A Remark on Stability Conditions on Smooth Projective Varieties

TL;DR

This paper proves that the bounded derived category of coherent sheaves on any smooth projective variety over complex numbers always admits Bridgeland stability conditions, confirming a fundamental conjecture in algebraic geometry.

Contribution

It establishes the existence of stability conditions on the derived category for all smooth projective varieties, a significant advancement in the field.

Findings

Existence of Bridgeland stability conditions on $ ext{D}^b(X)$ for all smooth projective varieties.

Confirms a key conjecture in the theory of stability conditions.

Provides a foundational result applicable to various problems in algebraic geometry.

Abstract

Let $X$ be a smooth projective variety over $\mathbb C$. In this paper, we prove that $\mathrm{D}^b(X)$, the bounded derived category of coherent sheaves on $X$, always admits stability conditions in the sense of Bridgeland.