Related papers: The volume of pseudoeffective line bundles and par…

In this paper, we study the dimension of cohomology of semipositive line bundles over Hermitian manifolds, and obtain an asymptotic estimate for the dimension of the space of harmonic $(0,q)$-forms with values in high tensor powers of a…

Let K be a compact semi-simple Lie group. We classify K-invariant Kaehler structures on the space Kc/(P,P), where Kc is the complexification of K, P is a parabolic subgroup of Kc, and (P,P) the commutator subgroup. For each Kaehler…

We introduce the concept of Bergman bundle attached to a hermitian manifold X, assuming the manifold X to be compact - although the results are local for a large part. The Bergman bundle is some sort of infinite dimensional very ample…

We give the sharp lower bound of the volume product of $n$-dimensional convex bodies which are invariant under a discrete subgroup $SO(K)=\{ g \in SO(n); g(K)=K \}$, where $K$ is an $n$-cube or $n$-simplex. This provides new partial results…

We study a tower of normal coverings over a compact K\"ahler manifold with holomorphic line bundles. When the line bundle is sufficiently positive, we obtain an effective estimate, which implies the Bergman stability. As a consequence, we…

Let $M=P(E)$ be the complex manifold underlying the total space of the projectivization of a holomorphic vector bundle $E \to \Sigma$ over a compact complex curve $\Sigma$ of genus $\ge 2$. Building on ideas of Fujiki, we prove that $M$…

We prove the existence of extremal, non-csc, K\"ahler metrics on certain unstable projectivised vector bundles $\P (E) \to M$ over a cscK-manifold $M$ with discrete holomorphic automorphism group, in certain adiabatic K\"ahler classes. In…

We prove that if M is a closed, connected, oriented, rationally inessential manifold, then the Hofer-Zehnder capacity of the unit disk bundle of the cotangent bundle of M is finite.

A recent celebrated theorem of Diverio-Trapani and Wu-Yau states that a compact K\"ahler manifold admitting a K\"ahler metric of quasi-negative holomorphic sectional curvature has an ample canonical line bundle, confirming a conjecture of…

In this paper we show that the realization in $L^p(X,\nu_\infty)$ of the nonsymmetric Ornstein-Uhlenbeck operator $L$ is sectorial for any $p\in(1,+\infty)$ and we provide an explicit sector of analyticity. Here $(X,\mu_\infty,H_\infty)$ is…

We study spectral behavior of the complex Laplacian on forms with values in the $k^{\text{th}}$ tensor power of a holomorphic line bundle over a smoothly bounded domain with degenerated boundary in a complex manifold. In particular, we…

For $k \in \mathbb{Z}_{>0}$, let $\mathcal{H}^{(k)}_{g,n}$ denote the vector bundle over $\mathfrak{M}_{g,n}$ whose every fiber consists of meromorphic $k$-differentials with poles of order at most $k-1$ on a fixed Riemman surface of genus…

We show the existence of uniformly bounded sequences of increasing numbers of orthonormal sections of powers $L^k$ of a positive holomorphic line bundle $L$ on a compact K\"ahler manifold $M$. In particular, we construct for each positive…

For $\phi$ a metric on the anticanonical bundle, $-K_X$, of a Fano manifold $X$ we consider the volume of $X$ $$ \int_X e^{-\phi}. $$ We prove that the logarithm of the volume is concave along continuous geodesics in the space of positively…

In this paper, we study the regular quantizations of K\"{a}hler manifolds by using the first two coefficients of Bergman function expansions. Firstly, we obtain sufficient and necessary conditions for certain Hermitian holomorphic vector…

We extend Siu's and Sampson's celebrated rigidity results to non-compact domains. More precisely, let $M$ be a smooth quasi-projective variety with universal cover $\tilde M$ and let $\tilde X$ be a symmetric space of non-compact type, a…

An important problem in geometric quantization is that of quantizing certain classes of Lagrangian submanifolds, so-called Bohr-Sommerfeld Lagrangian submanifolds, equipped with a smooth half-density. A procedure for this in the complex…

Let $X$ be a projective manifold, and $D$ be a normal crossing divisor of $X$. By Jost-Zuo's theorem that if we have a reductive representation $\rho$ of the fundamental group $\pi_{1}(X^{*})$ with unipotent local monodromy, where…

For a projective variety $X$ defined over a non-Archimedean complete non-trivially valued field $k$, and a semipositive metrized line bundle $(L, \phi)$ over it, we establish a metric extension result for sections of $L^{\otimes n}$ from a…

The spaces $H^0(M, L^N)$ of holomorphic sections of the powers of an ample line bundle $L$ over a compact K\"ahler manifold $(M,\omega)$ have been generalized by Boutet de Monvel and Guillemin to spaces $H^0_J(M, L^N)$ of `almost…