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

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.