Title: On existence and regularity of some complex Hessian equations on Kahler and transverse Kahler manifolds
Speaker: P. Sivaram (IISc Mathematics)
Date: 08 February 2024
Time: 3:30 pm
Venue: Hybrid - Microsoft Teams (online) and LH-3 LH-1, Mathematics Department
The thesis consists of two parts. In the first part, we study the modified J-flow, introduced by Li-Shi. Analogous to the Lejmi-Szekelyhidi conjecture for the J-equation,
Takahashi has conjectured that the solution to the modified J-equation exists if and only if some intersection numbers are positive and has verified the conjecture for
toric manifolds. We study the behaviour of the modified J-flow on the blow-up of the projective spaces for rotationally symmetric metrics using the Calabi ansatz and obtain
another proof of Takahashi’s conjecture in this special case. Furthermore, we also study the blow-up behaviour of the flow in the unstable case ie. when the positivity conditions fail.
We prove that the flow develops singularities along a co-dimension one sub-variety. Moreover, away from this singular set, the flow converges to a solution of the modified J-equation,
albeit with a different slope.
In the second part, we will describe a new proof of the regularity of conical Ricci flat metrics on Q-Gorenstein T-varieties. Such metrics arise naturally as singular
models for Gromov-Hausdorff limits of Kahler-Einstein manifolds. The regularity result was first proved by Berman for toric manifolds and by Tran-Trung Nghiem in general.
Nghiem adapted the pluri-potential theoretic approach of Kolodziej to the transverse Kahler setting. We instead adapt the purely PDE approach to $L^\infty$ estimates due to
Guo-Phong et al. to the transverse Kahler setting, and thereby obtain a purely PDE proof of the regularity result.