Descent theory of simple sheaves on $C_1$-fields

TL;DR

This paper proves that simple sheaves on a projective variety over a $C_1$-field descend from Galois extensions, establishing a correspondence between geometrically stable sheaves and $K$-rational points of moduli spaces.

Contribution

It demonstrates descent of simple sheaves over $C_1$-fields and links stable sheaves to rational points on moduli spaces.

Findings

Simple sheaves descend from Galois extensions.

A bijection exists between stable sheaves and rational points.

The result applies to varieties over any characteristic $C_1$-field.

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.