Positivity properties of the deformed Hermitian-Yang--Mills and related equations

Abstract

In the first part of the talk, we briefly introduce the deformed Hermitian-Yang-Mills (dHYM) equation and discuss a result regarding the solvability of the ​twisted dHYM equation on compact Kähler three-folds with slightly negative twisting functions. We then indicate how this result, along with a theorem of Chen and methods introduced by Datar–Pingali, can be used to show that the twisted dHYM equation on compact, projective manifolds can be solved if certain non-uniform numerical positivity conditions analogous to the ones used in the Demailly–Paun characterization of Kähler cones are satisfied. As a corollary, one obtains another proof, in the projective case, of a theorem of Chu–Lee–Takahashi addressing a conjecture of Collins–Jacob–Yau.

For the second part of the talk, we turn our attention to Monge-Ampère-positivity (MA-positivity), a notion of positivity introduced by Pingali for the study of a generalization of the complex Monge-Ampère equation to vector bundles. In particular, preservation of MA-positivity along a continuity path turns out to be crucial in proving the existence of solutions to the vector bundle Monge-Ampère (vbMA) equation. We discuss the preservation of MA-positivity for rank-two holomorphic bundles over complex surfaces and rank-two vortex bundles over complex three-folds. Lastly, we mention the existence of counterexamples to an algebraic version of MA-positivity preservation for vector bundles of rank-three and higher over complex manifolds of dimension greater than one.