Poincar\'e type J-equation

TL;DR

This paper develops a new approach using a two-parameter continuity path to solve the J-equation with Poincaré type singularities on Kähler manifolds, linking solutions to divisor geometry.

Contribution

It introduces a novel continuity method for the J-equation with Poincaré singularities and establishes solvability criteria related to divisor properties and K-energy bounds.

Findings

Solvability characterized by a two-parameter continuity path.

Classical subsolution condition ensures solutions on Kähler surfaces.

Existence of solutions implies related divisor equations.

Abstract

We introduce a two-parameter continuity path for the J-equation and use it to characterize the solvability of the J-equation for K\"ahler metrics with Poincar\'e type singularities along a divisor $D$, allowing simple normal crossings and self-intersections. On K\"ahler surfaces, we show that the classical subsolution condition in the smooth setting implies solvability in the Poincar\'e type setting for any smooth divisor $D$. As a consequence, if $X$ contains no curves of negative self-intersections and $K_X[D]$ is ample, then the K-energy is bounded from below on any Poincar\'e type K\"ahler class. In the smooth divisor case, we further analyze the asymptotic behavior of solutions near $D$, and show that existence of a Poincar\'e type solution implies existence of a solution to the J-equation on $D$.