The deformed Hermitian-Yang-Mills equation on the blowup of $\mathbb P^n$

TL;DR

This paper investigates the deformed Hermitian-Yang-Mills equation on the blowup of complex projective space, providing solutions under symmetry conditions and supporting a broader conjecture in Kähler geometry.

Contribution

It introduces a method to solve the equation via an ODE approach under symmetry and algebraic stability, advancing understanding of the conjecture for compact Kähler manifolds.

Findings

Equation reduces to an ODE with symmetry

Solutions exist under algebraic stability conditions

Supports conjecture relating to Kähler manifolds

Abstract

We study the deformed Hermitian-Yang-Mills equation on the blowup of complex projective space. Using symmetry, we express the equation as an ODE which can be solved using combinatorial methods if an algebraic stability condition is satisfied. This gives evidence towards a conjecture of the first author, T.C. Collins, and S.-T. Yau on general compact Kahler manifolds.