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 K\"{a}hler metric, always admits a solution. In particular, we describe the Lagrangian phase, with respect to any invariant K\"{a}hler metric, of every closed invariant $(1,1)$-form in terms of Lie theory. Building on this, we characterize all supercritical and hypercritical homogeneous solutions of the dHYM equation using the Cartan matrix associated with the underlying complex simple Lie algebra. Further, we provide an explicit formula, in terms of Lie theory, for the slope of holomorphic vector bundles over rational homogeneous varieties. Using this formula, we derive a new criterion for slope semistability through restrictions of holomorphic vector bundles to the generators of the associated cone of curves. Moreover, we provide a new characterization, in terms of central charges defined by rational curves, for slope (semi)stability of holomorphic vector bundles over rational homogeneous varieties. As an application of our main results, we describe all supercritical and hypercritical homogeneous solutions of the dHYM equation on the Fano threefold defined by the Wallach flag manifold ${\mathbb{P}}(T_{{\mathbb{P}^{2}}})$. In addition, 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. In this last case, we describe explicitly all associated Hermitian Yang-Mills instantons. We also present some new insights that explore the interplay between intersection theory and number theory.