Continuity of delta invariants and twisted Kähler--Einstein metrics

Continuity of Monge-Amp{è}re Potentials in Big Cohomology Classes

Extending DiNezza-Lu's approach to the setting of big cohomology classes, we prove that solutions of degenerate complex Monge-Amp{è}re equations on compact K{ä}hler manifolds are continuous on a Zariski open set. This allows us to show that singular K{ä}hler-Einstein metrics on log canonical varieties of general type have continuous potentials on the ample locus outside of the non-klt part.

Moment polytopes in real symplectic geometry I

Let Z be the real part of a K{ä}hler Hamiltonian manifold M. The O'Shea-Sjamaar's Theorem tells us that the moment polytope Delta(Z) corresponds to the anti-invariant part of the Kirwan polytope Delta(M). The purpose of the present paper is to explain how to parameterize the equations of the facets of Delta(Z) in terms of real Ressayre's pairs of Z.

Kähler-Einstein metrics on group compactifications

We obtain a necessary and sufficient condition of existence of a K{ä}hler-Einstein metric on a $G\times G$-equivariant Fano compactification of a complex connected reductive group $G$ in terms of the associated polytope. This condition is not equivalent to the vanishing of the Futaki invariant. The proof relies on the continuity method and its translation into a real Monge-Amp{è}re equation, using the invariance under the action of a maximal compact subgroup $K\times K$.

klt对上的Demailly-Kollár连续性及其在alpha和delta不变量中的应用

Demailly-Kollár continuity on klt pairs, and applications to alpha and delta invariants

该研究建立了klt对上的Demailly-Kollár型连续性定理，证明了两类alpha与delta不变量的版本相等，解决了相关猜想并给出了扭曲Kähler-Einstein方程可解性的新判据。

Twisted and Singular gravitating vortices

We introduce the notion of twisted gravitating vortex on a compact Riemann surface. If the genus of the Riemann surface is greater than 1 and the twisting forms have suitable signs, we prove an existence and uniqueness result for suitable range of the coupling constant generalizing the result of arXiv:1510.03810v2 in the non twisted setting. It is proved via solving a continuity path deforming the coupling constant from 0 for which the system decouples as twisted Kähler-Einstein metric and twisted vortices. Moreover, specializing to a family of twisting forms smoothing delta distribution terms…