Complex Monge-Amp\`ere equation for positive $(p,p)$-forms on compact K\"ahler manifolds

TL;DR

This paper introduces a new complex Monge-Ampère equation for differential forms on compact Kähler manifolds, proves existence and uniqueness of solutions for certain cases, and extends geometric flows to higher-order forms with convergence results.

Contribution

It generalizes the classical Monge-Ampère equation to higher-degree forms and develops a new geometric flow extending the K"ahler-Ricci flow to these forms.

Findings

Existence of smooth solutions for the equation when 1 ≤ p < n.

Extension of the K"ahler-Ricci flow to (p,p)-forms.

Convergence of the flow under certain conditions.

Abstract

A complex Monge-Amp\`ere equation for differential $(p,p)$-forms is introduced on compact K\"ahler manifolds. For any $1 \leq p < n$, we show the existence of smooth solutions unique up to adding constants. For $p=1$, this corresponds to the Calabi-Yau theorem proved by S. T. Yau, and for $p=n-1$, this gives the Monge-Amp\`ere equation for $(n-1)$ plurisubharmonic functions studied by Tosatti-Weinkove. For other $p$ values, this defines a non-linear PDE that falls outside of the general framework of Caffarelli-Nirenberg-Spruck. Further, we define a geometric flow for higher-order forms that preserves their cohomology classes, and extends the K\"ahler-Ricci flow naturally to $(p,p)$-forms. As a consequence of our main theorem, we show that this flow exists in a maximal time interval and can be shown to converge under some assumptions. A modified flow is introduced and the convergence of the associated normalized flow is shown.