Thời gian: 16:30, thứ năm, 27/10/202

Tóm tắt: Since Yau's resolution of the Calabi conjecture, the problem of finding canonical metrics (for example Kähler-Einstein metrics) on compact Kähler manifolds has been crucial in the development of Kähler geometry. The problem boils down to solving a complex Monge-Ampère equation. On manifolds with non-negative first Chern class, there is no obstruction to the existence of Kähler-Einstein metrics, as shown by Yau and Aubin. However, on Fano manifolds (i.e. manifolds with positive first Chern class), it was conjectured by Yau-Tian-Donaldson that a purely algebro-geometric obstruction, called K-stability, would be equivalent to the existence. This conjecture was proved by Chen-Donaldson-Sun in 2015.

In the singular landscape, the problem of finding Kähler-Einstein metrics on compact Kähler varieties with mild singularities and non-positive first Chern class was solved by Eyssidieux-Guedj-Zeriahi, largely extending Yau and Aubin's theorem. An analog of YTD conjecture for mildly singular Fano varieties was recently established by Chi Li. In the non-compact case, various authors, notably Collins-Székelyhidi, have established an equivalence between a suitable notion of K-stability and the existence of conical Calabi-Yau metrics on a class of normal affine varieties.

Despite all the advances, the K-stability condition remains very hard to check in practice. However, on low-complexity varieties admitting simple classification, such as spherical varieties, the complex geometry is translated to convex geometry, and K-stability becomes a purely combinatorial condition. For example, a toric Fano manifold is Kähler-Einstein if and only the barycenter of its moment polytope is 0. The goal of this talk is to present a new result concerning the equivalence between a volume minimization principle for Euclidean cones and the existence of conical Calabi-Yau metrics on a class of spherical cones, using an analytic approach. After a review on the Luna-Vust theory of spherical embeddings, I will give a combinatorial classification of spherical cones, and explain the result.

Hình thức: Offline tại phòng 602 và online qua google meet, cụ thể https://meet.google.com/yep-kbzk-eao?pli=1&authuser=1