Stability for Line Bundles and Deformed Hermitian-Yang-Mills Equation on Some Elliptic Surfaces

TL;DR

This paper investigates the relationship between twisted ampleness and stability conditions for line bundles on certain elliptic surfaces, providing new insights into the existence of solutions to the deformed Hermitian-Yang-Mills equation.

Contribution

It establishes a link between twisted ampleness and Bridgeland stability for line bundles on Weierstrass elliptic K3 surfaces, answering a specific open question.

Findings

Twisted ampleness implies Bridgeland stability for certain line bundles.

Results apply to line bundles with fiber degree 1 on elliptic K3 surfaces.

Provides conditions under which the deformed Hermitian-Yang-Mills equation has solutions.

Abstract

We study the twisted ampleness criterion due to Collins, Jacob and Yau on surfaces, which is equivalent to the existence of solutions to the deformed Hermitian-Yang-Mills (dHYM) equation. When $X$ is a Weierstrass elliptic K3 surface, and $\omega$ an ample class such that $\omega$ lies in the span of a section class and the fiber class, we show that for a class of line bundles $L$ with fiber degree 1 and $\omega c_1(L)>0$, the twisted ampleness of $L$ respect to $\omega$, always implies the $\sigma_{\omega, 0}$-stability (Bridgeland stability) of $L$. This answers a question by Collins and Yau for a class of examples.