Let M be a (bounded or not) domain of C^n which is complete with respect to a Kähler metric, or more generally, a complete Kähler manifold with trivial canonical bundle. Let f be a linearly nondegenerate meromorphic map from M to the complex projective space P^m. Under an assumption on the positivity of the pull-back by f of the Fubini-Study form on P^m, we prove that f can not omit a certain number of hyperplanes in subgeneral position in P^m. This is deduced directly from a non-integrated defect relation for such f which generalizes that obtained by Fujimoto in the case where M is a ball.
The purpose of this paper has twofold. The first is to prove a unicity theorem for meromorphic mappings of a complete Kähler manifold M in P^n(C) sharing few hypersurfaces. The second is to give a unicity theorem for the case of differential nondegenerate meromorphic mappings sharing 2N-n+3+2ρ(2N-n+1) hyperplanes in N subgeneral position.
The purpose of this paper is to prove the finiteness theorems for meromorphic mappings of a complete connected Kähler manifold into projective space sharing few hyperplanes in subgeneral position without counting multiplicity, where all zeros with multiplicities more than a certain number are omitted. Our results are extensions and generalizations of some recent ones.
Let $X$ be a compact complex non-Kähler manifold and $f$ a dominant meromorphic self-map of $X$. Examples of such maps are self-maps of Hopf manifolds, Calabi-Eckmann manifolds, non-tori nilmanifolds and their blowups. We prove that if $f$ has a dominant topological degree, then $f$ possesses an equilibrium measure $μ$ satisfying well-known properties as in the Kähler case. The key ingredients are the notion of weakly d.s.h. functions substituting d.s.h. functions in the Kähler case and the use of suitable test functions in Sobolev spaces. A large enough class of holomorphic self-maps with dom…
In this paper, a non-integrated defect relation for meromorphic maps from complete Kähler manifolds $M$ into smooth projective algebraic varieties $V$ intersecting hypersurfaces located in $k$-subgeneral position is proved. The novelty of this result lies in that both the upper bound and the truncation level of our defect relation depend only on $k$, $\dim_{\,\mathbb{C}}(V)$ and the degrees of the hypersurfaces considered. In addition, this defect relation recovers Hirotaka Fujimoto [Theorem 1.1; MR0884636 (88m:32049); Japan. J. Math. (N.S.) 11 (1985), no. 2, 233-264.] when subjected to the sa…
Let $M$ be a complete Kähler manifold, whose universal covering is biholomorphic to a ball $\mathbb B^m(R_0)$ in $\mathbb C^m$ ($0<R_0\le +\infty$). Our first aim in this paper is to study the algebraic dependence problem of differentiably meromorphic mappings. We will show that if $k$ differentibility nondegenerate meromorphic mappings $f^1,\ldots,f^k$ of $M$ into $\mathbb P^n(\mathbb C)\ (n\ge 2)$ satisfying the condition $(C_ρ)$ and sharing few hyperplanes in subgeneral position regardless of multiplicity then $f^1\wedge\cdots\wedge f^k\equiv 0$. For the second aim, we will show that there …
In this paper, we establish a truncated non-integrated defect relation for meromorphic mappings from a complete Kähler manifold into a projective variety intersecting a family of hypersurfaces located in subgeneral position, where the truncation level of the defect is explicitly estimated. Our result generalizes and improves previous ones. In particular, when the family of hypersurfaces located in general position, our theorem will implies the previous result of Min Ru-Sogome. In the last part of this paper we will apply ours to study the distribution of the Gauss map of minimal surfaces.
In this article, we establish a truncated non-integrated defect relation for meromorphic mappings from an $m$-dimensional complete Kähler manifold into $\mathbb P^n(\mathbb C)$ intersecting $q$ hypersurfaces $Q_1,...,Q_q$ in $k$-subgeneral position of degree $d_i$, i.e., the intersection of any $k+1$ hypersurfaces is emptyset. We will prove that $$ \sum_{i=1}^qδ_f^{[u-1]}(Q_i)\le (k-n+1)(n+1)+ε+\frac{ρu(u-1)}{d}, $$ where $u$ is explicitly estimated and $d$ is the least common multiple of $d_i'$s. Our result generalizes and improves previous results. In the last part of this paper we will appl…
Let $M$ be a complete Kähler manifold, whose universal covering is biholomorphic to a ball $\mathbb B^m(R_0)$ in $\mathbb C^m$ ($0<R_0\le +\infty$). In this article, we will show that if three meromorphic mappings $f^1,f^2,f^3$ of $M$ into $\mathbb P^n(\mathbb C)\ (n\ge 2)$ satisfying the condition $(C_ρ)$ and sharing $q\ (q> C+ρK)$ hyperplanes in general position regardless of multiplicity with certain positive constants $K$ and $C <2n$ (explicitly estimated), then there are some algebraic relation between them. A degeneracy theorem for $k\ (2\le k\le n+1)$ meromorphic mappings sharing hyperp…
Let $M$ be a complete Kähler manifold, whose universal covering is biholomorphic to a ball $\mathbb B^m(R_0)$ in $\mathbb C^m$ ($0<R_0\le +\infty$). In this article, we will show that if three meromorphic mappings $f^1,f^2,f^3$ of $M$ into $\mathbb P^n(\mathbb C)\ (n\ge 2)$ satisfying the condition $(C_ρ)$ and sharing $q\ (q> 2n+1+α+ρK)$ hyperplanes in general position regardless of multiplicity with certain positive constants $K$ and $α<1$ (explicitly estimated), then $f^1=f^2$ or $f^2=f^3$ or $f^3=f^1$. Moreover, if the above three mappings share the hyperplanes with mutiplicity counted to l…
The Kähler submanifolds between the ball bundles and the complex Euclidean space
School of Mathematics and Statistics, Henan Normal University（河南师范大学数学与统计学院）；Department of Mathematics, Northwest University（西北大学数学系）；School of Mathematical Sciences, Capital Normal University（首都师范大学数学科学学院）
In 1985 Diederich and Ohsawa proved that every disc bundle over a compact Kähler manifold is weakly 1-complete. In this paper, under certain conditions we generalize this result to the case of fiber bundles over compact Kähler manifolds whose fibers are bounded symmetric domains. Moreover if the bundle is obtained by the diagonal action on the product of irreducible bounded symmetric domains, we show that it is hyperconvex.
It is a consequence of the Jacobi Inversion Theorem that a line bundle over a Riemann surface M of genus g has a meromorphic section having at most g poles, or equivalently, the divisor class of a divisor D over M contains a divisor having at most g poles (counting multiplicities). We explore various analogues of these ideas for vector bundles and associated matrix divisors over M. The most explicit results are for the genus 1 case. We also review and improve earlier results concerning the construction of automorphic or relatively automorphic meromorphic matrix functions having a prescribed nu…
We prove that the normal bundle to the Levi foliation of a smooth Levi flat hypersurface does not admit a Hermitian metric with positive curvature along the leaves in compact Kähler manifolds of dimension at least 3. This represents an answer to a conjecture of Marco Brunella.
We show that an $n$-dimensional compact Kähler manifold $X$ admitting a non-degenerate meromorphic map $f:{\bf C}^n\to X$ of order $ρ_f<2$ is rationally connected.
Let $f$ be a meromorphic correspondence on a compact Kähler manifold $X$ of dimension $k$. Assume that its topological degree is larger than the dynamical degree of order $k-1$. We obtain a quantitative regularity of the equilibrium measure of $f$ in terms of its super-potentials.
For a holomorphic vector bundle over a compact Kähler orbifold, the slope stability of the bundle is shown to be equivalent to the existence of a Hermitian-Einstein metric or to the properness of a certain functional introduced by Donaldson.
This paper concerns obstruction flatness of hypersurfaces $Σ$ that arise as unit sphere bundles $S(E)$ of Griffiths negative Hermitian vector bundles $(E, h)$ over Kähler manifolds $(M, g).$ We prove that if the curvature of $(E, h)$ satisfies a splitting condition and $(M,g)$ has constant Ricci eigenvalues, then $S(E)$ is obstruction flat. If, in addition, all these eigenvalues are strictly less than one and $(M,g)$ is complete, then we show that the corresponding ball bundle admits a complete Kähler-Einstein metric.
The subject of this paper is the explicit momentum construction of complete Einstein metrics by ODE methods. Using the Calabi ansatz, further generalized by Hwang-Singer, we show that there are non-trivial complete conformally Kähler Einstein metrics on certain Hermitian holomorphic vector bundles and their subbundles over complete Kähler-Einstein manifolds. In special cases, we give the explicit expressions of of these metrics. These examples show that there is a compact Kähler manifold $M$ and its subvariety $N$ whose codimension is greater than 1 such that there is a complete conformally Kä…
In this paper, we prove the Miyaoka-Yau inequality for compact Kähler manifolds with semi-positive canonical bundle. The key point of the proof is the estimate for the $L^2$-norm of the scalar curvature along the Kähler-Ricci flow.
We study ordinary differential equations in the complex domain given by meromorphic vector fields on Kähler compact complex surfaces. We prove that if such an equation has a maximal single valued solution with Zariski-dense image (in particular, if it has an entire one) then, up to a bimeromorphic transformation, either the vector field is holomorphic or it preserves a fibration.
Let $X$ be a compact complex manifold in the Fujiki class $\mathscr{C}$. We study the compactification of $\operatorname{Aut}^0(X)$ given by its closure in Barlet cycle space. The boundary points give rise to non-dominant meromorphic self-maps of $X$. Moreover convergence in cycle space yields convergence of the corresponding meromorphic maps. There are analogous compactifications for reductive subgroups acting trivially on $\operatorname{Alb} X$. If $X$ is Kähler, these compactifications are projective. Finally we give applications to the action of $\operatorname{Aut}(X)$ on the set of probab…
In this paper we study holomorphic vector bundles with singular Hermitian metrics whose curvature are Hermitian matrix currents. We obtain an extension theorem for holomorphic jet sections of nef holomorphic vector bundle on compact Kähler manifolds. Using it we prove that Fano manifolds with strong Griffiths nef tangent bundles are rational homogeneous spaces.
Obstruction flatness of a strongly pseudoconvex hypersurface $Σ$ in a complex manifold refers to the property that any (local) Kähler-Einstein metric on the pseudoconvex side of $Σ$, complete up to $Σ$, has a potential $-\log u$ such that $u$ is $C^\infty$-smooth up to $Σ$. In general, $u$ has only a finite degree of smoothness up to $Σ$. In this paper, we study obstruction flatness of hypersurfaces $Σ$ that arise as unit circle bundles $S(L)$ of negative Hermitian line bundles $(L, h)$ over Kähler manifolds $(M, g).$ We prove that if $(M,g)$ has constant Ricci eigenvalues, then $S(L)$ is obst…
On a compact Kahler manifold, one can define global invariants by integrating local invariants of the metric. Assume that a global invariant thus obtained depends only on the Kahler class. Then we show that the integrand can be decomposed into a Chern polynomial (the integrand of a Chern number) and divergences of one forms, which do not contribute to the integral. We apply this decomposition formula to describe the asymptotic expansion of the Bergman kernel for positive line bundles and to show that the CR Q-curvature on a Sasakian manifold is a divergence.
We extend the Dinh-Sibony notion of densities of currents to the setting where the ambient manifold is not necessarily Kähler and study the intersection of analytic sets from the point of view of densities of currents. As an application, we introduce the notion of exotic periodic points of a meromorphic self-map. We then establish the expected asymptotic for the sum of the number of isolated periodic points and the number of exotic periodic points for holomorphic self-maps with a simple action on the cohomology groups on a compact Kähler manifold. We also show that the algebraic entropy of mer…
We show that the orbifold fundamental group of an effective compact K{ä}hler orbifold with nef anticanonical bundle has polynomial growth, which generalizes M.P ăun's results for manifolds [P ău97, Theorem 1,Theorem 2]
We shall construct a natural Higgs bundle structure on the complexified Kähler cone of a compact Kähler manifold, which can be seen as an analogy of the classical Higgs bundle structure associated to a variation of Hodge structure. In the proof of the flat-ness of our Higgs bundle, we find a commutator identity that can be used to decode the variational properties of the polarized Hodge-Lefschetz module structure on the fibres of our Higgs bundle. Thus we can use a generalized version of Lu's Hodge metric to study the curvature property of the complexified Kähler cone. In particular, it implie…
