K-稳定性与典则度量存在性的研究

此项目以Yau-Tian-Donaldson猜想为出发点，主要关注复代数几何中的K-稳定性及其相应典则度量的存在性。我们将寻找合适的代数不变量来判断K-稳定性，同时也将寻找合适的解析不变量刻画典则度量的存在性，并进一步建立两者的等式关系。具体地，我们将研究具有klt奇点的，以及具有非平凡全纯向量场的代数簇上的delta不变量以及Moser-Trudinger不等式最佳指数之间的关系；也将研究一般极化流形上的K-稳定性的代数判别法以及常数量曲率Kahler度量存在性的解析判别法。

Motivated by the Yau-Tian-Donaldson conjecture, this project mainly focuses on the notion of K-stability in complex algebraic geometry and the existence of the corresponding canonical metrics. We will search for algebraic invaraints to test K-stability, meanwhile also search for suitable analytic invariants to characterize the existence of canonical metrics, and further establish an equivalent relation between these two. More specifically, we will study the relation between the delta invariant and the optimal exponent of the Moser-Trudinger inequality on varieties with klt singularities and also with non-trivial holomorphic vector fields; we will also study algebraic criteria for K-stability and analytic criteria for the existence of constant scalar curvature Kahler metrics on general polarized manifolds.