Deformed Hermitian Yang–Mills equation on rational homogeneous varieties

Abstract

In this paper, we show that the deformed Hermitian Yang–Mills (dHYM) equation on a rational homogeneous variety, equipped with any invariant Kähler metric, always admits a solution. Unlike the known existence results for projective varieties, our result shows that in the homogeneous setting no additional hypotheses on the phase angle (such as the supercritical condition) is necessary to solve the dHYM equation. Moreover, we describe the Lagrangian phase, with respect to any invariant Kähler metric, of every closed invariant $(1,1)$-form in terms of Lie theory. Additionally, we provide an explicit formula, in terms of Lie theory, for the slope of torsion-free coherent sheaves on rational homogeneous varieties. Using this formula, we establish a new criterion for the slope semistability of holomorphic vector bundles over these varieties by restricting the bundle to the generators of the associated cone of curves. Furthermore, we provide a new characterization of slope (semi)stability for holomorphic vector bundles over rational homogeneous varieties in terms of central charges defined by rational curves. As a byproduct, we introduce a constructive method to obtain non-trivial examples of Hermitian–Einstein metrics on certain holomorphic vector bundles over $\mathbb{P}(T_{{\mathbb{P}}^2})$ from solutions of linear Diophantine equations. Also, motivated by the problem of constructing line bundles with prescribed slope, we present some new insights that explore the interplay between elementary number theory and combinatorics via Schubert calculus.