The primitive cohomology of the theta divisor of an abelian fivefold (Q2832764)

We find here a proof of the general Hodge conjecture [\textit{A. Grothendieck}, Topology 8, 299--303 (1969; Zbl 0177.49002)], for the particular case of the cohomology of a smooth \(\Theta\) divisor on a general p.p. abelian variety \(A\) of genus \(5\). Recall that Grothendieck's \(GHC(X,m,p)\) is the claim that for every \(\mathbb{Q}\)-Hodge substructure \(V\) of \(H^m(X,\mathbb{Q})\) with level \(\leq m-2p\) there exists a subvariety \(Z\subset X\) of pure codimension \(p\) such that \(V\subset \ker\{H^m(X,\mathbb{Q})\rightarrow H^m(X-Z,\mathbb{Q})\}\). For general \(A\) of any genus \(g\) the inclusion \(j: \Theta \to A \) as a smooth ample divisor yields the fact that the cohomology of \(\Theta\) is determined by the cohomology of \(A\), but for degree \(g-1\). Consider the primal cohomology of \(\Theta\), which is defined to be kernel \(\mathbb K\) of \(H^{g-1} (\Theta , \mathbb Z) \rightarrow H^{g+1} (A, \mathbb Z)\). Since \(GHC\) is known to hold for a general abelian variety [\textit{F. Hazama}, Compos. Math. 93, No. 2, 129--137 (1994; Zbl 0848.14003)] the issue then is to prove \(GHC\) for the primal cohomology. By using residues one can compute that the Hodge structure of the primal cohomology is of level \(g-3\). There is a second Hodge structure \(\mathbb H\) inside \(H^{g-1} (\Theta , \mathbb Z) \), also of level \(g-3\), which contains \(\mathbb K\). The point is that \(GHC\) holds for \(\mathbb H\) exactly if it holds for \(\mathbb K\). The situation for \(g \leq 3\) is easy to settle, while the case \(g=4\) is the content of [\textit{E. Izadi} and \textit{D. van Straten}, J. Algebr. Geom. 4, No. 3, 557--590 (1995; Zbl 0862.14030)]. The method of the present paper, \(g=5\), is to produce a surface \(S\) which pametrizes a family of curves inside \(\Theta\) such that the correspondence thus obtained determines a surjection from \(H^{2}(S)\) to \(\mathbb H\), modulo an irrilevant contribution coming from \(H^{2} (A) \). The strategy is to control the geometry by using the construction of \(A\) as a Prym variety. So there is in the picture a smooth curve \(X\) of genus \(6\) with an étale double cover \(\tilde X\) of genus 11. The canditate surface \(S\) and the family of curves which it parametrizes are thus produced, with careful and detailed work, starting from the surface \( W^1_5\) in \(\mathrm{Pic}^5 (X)\).NEWLINENEWLINEThe problem is to be able to decide surjectivity. The idea is to deform the situation to a good reduction of \(A\), to the case when it becomes the Jacobian \(J\) of a curve \(C\) of genus \(5\). The Prym construction is in this case performed by using the Wirtinger's cover method, so that \(X\) becomes \(C_{pq}\) (the nodal curve with exactly one node, obtained from \(C\) by gluing \(p\) with \(q\)) and \( \tilde X\) is reducible, made of two copies of \(C\).NEWLINENEWLINEThe tool for controlling in general the cohomology of varieties by means of stable degenerations along base curves is a highly efficient machine, whose construction is due to the wonderful ideas of several people. For this matter the authors refer to \textit{D. R. Morrison}'s excellent exposition [Ann. Math. Stud. 106, 101--119 (1984; Zbl 0576.32034)].NEWLINENEWLINEThe issue is then to put the machinery in action, for this purpose one needs to have concrete knowledge of the central degeneration of the objects which enter the picture, because it is they which carry all the information required to get hold of the smooth fibres. This work has been performed for the situation at hand here, it could be done because the degeneration from \(X\) to \(C_{pq}\) is such that one can explicitly describe all the complicated central fibres which turn out in performing the process of the various stable reduction required. The techniques entail extensive use of the rich knowledge which the authors command of the geometry of special divisors on curves and of the theory of Prym varieties.