I will discuss a joint work with Olivier Biquard about degenerating conic Kähler-Einstein metrics by letting the cone angle go to zero. In the case where one is given a smooth anticanonical divisor $D$ in a Fano manifold $X$, I will explain how the complete Ricci flat Tian-Yau metric on $X \smallsetminus D$ appears as rescaled limit of such conic KE metrics. 

I will construct the differential geometric, gauge-theoretic, and duality covariant model of classical four-dimensional Abelian gauge theory on an orientable four-manifold of arbitrary topology. I will do so by implementing the Dirac-Schwinger-Zwanziger (DSZ) integrality condition in classical Abelian gauge theories with general duality structure and interpreting the associated sheaf cohomology groups geometrically. As a result, I will obtain that four-dimensional Abelian gauge theories are theories of connections on Siegel bundles, namely principal bundles whose structure group is the generically non-abelian disconnected group of automorphisms of an integral affine symplectic torus. This differential-geometric model includes the electric and magnetic gauge potentials on an equal footing and describes the equations of motion through a first-order polarized self-duality condition for the curvature of a connection. This condition is reminiscent of the theory of four-dimensional Euclidean instantons, even though we consider a two-derivative theory in Lorentzian signature. Finally, I will elaborate on various applications of this differential-geometric model, including a mathematically rigorous description of electromagnetic duality in Abelian gauge theory and the reduction of the polarized self-duality condition to a Riemannian three-manifold, which gives as a result a new type of Bogomolny equation.

We describe an approach to Bialynicki-Birula theory for holomorphic $\mathbb{C}^*$ actions on complex analytic spaces and Morse-Bott theory for Hamiltonian functions for the induced circle actions. A key principle is that positivity of a suitably defined "virtual Morse-Bott index" at a critical point of the Hamiltonian function implies that the critical point cannot be a local minimum even when it is a singular point in the moduli space. Inspired by Hitchin’s 1987 study of the moduli space of Higgs monopoles over Riemann surfaces, we apply our method in the context of the moduli space of non-Abelian monopoles or, equivalently, stable holomorphic pairs over a closed, complex, Kaehler surface. We use the Hirzebruch-Riemann-Roch Theorem to compute virtual Morse-Bott indices of all critical strata (Seiberg-Witten moduli subspaces) and show that these indices are positive in a setting motivated by a conjecture that all closed, smooth four-manifolds of Seiberg-Witten simple type (including symplectic four-manifolds) obey the Bogomolov-Miyaoka-Yau inequality.

Pluripotential theory studies plurisubharmonic functions and complex Monge-Ampère equations on complex manifolds, and has played a key role in recent progress on Kähler-Einstein and constant scalar curvature Kähler metrics. This theory admits a non-Archimedean analogue over Berkovich spaces, that can be used to study K-stability. The purpose of this talk is to provide an introduction to this circle of ideas, and to discuss more specifically recent joint work with Mattias Jonsson studying Green's functions in this context.