Descent theory of simple sheaves on $C_1$-fields

Abstract

Let $K$ be a $C_1$-field of any characteristic and $X$ a projective variety over $K$. In this article we prove that for a finite Galois extension $L$ of $K$, a simple sheaf with covering datum on $X{\times }_{K}L$ descends to a simple sheaf on $X$. As a consequence, we show that there is a $1-1$ correspondence between the set of geometrically stable sheaves on $X$ with fixed Hibert polynomial $P$ and the set of $K$-rational points of the corresponding moduli space.

Appeared in the Proceedings of the ICM 2018 satellite 'Moduli spaces in Algebraic Geometry and Applications'

References in corpus (2)

• A pathological case of the $C_1$ conjecture in mixed characteristic
• Examples of varieties with index one on $C_1$-fields