Related papers: Complex Hessian equations with prescribed singular…

In this paper, we concern with the existence of solutions of the complex $m-$Hessian type equation $-\chi(u)H_{m}(u)=\mu$ in the class $\mathcal{E}_{m,\chi}(f,\Omega)$ if there exists subsolution in this class, where the given boundary…

We prove that on a compact K\"ahler manifold, the non-pluripolar Monge-Amp\`ere mass of a $\theta$-psh function decreases as the singularities increase. This was conjectured by Boucksom-Eyssidieux-Guedj-Zeriahi who proved it under the…

We prove that when Hodge theory survives on non-compact symplectic manifolds, a compact symplectic Lie group action having fixed points is necessarily Hamiltonian, provided the associated almost complex structure preserves the space of…

The $J$-equation proposed by Donaldson is a complex Hessian quotient equation on K\"ahler manifolds. The solvability of the $J$-equation is proved by Song-Weinkove to be equivalent to the existence of a subsolution. It is also conjectured…

We solve the Dirichlet problem for $k$-Hessian equations on compact complex manifolds with boundary, given the existence of a subsolution. Our method is based on a second order a priori estimate of the solution on the boundary with a…

We prove strong hypercontractivity (SHC) inequalities for logarithmically subharmonic functions on $\RR^n$ and different classes of measures: Gaussian measures on $\RR^n$, symmetric Bernoulli and symmetric uniform probability measures on…

Let $X$ be a compact K\"ahler manifold. Given a big cohomology class $\{\theta\}$, there is a natural equivalence relation on the space of $\theta$-psh functions giving rise to $\mathcal S(X,\theta)$, the space of singularity types of…

We study viscosity solutions to complex hessian equations. In the local case, we consider $\Omega$ a bounded domain in $\mathbb{C}^n,$ $\beta$ the standard K\"{a}hler form in $\mathcal{C}^n$ and $1\leq m\leq n.$ Under some suitable…

We show that compact K\"ahler manifolds have the rational cohomology ring of complex projective space provided a weighted sum of the lowest three eigenvalues of the K\"ahler curvature operator is positive. This follows from a more general…

Let $(M, g)$ be a closed Riemannian manifold of dimension $n \geq 3$, and let $h \in C^1(M)$ be such that the operator $\Delta_g + h$ is coercive. Fix $x_0 \in M$ and $s \in (0, 2)$. We obtain uniform bounds on the solutions of the critical…

We survey the Dirichlet problem for the complex Homogeneous Monge-Amp\`ere Equation, both in the case of domains in $\mathbb C^n$ and the case of compact K\"ahler manifolds parametrized by a Riemann surface with boundary. We then give a…

We give a sufficient condition on a sequence of uniformly bounded $\omega$-plurisubharmonic functions, $\omega$ being a Hermitian metric, for which the sequence of associated Monge-Amp\`ere measures converges weakly. This criterion can be…

For a K\"{a}hler manifold endowed with a weighted measure $e^{-f}\,dv,$ the associated weighted Hodge Laplacian $\Delta _{f}$ maps the space of $(p,q)$-forms to itself if and only if the $(1,0)$-part of the gradient vector field $\nabla f$…

We prove that if a positive closed current is bounded by another one with bounded, continuous or Hoelder continuous super-potentials, then it inherits the same property. There are two different methods to define wedge-products of positive…

Let M be a simply-connected complete Kahler manifold whose sectional curvature is bounded between two negative numbers. In this paper we prove the existence of non-constant bounded holomorphic functions on M if the complex dimension of M is…

We prove that quasi-plurisubharmonic envelopes with prescribed analytic singularities in suitable big cohomology classes on compact K\"ahler manifolds have the optimal $C^{1,1}$ regularity on a Zariski open set. This also proves regularity…

In this article, we discuss the spaces of harmonic forms $\mathcal{H}^{\bullet}_{d}$ over a closed almost K\"{a}hler manifold $(X, J,\omega)$. We show that if the almost complex structure $J$ on the almost K\"{a}hler manifold $X$ is not too…

The Hitchin-Simpson equations are first-order non-linear equations for a pair consisting of a connection and a Higgs field. In this paper, we study the behavior of sequences of solutions to the Hitchin-Simpson equations on closed K\"ahler…

We prove a regularity result for the Monge--Amp\`ere equations on compact Kaehler manifolds with degenerate rhs member.

We prove that for every compact K\"ahler manifold $X$ there exists an $L$-infinity morphism, lifting the usual cup product in cohomology, from the Kodaira-Spencer differential graded Lie algebra to the suspension of the space of linear…