3,013 research outputs found
The Tate conjecture for K3 surfaces in odd characteristic
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
On automorphisms of moduli spaces of parabolic vector bundles
Fix general points , and a weight
vector of real numbers . Consider the moduli space parametrizing rank
two parabolic vector bundles with trivial determinant on which are semistable with respect to . Under
some conditions on the weights, we determine and give a modular interpretation
for the automorphism group of the moduli space . It
is isomorphic to for some
, and is generated by admissible elementary
transformations of parabolic vector bundles. The largest of these automorphism
groups, with , occurs for the central weight . The corresponding moduli space
is a Fano variety of dimension , which is
smooth if is odd, and has isolated singularities if is even.Comment: 13 page
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