Positivity properties of the vector bundle Monge-Amp\`ere equation

TL;DR

This paper investigates positivity properties of solutions to a vector bundle Monge-Ampère equation, establishing conditions under which positivity is preserved for certain ranks and dimensions, with implications for complex geometry.

Contribution

It demonstrates positivity preservation for rank-two bundles over complex surfaces and explores limitations for higher ranks and dimensions, setting up a continuity method for symmetric bundles.

Findings

Positivity preservation holds for rank-two bundles over complex surfaces.

Higher rank bundles do not necessarily preserve positivity at algebraic solutions.

A continuity path shows openness of positivity preservation in specific symmetric cases.

Abstract

We study MA-positivity, a notion of positivity relevant to a vector bundle version of the complex Monge--Amp\`ere equation introduced in an earlier work, and show that for rank-two holomorphic bundles over complex surfaces, MA-semi-positive solutions of the vector bundle Monge--Amp\`ere (vbMA) equation are also MA-positive. For vector bundles of rank-three and higher, over complex manifolds of dimension greater than one, we show that this positivity-preservation property need not hold for an algebraic solution of the vbMA equation treated as a purely algebraic equation at a given point. Finally, we set up a continuity path for certain classes of highly symmetric rank-two vector bundles over complex three-folds and prove a restricted version of positivity preservation which is nevertheless sufficient to prove openness along this continuity path.