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 Takahashiâ€™s conjecture in this special case. Furthermore, we also study the blow-up behaviour of the flow in the unstable case i.e. 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-Goerenstein 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.

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Last updated: 13 Sep 2024