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Register a new userAsymptotic estimate of cohomology groups valued in pseudo-effective line bundles
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In this paper, we study questions of Demailly and Matsumura on the asymptotic behavior of dimensions of cohomology groups for high tensor powers of (nef) pseudo-effective line bundles over non-necessarily projective algebraic manifolds. By generalizing Sius $partialoverline{partial}$-formula and Berndtssons eigenvalue estimate of $overline{partial}$-Laplacian and combining Bonaveros technique, we obtain the following result: given a holomorphic pseudo-effective line bundle $(L, h_L)$ on a compact Hermitian manifold $(X,omega)$, if $h_L$ is a singular metric with algebraic singularities, then $dim H^{q}(X,L^kotimes Eotimes mathcal{I}(h_L^{k}))leq Ck^{n-q}$ for $k$ large, with $E$ an arbitrary holomorphic vector bundle. As applications, we obtain partial solutions to the questions of Demailly and Matsumura.
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We shall give an explicit estimate of the lower bound of the Bergman kernel associated to a positive line bundle. In the compact Riemann surface case, our result can be seen as an explicit version of Tians partial $C^0$-estimate.
This paper is a continuation of our article (European J. Math., https://doi.org/10.1007/s40879-020-00419-8). The notion of a poor complex compact manifold was introduced there and the group $Aut(X)$ for a $P^1$-bundle over such a manifold was proven to be very Jordan. We call a group $G$ very Jordan if it contains a normal abelian subgroup $G_0$ such that the orders of finite subgroups of the quotient $G/G_0$ are bounded by a constant depending on $G$ only. In this paper we provide explicit examples of infinite families of poor manifolds of any complex dimension, namely simple tori of algebraic dimension zero. Then we consider a non-trivial holomorphic $P^1$-bundle $(X,p,Y)$ over a non-uniruled complex compact Kaehler manifold $Y$. We prove that $Aut(X)$ is very Jordan provided some additional conditions on the set of sections of $p$ are met. Applications to $P^1$-bundles over non-algebraic complex tori are given.
Let $X$ be a compact connected CR manifold with a transversal CR $S^1$-action of dimension $2n-1$, which is only assumed to be weakly pseudoconvex. Let $Box_b$ be the $overline{partial}_b$-Laplacian. Eigenvalue estimate of $Box_b$ is a fundamental issue both in CR geometry and analysis. In this paper, we are able to obtain a sharp estimate of the number of eigenvalues smaller than or equal to $lambda$ of $Box_b$ acting on the $m$-th Fourier components of smooth $(n-1,q)$-forms on $X$, where $min mathbb{Z}_+$ and $q=0,1,cdots, n-1$. Here the sharp means the growth order with respect to $m$ is sharp. In particular, when $lambda=0$, we obtain the asymptotic estimate of the growth for $m$-th Fourier components $H^{n-1,q}_{b,m}(X)$ of $H^{n-1,q}_b(X)$ as $m rightarrow +infty$. Furthermore, we establish a Serre type duality theorem for Fourier components of Kohn-Rossi cohomology which is of independent interest. As a byproduct, the asymptotic growth of the dimensions of the Fourier components $H^{0,q}_{b,-m}(X)$ for $ min mathbb{Z}_+$ is established. Compared with previous results in this field, the estimate for $lambda=0$ already improves very much the corresponding estimate of Hsiao and Li . We also give appilcations of our main results, including Morse type inequalities, asymptotic Riemann-Roch type theorem, Grauert-Riemenscheider type criterion, and an orbifold version of our main results which answers an open problem.
We introduce several families of filtrations on the space of vector bundles over a smooth projective variety. These filtrations are defined using the large k asymptotics of the kernel of the Dolbeault Dirac operator on a bundle twisted by the kth power of an ample line bundle. The filtrations measure the failure of the bundle to admit a holomorphic structure. We study compatibility under the Chern isomorphism of these filtrations with the Hodge filtration on cohomology.
There is a standard method to calculate the cohomology of torus-invariant sheaves $L$ on a toric variety via the simplicial cohomology of associated subsets $V(L)$ of the space $N_{mathbb R}$ of 1-parameter subgroups of the torus. For a line bundle $L$ represented by a formal difference $Delta^+-Delta^-$ of polyhedra in the character space $M_{mathbb R}$, [ABKW18] contains a simpler formula for the cohomology of $L$, replacing $V(L)$ by the set-theoretic difference $Delta^- setminus Delta^+$. Here, we provide a short and direct proof of this formula.
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