We study the limit distribution of zeros of certain sequences of holomorphic sections of high powers $L^N$ of a positive holomorphic Hermitian line bundle $L$ over a compact complex manifold $M$. Our first result concerns `random' sequences of sections. Using the natural probability measure on the space of sequences of orthonormal bases $\{S^N_j\}$ of $H^0(M, L^N)$, we show that for almost every sequence $\{S^N_j\}$, the associated sequence of zero currents $1/N Z_{S^N_j}$ tends to the curvature form $ω$ of $L$. Thus, the zeros of a sequence of sections $s_N \in H^0(M, L^N)$ chosen independently and at random become uniformly distributed. Our second result concerns the zeros of quantum ergodic eigenfunctions, where the relevant orthonormal bases $\{S^N_j\}$ of $H^0(M, L^N)$ consist of eigensections of a quantum ergodic map. We show that also in this case the zeros become uniformly distributed.
We generalize a construction of Hitchin to prove that, given any compact Kähler manifold $M$ with positive holomorphic sectional curvature and any holomorphic vector bundle $E$ over $M$, the projectivized vector bundle ${\mathbb P}(E)$ admits a Kähler metric with positive holomorphic sectional curvature.
We prove an $L^2$ extension theorem of Ohsawa-Takegoshi type for extending holomorphic sections of line bundles from a subvariety which is given as a maximal log-canonical center of a pair and is of general codimension in a projective variety. Our method of proof indicates that such a setting is the most natural one in a sense, for general $L^2$ extension of line bundle sections.
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.
We show that if a compact complex manifold admits a Kähler metric whose holomorphic sectional curvature is everywhere non positive and strictly negative in at least one point, then its canonical bundle is positive.
We prove a sharp Ohsawa-Takegoshi-Manivel type extension result for twisted holomorphic sections of singular hermitian line bundles over almost Stein manifolds. We establish as corollaries some extension results for pluri-twisted holomorphic sections of singular hermitian line bundles over projective manifolds.
We establish the equidistribution of zeros of random holomorphic sections of powers of a semipositive singular Hermitian line bundle, with an estimate of the convergence speed.
We prove that a smooth complex projective threefold with a Kähler metric of negative holomorphic sectional curvature has ample canonical line bundle. In dimensions greater than three, we prove that, under equal assumptions, the nef dimension of the canonical line bundle is maximal. With certain additional assumptions, ampleness is again obtained. The methods used come from both complex differential geometry and complex algebraic geometry.
Using $L^2$-methods for the $\bar\partial$-equation we prove that the Ohsawa-Takegoshi extension theorem also holds for holomorphic sections of a vector bundle, over compact Kähler manifolds. We then proceed to show that the conditions that are needed are more liberal than the ones one would need if one instead reduced the extension problem to line bundles through the usual algebraic geometric procedure of studying the projective bundle associated with the vector bundle.
Our interest is a regularity of a minimal singular metric of a line bundle. One main conclusion of our general result in this paper is the existence of continuous Hermitian metrics with semi-positive curvatures on the so-called Zariski's example of a line bundle defined over the blow-up of $\mathbb{P}^2$ at some twelve points. This is an example of a line bundle which is nef, big, not semi-ample, and whose section ring is not finitely generated. We generalize this result to the higher dimensional case when the stable base locus of a line bundle is a smooth hypersurface with a holomorphic tubul…
We prove the convergence of the normalized Fubini-Study measures and the logarithms of the Bergman kernels of various Bergman spaces of holomorphic and weakly holomorphic sections associated to a singular Hermitian holomorphic line bundle on an algebraic curve. Using this, we study the asymptotic distribution of the zeros of random sequences of sections in these spaces.
We first present a Skoda type division theorem for holomorphic sections of line bundles on a projective variety which is essentially the most general, compared to previous ones. It is derived from Varolin's theorem as a corollary. Then we revisit Geometric Effective Nullstellensatz and observe that even this general Skoda division is far from sufficient to yield stronger Geometric Effective Nullstellensatz such as `vanishing order $1$ division', which could be used for finite generation of section rings by the basic finite generation lemma. To resolve this problem, we develop a notion of pseud…
We study critical points of holomorphic sections of $\ocal(m)$ on $\CP^n$. For quadrics, we give a complete discription of their critical points. When $n=1$, we prove a spherical Gauss-Lucas theorem. For general situation, we prove that a general section has all its critical points isolated and non-degenerate.
Fix a point $t_0$ in the circle $S^1$. The space $J^k(t_0, \mathbb{P}^1)$ of $k$-jets at $t_0$ of $C^{\infty}$ maps from $S^1$ to the Riemann sphere $\mathbb{P}^1$ is a $k+1$ dimensional complex algebraic manifold. We identify a class of holomorphic sections of line bundles on $J^k(t_0, \mathbb{P}^1)$.
Let $(X, ω)$ be a weakly pseudoconvex Kähler manifold, $Y \subset X$ a closed submanifold defined by some holomorphic section of a vector bundle over $X,$ and $L$ a Hermitian line bundle satisfying certain positivity conditions. We prove that for any integer $k\geq 0,$ any section of the jet sheaf $L\otimes {\cal O}_X/{\cal I}_Y^{k+1},$ which satisfies a certain $L^2$ condition, can be extended into a global holomorphic section of $L$ over $X$ whose $L^2$ growth on an arbitrary compact subset of $X$ is under control. In particular, if $Y$ is merely a point, this gives the existence of a global…
In this paper we study the deformation problem of pairs consisting of a Riemann surface and a holomorphic line bundle over that surface, and also sections thereof. We emphasize a constructive approach throughout and work and use covering space techniques. In particular, we also describe the limits of such degenerations as the boundary of Teichmüller space is approached, and review the construction of augmented Teichmüller space in great detail.
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 Kähler manifold $M$. In particular, we construct for each positive integer $k$, orthonormal sections $s^k_1,\dots,s^k_{n_k}$ in $H^0(M,L^k)$, $n_k\geβ\dim H^0(M,L^k)$, such that $\{s^k_j\}$ is a uniformly bounded family, where $β$ is an explicit positive constant depending only on the dimension of $M$. For $m=1$, we can take $β=.99564$.
By using asymptotic Morse inequalities we give a lower bound for the space of holomorphic sections of high tensor powers in a positive line bundle over a q-concave domain. The curvature of the positive bundle induces a hermitian metric on the manifold. The bound is given explicitely in terms of the volume of the domain in this metric and a certain integral on the boundary involving the defining function and its Levi form. As application we study the perturbattion of the complex structure of a q-concave manifold.
Let $X$ be a compact generalized Sasakian CR manifold of dimension $2n-1$, $n\geqslant2$, and let $L$ be a generalized Sasakian CR line bundle over $X$ equipped with a rigid semi-positive Hermitian fiber metric $h^L$. In this paper we prove that if $h^L$ is positive at some point of $X$ and conditions Y(0) and Y(1) hold at each point of $X$, then $L$ is big.
Let $X$ be a compact normal complex space of dimension $n$, and $L$ be a holomorphic line bundle on $X$. Suppose $Σ=(Σ_1,\ldots,Σ_\ell)$ is an $\ell$-tuple of distinct irreducible proper analytic subsets of $X$, $τ=(τ_1,\ldots,τ_\ell)$ is an $\ell$-tuple of positive real numbers, and consider the space $H^0_0 (X, L^p)$ of global holomorphic sections of $L^p:=L^{\otimes p}$ that vanish to order at least $τ_{j}p$ along $Σ_{j}$, $1\leq j\leq\ell$. We find necessary and sufficient conditions which ensure that $\dim H^0_0(X,L^p)\sim p^n$, analogous to Ji-Shiffman's criterion for big line bundles. W…
In this paper we study the problem of extension of holomorphic sections of line bundles/vector bundles from reduced unions of strata of divisors. An extension theorem of Ohsawa--Takegoshi type is proved. As consequences we deduce several qualitative results on extension from snc divisors and generic global generation of vector bundles.
In this paper, we obtain optimal $L^2$ extension of holomorphic sections of a holomorphic vector bundle from subvarieties in weakly pseudoconvex Kähler manifolds. Moreover, in the case of line bundle the Hermitian metric is allowed to be singular.
For a projective variety $X$ defined over a non-Archimedean complete non-trivially valued field $k$, and a semipositive metrized line bundle $(L, ϕ)$ over it, we establish a metric extension result for sections of $L^{\otimes n}$ from a sub-variety $Y$ to $X$. We form normed section algebras from $(L, ϕ)$ and study their Berkovich spectra. To compare the supremum algebra norm and the quotient algebra norm on the restricted section algebra $V(L_{X|Y})$, two different methods are used: one exploits the holomorphic convexity of the spectrum, following an argument of Grauert; another relies on fin…
The construction of sections of bundles with prescribed jet values plays a fundamental role in problems of algebraic and complex geometry. When the jet values are prescribed on a positive dimensional subvariety, it is handled by theorems of Ohsawa-Takegoshi type which give extension of line bundle valued square-integrable top-degree holomorphic forms from the fiber at the origin of a family of complex manifolds over the open unit 1-disk when the curvature of the metric of line bundle is semipositive. We prove here an extension result when the curvature of the line bundle is only semipositive o…
We prove a parametric Oka principle for equivariant sections of a holomorphic fibre bundle $E$ with a structure group bundle $\mathscr G$ on a reduced Stein space $X$, such that the fibre of $E$ is a homogeneous space of the fibre of $\mathscr G$, with the complexification $K^\mathbb C$ of a compact real Lie group $K$ acting on $X$, $\mathscr G$, and $E$. Our main result is that the inclusion of the space of $K^\mathbb C$-equivariant holomorphic sections of $E$ over $X$ into the space of $K$-equivariant continuous sections is a weak homotopy equivalence. The result has a wide scope; we describ…
In this paper, we study holomorphic rank-2 vector bundles on non-K\" ahler elliptic surfaces. Our main tool for analysing these bundles is of course the spectral cover. However, given the non-Kähler condition, the elliptic surfaces we are considering do not have sections and gerbes naturally arise in this context. The spectral construction presented in this paper is a modification of the Fourier-Mukai transform for elliptic fibrations without a section. After examining some of the properties of this Fourier-Mukai transform, we give a complete classification of vector bundles on these surfaces.
A complex manifold Y satisfies the Convex Approximation Property (CAP) if every holomorphic map from a neighborhood of a compact convex set K in a complex Euclidean space C^n to Y can be approximated, uniformly on K, by entire maps from C^n to Y. If X is a reduced Stein space and Z is a stratified holomorphic fiber bundle over X all of whose fibers satisfy CAP, then sections of Z over X enjoy the Oka property with (jet) interpolation and approximation. Previously this has been proved by the author in the case when X is a Stein manifold without singularities (Ann. Math., 163 (2006), 689-707, ma…
Given a holomorphic vector bundle $E$ over a compact Riemann surface $M$, and an open set $D$ in $M$, we prove that the Bergman space of holomorphic sections of the restriction of $E$ to $D$ must either coincide with the space of global holomorphic sections of $E$, or be infinite dimensional. Moreover, we characterize the latter entirely in terms of potential-theoretic properties of $D$.
We give a proof of the following theorem of M. Gromov (Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc., 2 (1989), 851-897). Let Z be a holomorphic fiber bundle over a Stein manifold. If the fiber of Z admits a dominating holomorphic spray then the inclusion of the space of global holomorphic sections of Z into the space of continuous sections is a weak homotopy equivalence. In the classical case when the fiber is a complex Lie group or a homogeneous space this was proved by H. Grauert (Math. Ann., 135 (1958), 263--273). For further results in this direction se…
