We solve the Dirichlet problem for complex Monge–Ampère equation near an isolate Klt singularity, which generalizes result of Eyssidieux et al. [Singular Kähler–Einstein metrics, J. Amer. Math. Soc. 22 (2009) 607–639], where is solved on singular varieties without boundary. As a corollary, we construct solutions to with isolated singularity strongly pseudoconvex domain [Formula: see text] conta...

In this paper we show that stability for holomorphic vector bundles are equivalent to the existence of solutions to certain system of Monge Amp ere equations parametrized by a parameter k. We solve this fully nonlinear elliptic system by singular perturbation technique and show that the vanishing of obstructions for the perturbation is given precisely by the stability condition. This can be int...

Extending DiNezza-Lu's approach to the setting of big cohomology classes, we prove that solutions degenerate complex Monge-Amp{\`e}re equations on compact K{\"a}hler manifolds are continuous a Zariski open set. This allows us show singular K{\"a}hler-Einstein metrics log canonical varieties general type have potentials ample locus outside non-klt part.

We define Chern and Segre forms, or rather currents, associated with a Griffiths positive singular hermitian metric $h$ analytic singularities on holomorphic vector bundle $E$. The currents are constructed as pushforwards of generalized Monge-Amp\`ere products the projectivization represent classes $E$, respectively, coincide forms $E$ $h$, where is smooth. Moreover, our by first three authors ...

Optimal mass transportation can be traced back to Gaspard Monge’s famous paper of 1781: ‘Mémoire sur la théorie des déblais et des remblais’. The problem there is to minimize the cost of transporting a given distribution of mass from one location to another. Since then, it has become a classical subject in probability theory, economics and optimization. At the end of the 80’s, the seminal work ...