ON THE INTEGRAL HODGE CONJECTURE FOR REAL THREEFOLDS OLIVIER WITTENBERG This is joint work with Olivier Benoist. 1.1. 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. 1.2. 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 3-fold. Does X contain a rational curve? a conic? [Hassett: yes (rational curve).] 1.4. ELW-invariants. 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. Integral Hodge conjecture for 1-cycles 2.1. Review over C. Let X/C be a smooth projective d-dimensional 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 Calabi-Yau 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). 2.2. Over the reals. We would like real analogues of the above statements. Let X/R be a smooth projective d-dimensional 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 G-equivariant 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 Calabi-Yau threefolds, (ii) IHC(X/R) holds for (geometrically) rationally connected varieties? 2
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) non-orientable. 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. 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. 5.2. 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 push-forward 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 Leray-type spectral sequence for the composition of functors G-equivariant sheaves on X(C) sheaves on S(C)/G Γ abelian groups 4
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