Pluriclosed flow and the geometrization of complex surfaces

TL;DR

This paper reviews pluriclosed flow on complex surfaces, discusses its long-term behavior, and proposes a conjectural geometrization approach that impacts surface topology and Kähler structure classification.

Contribution

It provides a comprehensive survey of pluriclosed flow, formulates a conjecture on its long-term behavior, and links it to complex surface topology and generalized Kähler structures.

Findings

Survey of existence and convergence results

Formulation of a conjecture on flow behavior

Implications for complex surface topology

Abstract

We recall fundamental aspects of the pluriclosed flow equation and survey various existence and convergence results, and the various analytic techniques used to establish them. Building on this, we formulate a precise conjectural description of the long time behavior of the flow on complex surfaces. This suggests an attendant geometrization conjecture which has implications for the topology of complex surfaces and the classification of generalized K\"ahler structures.