A note on the supercritical deformed Hermitian-Yang-Mills equation

Abstract

We show that on a compact Kähler manifold all real $(1,1)$-classes admitting solutions to the supercritical deformed Hermitian-Yang-Mills equation form a both open and closed subset of those which satisfy the numerical condition proposed by Collins-Jacob-Yau. More importantly, we show by examples that it can be a proper subset. This disproves a conjecture made by Collins-Jacob-Yau.

arXiv comment6 pages. v6: easier examples of the main theorem are given following the suggestion of Mao Sheng; minor corrections; v5-v1:references added and the discussion section is rewritten; acknowledge to V.P. Pingali is added; the proof of openness is modified; reference added and the claim of disconnectedness, for which the previous argument contains a mistake, is deleted