A long-standing conjecture in complex geometry says that a compact Hermitian manifold with constant holomorphic sectional curvature must be Kähler when the constant is non-zero and must be Chern flat when the constant is zero. The conjecture is known in complex dimension 2 by the work of Balas-Gauduchon in 1985 (when the constant is zero or negative) and by Apostolov-Davidov-Muskarov in 1996 (when the constant is positive). For higher dimensions, the conjecture is still largely unknown. In this article, we restrict ourselves to pluriclosed manifolds, and confirm the conjecture for the special case of Strominger Kähler-like manifolds, namely, for Hermitian manifolds whose Strominger connection (also known as Bismut connection) obeys all the Kähler symmetries.

The logarithmic capacitary Minkowski problem asks for necessary and sufficient conditions on a finite Borel measure on the unit sphere so that it is the logarithmic capacitary measure of a convex body. This article completely solves the case of discrete measures whose support sets are in general position.

We present some improvements of the Li-Yau heat kernel estimate on a Riemannian manifold with Ricci curvature bounded below. As a consequence we prove a gradient estimate to the heat kernel with an optimal leading term.

We review the recent proof of the N. Takahashi’s conjecture on genus 0 Gromov-Witten invariants of (P², E), where E is a smooth cubic curve in the complex projective plane P². The main idea is the use of the algebraic notion of scattering diagram as a bridge between the world of Gromov-Witten invariants of (P², E) and the world of moduli spaces of coherent sheaves on P². Using this bridge, the N. Takahashi’s conjecture can be translated into a manageable question about moduli spaces of coherent sheaves on P². This survey is based on a three hours lecture series given as part of the Beijing-Zurich moduli workshop in Beijing, 9-12 September 2019.