Abstract: The dHYM equation is a special type of complex Hessian equations which has connection to mirror symmetry in string theory. Recently, in the smooth setting it is shown that there exists a smooth solution of the “super-critical” dHYM equation on compact K\”ahler manifolds if and only if certain Nakai-Moishezon type criterion holds. In this talk, we will focus when the NM-type criterion fails – which is the so-called “unstable” case. We will show the existence and uniqueness of solutions of the “weak” dHYM equation, where the wedge product is replaced by the non-pluripolar product. We will also discuss the convergence of the (dHYM) cotangent flow in the unstable case. Based on a joint work with Prof. Ved Datar (IISc, Bengaluru) and Prof. Jian Song (Rutgers University).

Short Biography: Ramesh Mete is currently an Integrated PhD student at Indian Institute of Science, Bengaluru, under the supervision of Prof. Ved V. Datar. He is interested in Complex Differential Geometry.

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