Hodge homotheties, algebraic classes, and Kuga-Satake varieties
| Autor: | Varesco, Mauro |
|---|---|
| Rok vydání: | 2023 |
| Předmět: | |
| DOI: | 10.48550/arxiv.2304.02519 |
| Popis: | We introduce in this paper the notion of Hodge homotheties of transcendental lattices of hyperk\"ahler manifolds and investigate the Hodge conjecture for these Hodge morphisms. Studying K3 surfaces with a symplectic automorphism, we prove the Hodge conjecture for the square of the general member of the first four-dimensional families of K3 surfaces with totally real multiplication of degree two. We then show the functoriality of the Kuga--Satake construction with respect to Hodge homotheties. This implies that, if the Kuga--Satake Hodge conjecture holds for two hyperk\"ahler manifolds, then every Hodge homothety between their transcendental lattices is algebraic after composing it with the Lefschetz isomorphism. In particular, we deduce that Hodge homotheties of transcendental lattices of hyperk\"ahler manifolds of generalized Kummer deformation type are algebraic. Comment: 22 pages, added Theorem 0.4 which proves the algebraicity of Hodge homotheties of transcendental lattices of hyperk\"ahler manifolds of generalized Kummer deformation type. Comments are welcome |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|