Surfaces generating the even primal cohomology of an abelian fivefold
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Mathematische Annalen
Surfaces generating the even primal cohomology of an abelian fivefold Jonathan Conder1 · Edward Dewey1 · Elham Izadi1 Received: 1 May 2020 / Revised: 2 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Given a very general abelian fivefold A and a principal polarization ⊂ A, we construct surfaces generating the algebraic part of the middle cohomology H 4 (, Q), and determine the intersection pairing between these surfaces. In particular, we obtain a new proof of the Hodge conjecture for H 4 (, Q) and show that it contains a copy of the root lattice of E 6 . Mathematics Subject Classification Primary 14C30; Secondary 14D06 · 14K12 · 14H40
Contents Introduction . . . . . . . . . . . . . . . Notation . . . . . . . . . . . . . . . . . 1 Prym varieties and Prym-embeddings 2 Invariants of the surfaces . . . . . . . 3 27 surfaces . . . . . . . . . . . . . . 4 27 curves . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . .
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Communicated by Vasudevan Srinivas.
B
Elham Izadi [email protected] Jonathan Conder [email protected] Edward Dewey [email protected]
1
Department of Mathematics, University of California San Diego, 9500 Gilman Drive # 0112, La Jolla, CA 92093-0112, USA
123
J. Conder et al.
Introduction Let A be a principally polarized abelian variety (ppav) of dimension g ≥ 4 with smooth symmetric theta divisor . By the Lefschetz hyperplane theorem and Poincaré Duality (see, e.g., [16]), the cohomology of is determined by that of A except in the middle dimension g − 1. The primitive cohomology of , in the sense of Lefschetz, is ∪[] Hprg−1 (, Z) := Ker H g−1 (, Z) −−−→ H g+1 (, Z) . The primal cohomology of is defined as (see [15,16]) K := Ker H
g−1
i∗
(, Z) −→ H
g+1
(A, Z) g−1
where i : → A is the inclusion. This is a Hodge substructure of Hpr (, Z) 1 2g of rank g! − g+1 g and level g − 3 (coniveau 1) while the primitive cohomology g−1
Hpr (, Z) has full level g − 1 (coniveau 0). The action of −1 splits KQ := K ⊗ Q into the direct sum of its invariant piece K+ Q and its anti-invariant piece K− . As shown in [17, Lemma 6.1], the Hodge structure Q
K(−1) has level g − 3 while the Hodge structure K(−1) has level g − 5 (coniveau 2). g−1 The primal cohomology K and its Hodge substructure K(−1) are therefore interesting test cases for the general Hodge conjecture. The general Hodge conjecture p
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