The Tate conjecture for K3 surfaces in odd characteristic
By Keerthi Madapusi Pera. 2013
164 citations recorded by the source corpus.
We show that the classical Kuga–Satake construction gives rise, away from characteristic \(2\), to an open immersion from the moduli of primitively polarized K3 surfaces (of any fixed degree) to a certain regular integral model for a Shimura variety of orthogonal type. This allows us to attach to every polarized K3 surface in odd characteristic an abelian variety such that divisors on the surface can be identified with certain endomorphisms of the attached abelian variety. In turn, this reduces the Tate conjecture for K3 surfaces over finitely generated fields of odd characteristic to a version of the Tate conjecture for certain endomorphisms on the attached Kuga–Satake abelian variety, which we prove. As a by-product of our methods, we also show that the moduli stack of primitively polarized K3 surfaces of degree \(2d\) is quasi-projective and, when \(d\) is not divisible by \(p^2\), is geometrically irreducible in characteristic \(p\). We indicate how the same method applies to prove the Tate conjecture for co-dimension \(2\) cycles on cubic fourfolds.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/6a729d1f38cc86c185b854ffc17b9442f62225b7
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Geometry and Topology
- D. Huybrechts
- C. Liedtke
- Wansu Kim
- Keerthi Madapusi Pera
- E. Markman
- Kieran G. O’Grady
- P. Stellari
- Emanuele Macrì
In Mathematical Physics
- D. Huybrechts
- Wansu Kim
- C. Liedtke
- Keerthi Madapusi Pera
- P. Stellari
- Emanuele Macrì
- E. Markman
- Keiji Oguiso
Citations in the corpus
- 2-ADIC INTEGRAL CANONICAL MODELS (2016, methodology): “We will assume that the reader is familiar with the methods of [17], and so our treatment will be a bit terse.”
- Lectures on K3 Surfaces (2016, methodology): “However, since global deformations to elliptic K3 surfaces are used, details would need to be checked in positive characteristic, but the irreducibility of the moduli space proved by Madapusi Pera in [385] should be enough.”
- Supersingular K3 surfaces are unirational (2013, background): “c and we discussed the converse direction above. The implication (2)⇒(3) holds in any characteristic, and its converse would follow from the Tate-conjecture for K3 surfaces, which is true in p = 3 by [MP13]. The equivalence (3)⇔(4) holds in every characteristic. In particular (see also Section 4.1),”