Abstract:

We report a recent joint work with Biao Ma. Fully non-linear PDEs play crucial roles in Kahler geometry. Classical Monge-Ampere equation solved by Yau and J-equation posed by Donaldson are 2 examples that have deep applications. the dHYM equation is another example. We purpose a general form of PDEs on Kahler manifolds that include many classical examples. Based on previous works of many, we pose proper positivity conditions under which these PDEs can be properly studied. We then establish analytical and algebraic criteria for the existence of solutions of PDEs, which extends many existing results on special cases.