THE INTEGRAL HODGE CONJECTURE FOR REAL THREEFOLDS OLIVIER WITTENBERG This is joint work with Olivier Benoist Work of Kollár. 1. Motivation Theorem 1.1 (Kollár). If X is a smooth projective (geometrically) rationally connected variety over R, then for every x X(R) there is a nonconstant morphism f : P 1 X such that f(0) = x. Question 1.2 (Kollár). Take X as above but suppose that X(R) =. Does X contain a geometrically rational curve? I.e., does there exist a map C X, if C denotes the conic x 2 + y 2 + z 2 = 0? This question has an affirmative answer for surfaces, for classification reasons Graber Harris Starr over R. Question 1.3. Is there a Graber Harris Starr theorem over R? 1.3. Lang s conjecture. Let C/R be a smooth projective curve with C(R) =. Then the function field R(C) is a C 1 field, i.e., every Fano hypersurface over R(C) has a rational point. Challenge 1.4. Let X P 4 R be a quartic 3fold. Does X contain a rational curve? a conic? [Hassett: yes (rational curve).] 1.4. ELWinvariants. Let X/R be a proper variety. The quantity gcd{χ(x, E) : E is a coherent sheaf on X with dim Supp E 1} is equal to 1 or 2. It is equal to 1 if and only if either X(R) or X contains a geometrically irreducible curve of even genus. So if we have a variety X/R with X(R) =, we want to know if it contains a geometrically irreducible curve of even genus. Remark 1.5. If X/R is a proper curve with X(R) =, then the arithmetic genus and the geometric genus have the same parity, so we can just speak of the parity of the genus of X. 1
2 2. Integral Hodge conjecture for 1cycles 2.1. Review over C. Let X/C be a smooth projective ddimensional variety. Definition 2.1. The expression IHC(X/C) is short hand for the cycle class map CH 1 (X) (X(C), Z(d 1)) Hdg 2d 2 is surjective. Here Z(m) is the twist (2πi) m Z and Hdg 2d 2 stands for Hodge classes, i.e., classes whose image in (X(C), C) has type (d 1, d 1). Some results: IHC(X/C) holds for surfaces. This is basically the Lefschetz (1, 1)theorem. IHC(X/C) fails for a very general hypersurface of degree 48 X P 4 C (Kollár). Theorem 2.2 (Voisin). (i) IHC(X/C) holds if X is a uniruled or CalabiYau threefold. (ii) IHC(X/C) holds if X is rationally connected, provided one assumes the Tate conjecture for surfaces over finite fields (relies on Schoen s work) Over the reals. We would like real analogues of the above statements. Let X/R be a smooth projective ddimensional variety. Our first task is to get a replacement for the cycle class map. For this we use equivariant cohomology. Letting G = Gal(C/R), one can define cohomology groups G (X(C), Z(d 1)) via the right derived functors of the invariant global sections functor on the category of Gequivariant sheaves. For d 2 there is a map G (X(C), Z(d 1)) res G (X(R), Z(d 1)) = H p (X(R), Z/2Z), 0pd 1 p d 1 mod 2 and the equality is a canonical decomposition. For α G (X(C), Z/2Z), we write (α p ) for its image under this map. Now define G (X(C), Z(d 1)) 0 = {α G (X(C), Z(d 1)) : α p = 0 for all p < d 1)}. Definition 2.3. The expression IHC(X/R) is short hand for the map is surjective. (d 1, d 1). CH 1 (X) G (X(C), Z(d 1)) 0 Hdg 2d 2 Here Hdg 2d 2 stands for classes whose image in (X(C), C) has type Proposition 2.4 (Krasnov). IHC(X/R) holds for surfaces. Conjecture/Question 2.5. Is it true that (i) IHC(X/R) holds for uniruled or CalabiYau threefolds, (ii) IHC(X/R) holds for (geometrically) rationally connected varieties? 2
3 3. Connections between IHC(X/C), IHC(X/R), curves of even genus, etc. Theorem 3.1. Let X/R be a smooth projective variety of dimension d 2. Assume that π ab 1 (X(C)) = 0. Then there are exact sequences: if X(R) = : (X(C), Z(d 1)) norm G (X(C), Z(d 1)) Z/2Z 0 ϕ norm CH 1 (X C ) CH 1 (X) where for an irreducible curve C we have { 1 if C is geometrically irreducible of even genus, ϕ([c]) = if X(R) : 0 otherwise. (X(C), Z(d 1)) norm G (X(C), Z(d 1)) 0 H 1 (X(R), Z/2Z) 0 ψ norm CH 1 (X C ) CH 1 (X) where ψ is the Borel Haefliger cycle map. Corollary 3.2. If in addition H 2 (X, O X ) = 0 and IHC(X C /C) holds, then IHC(X/R) holds if and only if either X(R) = and X contains a curve of even genus, or X(R) and the Borel Haefliger cycle map is surjective. Corollary 3.3. If X/R is a surface with p g = 0 and π ab 1 (X(C)) = 0 then X contains a curve of even genus and the map CH 1 (X) H 1 (X(R), Z/2Z) is surjective. Question 3.4. If X/R is geometrically rationally connected, is H 1 (X(R), Z/2Z) generated by classes of rational curves? Remark 3.5. When H 2 (X, O X ) 0 or π1 ab (X(C) 0 the corollaries may fail: (Kollár s example) If X P 3 R is a very general quartic K3 surface with no real points, then Pic X = Z.O(1), and the generator has genus 3. Hence X does not contain a curve of even genus. However IHC(X/R) is satisfied because X is a surface. Mangolte and van Hamel have shown that when X/R is an Enriques surface, the map CH 1 (X) H 1 (X(R), Z/2Z) is surjective if and only if X(R) is orientable. Moreover it can happen that X(R) nonorientable. However IHC(X/R) holds, as X is a surface. (One can also construct examples of surfaces with H 2 (X, O X ) = 0, no real point and no curve of even genus Campedelli surfaces.) 3
4 4. Theorems on threefolds Theorem 4.1. Let X be a smooth Fano threefold. If X(R) = then IHC(X/R) holds, i.e., X contains a geometrically irreducible curve of even genus. Corollary 4.2. Quartic threefolds over R contain geometrically irreducible curves of even genus. Theorem 4.3. Let f : X S be a morphism between smooth projective varieties over R with dim S = 2. Suppose that the generic fiber is a conic. Then IHC(X/R) holds. Moreover, if S = P 2, then H 1 (X(R), Z/2Z) is generated by classes of rational curves. Corollary 4.4. If X is a cubic threefold, then H 1 (X(R), Z/2Z) is generated by classes of rational curves. Theorem 4.5. Let f : X C be a morphism between smooth projective varieties over R with dim C = 1. Suppose that the generic fiber is a del Pezzo surface of degree δ. Then IHC(X/R) holds if δ {3, 5, 6, 7, 8, 9}. It also holds if δ {2, 4} and X(R) = but C(R). 5. Some words about the proofs 5.1. Sketch of the proof of Theorem 4.1. Focus on the case K X is very ample: the remaining cases are easy, or can be reduced to this case. The idea is to mimic Voisin s proof over C. Use very ampleness of K X to embed X P N. Let H P N be a hyperplane, and let S = X H. We have commutative diagram H 2 G(S(C), Z(1)) Gysin H 4 G(X(C), Z(2)) CH 1 (S) CH 1 (X) where the top map is surjective. We want to prove that the right map is surjective. The idea is to vary H to produce enough Hodge classes in HG 2 (S(C), Z(1)) that they fill H4 G (X(C), Z(2)). As H is linearly equivalent to K X we have h 2 (S, O S ) = 1, which simplifies the proof Sketch of the proof of Theorem 4.3. We only briefly describe the ingredients of the proof. First, the map f gives rise to a pushforward map H 4 G(X(C), Z(2)) Hdg 4 H 2 G(S(C), Z(1)) Hdg 2. Use the Lefschetz (1, 1)theorem to see we have algebraic classes on S. We want to lift them, but there is a topological obstruction; we need these classes to be algebraic over R. To this end, one uses approximation theorems and the EPT theorem. Second, one analyses the Leraytype spectral sequence for the composition of functors Gequivariant sheaves on X(C) sheaves on S(C)/G Γ abelian groups 4
5 to prove that a class in the kernel of the pushforward map is supported over a codimension 1 closed subset of S. Finally, using the conic bundle structure, one proves algebraicity of equivariant homology of f 1 (Z) where Z S is a closed subset of codimension 1. 5
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