Informal Geometric Analysis Archives for Academic Year 2021

Degenerating conic Kähler-Einstein metrics

When: Tue, August 31, 2021 - 3:30pm
Where: Math3206 AND
Speaker: Henri Guenancia (CNRS/Toulouse) -
Abstract: I will discuss a recent joint work with Olivier Biquard about conic Kähler-Einstein metric with cone angle going to zero. We study two situations, one in negative curvature (toroidal compactifications of ball quotients) and one in positive curvature (on Fano manifolds endowed with a smooth anticanonical divisor) leading up to the resolution of a question asked by Donaldson in 2011.

Complex Monge-Ampere equation with solutions in finite energy classes

When: Tue, September 28, 2021 - 3:30pm
Speaker: Duc Viet Vu (University of Cologne) -
Abstract: The notion of pluricomplex energy was introduced by U. Cegrell in 1988. Since then it has played an important role in complex Monge-Ampere equations. I present a recent joint work with Do Duc Thai in which we characterize the class of probability measures on a compact Kahler manifold such that the associated Monge-Ampere equation has a solution of finite energy. Such a characterization was previously only known for particular cases.

Metric SYZ conjecture

When: Tue, October 5, 2021 - 3:30pm
Speaker: Yang Li (MIT) -
Abstract: I will discuss some recent progress on the metric version of the SYZ conjecture, which says that for polarized Calabi-Yau manifolds near the large complex structure limit, then on 99% of the manifold there exists a special lagrangian torus fibration. This is unconditionally proved in the case of Fermat hypersurface families, and conditionally proved in general assuming a conjecture in non-archimedean geometry. Time permitting I will try to say a few words about the relative merits of the two methods.

Legendre transforms, convex bodies, and plurisubharmonic metrics

When: Tue, November 2, 2021 - 3:30pm
Speaker: Remi Reboulet (Institut Fourier) -
Abstract: We begin by explaining the correspondence between convex functions on integral polytopes and plurisubharmonic (i.e. "generalized convex") metrics on polarized toric varieties. Under this correspondence, geodesics in the space of toric psh metrics are transformed into affine segments of convex functions. We then show how this result can be extended to more general geodesics of plurisubharmonic metrics in the non-toric case, using a construction of Witt Nyström. If time permits, we will also look into some non-Archimedean aspects of this generalized result, applied to geodesic rays.

Kahler-Einstein metric near isolated log canonical singularity

When: Tue, November 9, 2021 - 3:30pm
Speaker: Xin Fu (UCI) -首页
Abstract: We construct Kahler-Einstein metrics with negative scalar curvature near
an isolated log canonical (non-log terminal) singularity and we continue
to describe the geometry of Kahler-Einstein metric with focus on complex
hyperbolic cusp. This is based on joint work with Ved Datar and Jian

Convergence of the Kähler-Ricci flow on varieties of general type

When: Tue, November 16, 2021 - 3:30pm
Speaker: Tat Dat Tô (Sorbonne) -
Abstract: We study the Kähler-Ricci flow on varieties of general type. We show that the normalized Kähler-Ricci flow exists at all times in the sense of viscosity, is continuous in an open Zariski set and converges to the singular Kähler-Einstein metric. This gives a partial answer to a question of Feldman-Ilmanen-Knopf on defining and constructing weak solutions of the Kähler-Ricci flow.