Elisabetta Barletta
We obtain a conceptually new differential geometric proof of P.F. Klembeck's result that the holomorphic sectional curvature of a strictly pseudoconvex domain approaches (in the boundary limit) the constant sectional curvature of the Bergman metric of the unit ball.
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In this paper we study the behaviour of the holomorphic sectional curvature (or Gaussian curvature) of the Bergman metric of planar annuli. The results are then utilized to construct a domain for which the curvature is divergent at one of its boundary points and moreover the limes superior is the maximal possible for the Bergman metric (2), whereas the limes inferior is $-\infty$.
We extend and simplify results of \cite{Din~2009} where the asymptotic behavior of the holomorphic sectional curvature of the Bergman metric in annuli is studied. Similarly as in \cite{Din~2009} the description enables us to construct an infinitely connected planar domain (in our paper it is a Zalcman type domain) for which the supremum of the holomorphic sectional curvature is two whereas its infimum is equal to $-\infty$.
On Point Separation of Bergman Space on Stein Manifolds with Constant Holomorphic Sectional Curvature
stein流形在特定曲率条件下，其bergman度量的定义与分离点性质密切相关，且与单位球体的biholomorphic性相关。
n this paper, we establish the sharp estimate of the Lipschitz continuity with respect to the Bergman metric. The obtained results are the improvement and generalization of the corresponding results of Ghatage, Yan and Zheng (Proc. Amer. Math. Soc., 129: 2039-2044, 2000).
Abundance of Bergman metrics with constant positive holomorphic sectional curvature
本文研究了是否存在复流形其Bergman度量局部等距于m倍的Fubini-Study度量的问题，构造了Reinhardt域家族并证明了其Bergman度量的等距性，同时指出此类复流形的几何分类不可行。
Holomorphic Bergman isometries between bounded domains and the extension of Lu's theorem
该研究探讨复流形间保持Bergman度量的全纯映射，得出等维情形下全纯Bergman等距映射的性质，推广Lu定理并给出相关应用。
We prove that for a bounded domain in $\mathbb C^n$ with the Bergman metric of constant holomorphic sectional curvature being biholomorphic to a ball is equivalent to the hyperconvexity or the exhaustiveness of the Bergman-Calabi diastasis. By finding its connection with the Bergman representative coordinate, we give explicit formulas of the Bergman-Calabi diastasis and show that it has bounded gradient. In particular, we prove that any bounded domain whose Bergman metric has constant holomorphic sectional curvature is Lu Qi-Keng. We also extend a theorem of Lu towards the incomplete situation…
Recently Herbig, McNeal, and Straube have showed that the Bergman projection of conjugate holomorphic functions is smooth up to the boundary on a class of pseudoconvex domains. We show that a further smoothing property holds on a family of Reinhardt domains; namely, the Bergman projection of conjugate holomorphic functions is holomorphic past the boundary.
Domains with Bergman metrics of constant curvature and Bergman-negligible subsets
研究了具有常负曲率Bergman度量的领域，证明其与单位球体减去测度为零的闭集biholomorphic，并讨论了相关函数扩展性质及应用。
Transformation formula for the Reduced Bergman kernel and its Application
本文证明了复平面有界域间真全纯对应及映射下约化Bergman核与加权约化Bergman核的变换公式，并给出了其应用。
Sobolev irregularity of the Bergman projection on a family of domains containing the Hartogs triangle is shown. On the Hartogs triangle itself, a sub-Bergman projection is shown to satisfy better Sobolev norm estimates than its Bergman projection.
Let $Ω\subset\mathbb{C}^n$ be a bounded domain with smooth boundary, whose Bergman projection $B$ maps the Sobolev space $H^{k_{1}}(Ω)$ (continuously) into $H^{k_{2}}(Ω)$. We establish two smoothing results: (i) the full Sobolev norm $\|Bf\|_{k_{2}}$ is controlled by $L^2$ derivatives of $f$ taken along a single, distinguished direction (of order $\leq k_{1}$), and (ii) the projection of a conjugate holomorphic function in $L^{2}(Ω)$ is automatically in $H^{k_{2}}(Ω)$. There are obvious corollaries for when $B$ is globally regular.
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 $D$.
In this paper we obtain the full asymptotic expansion of the Bergman-Hodge kernel associated to a high power of a holomorphic line bundle with non-degenerate curvature. We also explore some relations with asymptotic holomorphic sections on symplectic manifolds
On a reasonable class of domains in $\CC^n$, we characterize those holomorphic functions which continue analytically past the boundary. Then we give some applications of this result to holomorphic mappings. In addition, some new results about the Lu Qi-Keng conjecture are treated.
We characterize operator-theoretic properties (boundedness, compactness, and Schatten class membership) of Toeplitz operators with positive measure symbols on Bergman spaces of holomorphic hermitian line bundles over Kähler Cartan-Hadamard manifolds in terms of geometric or operator-theoretic properties of measures.
Equivalent Bergman Spaces with Inequivalent Weights
该研究证明每个加权平方可积全纯函数空间都存在使Bergman核有零点的等价权重，还给出了一族对应Bergman核有无穷多零点的$L^1(\u2102)$径向权重，丰富了Bergman空间与权重关系的理论。
We prove the asymptotic of the logarithmic Bergman kernel. And as an application, we calculate the conditional expectation of density of zeros of Gaussian random sections of powers of a positive line bundle that vanish along a fixed smooth subvariety.
The definition of classical holomorphic function spaces such as the Hardy space or the Dirichlet space on the Hartogs triangle is not canonical. In this paper we introduce a natural family of holomorphic function spaces on the Hartogs triangle which includes some weighted Bergman spaces, a candidate Hardy space and a candidate Dirichlet space. For the weighted Bergman spaces and the Hardy space we study the $L^p$ mapping properties of Bergman and Szegő projection respectively, whereas for the Dirichlet space we prove it is isometric to the Dirichlet space on the bidisc.
Biholomorphic rigidity and symmetry breaking for domains with Bergman metric of constant positive holomorphic sectional curvature
本文针对Hayashida-Kamimoto构造的具有常正全纯截面曲率2的Bergman度量的Reinhardt区域，证明三明治原理并构造大量不等价光滑非Reinhardt区域，还分类双全纯映射并确定角扰动族的自同构群。
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.
The aim of this study is to understand to what extent a 1-convex domain with Levi-flat boundary is capable of holomorphic functions with slow growth. This paper discusses a typical example of such domain, the space of all the geodesic segments on a hyperbolic compact Riemann surface. Our main finding is an integral formula that produces holomorphic functions on the domain from holomorphic differentials on the Riemann surface. This construction can be seen as a non-trivial example of $L^2$ jet extension of holomorphic functions with optimal constant. As its corollary, it is shown that the weigh…
The explicit complete Einstein-Kähler metric on the second type Cartan-Hartogs domain $Y_{II}(r,p;K)$ is obtained in this paper when the parameter $K$ equals $\frac p2+\frac 1{p+1}$. The estimate of holomorphic sectional curvature under this metric is also given which intervenes between $-2K$ and $-\frac{2K}p$ and it is a sharp estimate. In the meantime we also prove that the complete Einstein-Kähler metric is equivalent to the Bergman metric on $Y_{II}(r,p;K)$ when $K=\frac p2+\frac 1{p+1}$.
We consider a proper holomorphic map form D to G domains in C^n and show that it induces a unitary isomorphism between the Bergman space A^2(G) and some subspace of A^2(D). Using this isomorphism we construct orthogonal projection onto that subspace and we derive Bells transformation formula for the Bergman kernel under proper holomorphic mappings. As a consequence of the formula we get that the tetrablock is not a Lu Qi-Keng domain.
Bergman representative coordinate, constant holomorphic curvature and a multidimensional generalization of Carathéodory's theorem
本文利用Bergman代表坐标和Calabi的diastasis，将Lu的定理推广至度量不完整且具常全纯曲率的有界伪凸域，并多维推广了Carathéodory定理，给出双全纯球边界为拓扑球面的充分条件。
Asymptotic behaviour of the Bergman kernel and metric
本文研究光滑有界拟凸域在指数平坦无限型边界点处的 Bergman 不变量，通过局部化与缩放方法证明了 Bergman 核、度量及全纯截曲率的非切向渐近极限。
We obtain several new characterizations for the standard weighted Bergman spaces $A^p_α$ on the unit ball of $\cn$ in terms of the radial derivative, the holomorphic gradient, and the invariant gradient.
On the dimension of Bergman spaces on $\mathbb{P}^1$
本文受 Szőke 结果启发，利用势论方法刻画了 $\mathbb{P}^1$ 上线丛限制到开集的全纯截面 Bergman 空间的维数。
The Grothendieck Theorem in Bergman Spaces
本文证明了 Bergman 空间中的 Grothendieck 定理，指出若一个全纯 $L^p$ 可积空间的闭子空间同时包含于全纯 $L^q$ 可积空间中（$p>1$，$q>p$），则该子空间的维数必为有限。
Bergman metrics as pull-backs of the Fubini-Study metric
本文通过将 Bergman 度量解释为无限维复射影空间 Fubini-Study 度量在 Bergman–Bochner 映射下的拉回，刻画恒常全纯截面曲率情形，并构造新例、提出猜想。
Herein, the Laplace transform representations for functions of weighted holomorphic Bergman spaces on the tube domains are developed. Then a weighted version of the edge-of-the-wedge theorem is derived as a byproduct of the main results.
We present some formulae related to the Chern-Ricci curvatures and scalar curvatures of special Hermitian metrics. We prove that a compact locally conformal Kähler manifold with constant nonpositive holomorphic sectional curvature is Kähler. We also give examples of complete non-Kähler metrics with pointwise negative constant but not globally constant holomorphic sectional curvature, and complete non-Kähler metric with zero holomorphic sectional curvature and nonvanishing curvature tensor.
We give the parameter version of localization theorem for Bergman metric near the boundary points of strictly pseudoconvex domains. The approximation theorem for square integrable holomorphic functions on such domains in the spirit of Graham-Kerzman is proved in the hereby paper, as well.
On the symmetrized bidisc G2 with the Bergman metric, the holomorphic sectional curvature is negatively pinched and the holomorphic bisectional curvature is not. The consequences in invariant metrics are provided.
We prove that a compact Hermitian manifold with semi-positive but not identically zero holomorphic sectional curvature has Kodaira dimension $-\infty$. As applications, we show that Kodaira surfaces and hyperelliptic surfaces can not admit Hermitian metrics with semi-positive holomorphic sectional curvature although they have nef tangent bundles.
Infinite-Dimensionality of Bergman Spaces on\ Complete $d$-Bounded Kähler Manifolds
本文证明在Gromov意义下$d$-有界的完全Kähler流形上，其Bergman空间（$L^2$全纯$n$-形式的Hilbert空间）是无限维的，方法结合Gromov线丛构造、参数一致$\bar{\partial}$估计与奇异权$L^2$方法。
Let $AL^{2}_ϕ(\mathbb{D})$ denote the closed subspace of $L^{2}(\mathbb{D},e^{-2ϕ}dλ)$ consisting of holomorphic functions in the unit disc ${\mathbb D}$. For certain class of subharmonic funcions $ϕ: {\mathbb D}\rightarrow{\mathbb D}$, we prove upper pointwise estimate for the Bergman kernel for $AL^{2}_ϕ(\mathbb{D})$
We~identify the standard weighted Bergman kernels of spaces of nearly holomorphic functions, in~the sense of Shimura, on~bounded symmetric domains. This also yields a description of the analogous kernels for spaces of ``invariantly-polyanalytic'' functions -- a~generalization of the ordinary polyanalytic functions on the ball which seems to be the most appropriate one from the point of view of holomorphic invariance. In~both cases, the~kernels turn out to be given by certain spherical functions, or equivalently Heckman-Opdam hypergeometric functions, and a conjecture relating some of these to …
We consider Bergman spaces and variations of them in one or several complex variables. For some domains we show that in these spaces the generic function is totally unbounded and hence non - extendable. We also show that the generic function is not - even locally - in Bergman spaces of higher order. Finally, in certain domains we consider the space of holomorphic functions whose derivatives up to some order extend continuously to the closure of the domain endowed with its natural topology. Generically, every function in this space is proven to be non - extendable, although bounded.
On diameter rigidity for Kähler manifolds with positive holomorphic sectional curvature
本文研究了正全纯截面曲率的凯勒流形的直径刚性性质。
Consider a very ample line bundle $ E \to X$ over a compact complex manifold, endowed with a hermitian metric of curvature $-i ω$, and the space $\mathcal{O}(E)$ of its holomorphic sections. The Fubini--Study map associates with positive definite inner products $\langle \, , \rangle$ on $\mathcal{O}(E)$ functions FS$(\langle \, ,\rangle) \in \mathcal{H}_ω=\{u \in C^{\infty}(X):ω+i\partial\overline{\partial} u >0\}$. We prove that FS is an injective immersion, but its image in general is not closed in $\mathcal{H}_ω$. To obtain a closed range, FS has to be extended to certain degenerate inner p…
Recently, Wu-Yau and Tosatti-Yang established the connection between the negativity of holomorphic sectional curvatures and the positivity of canonical bundles for compact Kähler manifolds. In this short note, we give anothe proof of their theorems by using the Kähler-Ricci flow.
Let X be a hermitian manifold and let L^k be a high power of a hermitian line bundle over X. Local versions of Demailly's holomorphic Morse inequalities are presented - after integration they yield the usual inequalities. The local weak inequalities hold on any hermitian manifold X, regardless of compactness and completeness. The proofs, which are elementary, are based on a new approach to pointwise Bergman kernel estimates, where the kernels are estimated by a model kernel in the standard complex space C^n.
Integrating holomorphic sectional curvatures
本文通过表示论方法计算Kähler度量全纯截面曲率的$L^2$-范数，重新证明其决定整个曲率张量，并进一步研究Hermitian度量情形及全纯双截面曲率的$L^2$-范数。
We give a quick direct proof of Wu's theorem on the subadditivity of Hermitian metrics of negative holomorphic sectional curvature.
If a bounded domain can be covered by the polydisk through a rational proper holomorphic map, then the Bergman projection is $L^p$-bounded for $p$ in a certain range depending on the ramified rational covering. This result can be applied to the symmetrized polydisk and to the Hartogs triangle with exponent $γ$.
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 study several connected problems of holomorphic function spaces on homogeneous Siegel domains. The main object of our study concerns weighted mixed norm Bergman spaces on homogeneous Siegel domains of type II. These problems include: sampling, atomic decomposition, duality, boundary values, boundedness of the Bergman projectors. Our analysis include the Hardy spaces, and suitable generalizations of the classical Bloch and Dirichlet spaces. One of the main novelties in this work is the generality of the domains under consideration, that is, homogeneous Siegel domains, extending many results …
In a previous paper, \cite{Berndtsson}, we have studied a property of subharmonic dependence on a parameter of Bergman kernels for a family of weighted $L^2$-spaces of holomorphic functions. Here we prove a result on the curvature of a vector bundle defined by this family of $L^2$-spaces itself, which has the earlier results on Bergman kernels as a corollary. Applying the same arguments to spaces of holomorphic sections to line bundles over a locally trivial fibration we also prove that if a holomorphic vector bundle, $V$, over a complex manifold is ample in the sense of Hartshorne, then $V\gr…
Kähler metrics of negative holomorphic (bi)sectional curvature on a compact relative Kähler fibration
本文研究了在特定条件下，如何在总空间上构造具有负全纯双截面曲率的Kähler度量。
Let $(X,ω)$ be a compact Kähler manifold. Let $(L,h)$ be a hermitian holomorphic line bundle over $X$, such that $Θ_{L,h}\geq -\varepsilonω$ for a small $\varepsilon>0$, $E$ be a holomorphic line bundle over $X$. For $k\in \mathbb{N}_+$, denote by $X_k:=(X,ω^k)$ the Kähler manifold $X$ with new scaled metric $ω^k=kω$. Estimates of the number of eigenvalues smaller than $λ$ of the $\debar$-Laplacian on forms on $X_k$ with values in $L^k\otimes E$ are presented for $0\leq λ<k$. In particular, when $λ=0$, we get a numeric bound for the cohomology groups.
Off-diagonal upper bounds are established away from the diagonal for the Bergman kernels associated to high powers of holomorphic line bundles over compact complex manifolds, asymptotically as the power tends to infinity. The line bundle is assumed to be equipped with a Hermitian metric with positive curvature form, which is infinitely differentiable but not necessarily real analytic. The bounds obtained are the best possible for this class of metrics.
Identifying holomorphic functions by moment conditions is a classical problem. This paper explores ideas introduced by Tumanov in a well known paper on the strip problem, and building on the author's previous work, to show that weighted Bergman projections can be computed via asymptotic formulas involving Szegő projections a fixed 1-dimensional family of curves. Various new operators are introduced; in addition, similar formulas are derived via dimension reduction, using the author's CR Hartogs theorem.
Contraction properties for holomorphic functions via isoperimetric stability on the Bergman ball
本文研究全纯函数在近常数情况下的局部收缩性质，通过伯格曼球上的等周稳定性结果，将加权伯格曼空间的缺陷不等式与全纯函数的收缩性质联系起来。
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 lower bound for the Tanaka-Webster holomorphic sectional curvature of strictly pseudoconvex real hypersurfaces that are "semi-isometrically" immersed in a Kähler manifold of nonnegative holomorphic sectional curvature under an appropriate convexity condition. This gives a partial answer to a question posed by Chanillo, Chiu, and Yang regarding the positivity of the Tanaka-Webster scalar curvature of the boundary of a strictly convex domain in $\mathbb{C}^2$ from 2012. In fact, the main result proves a stronger positivity property, namely the $\frac12$-positivity in the sense o…
Meromorphic Bergman spaces
本文在带点单位圆盘上引入推广经典Bergman空间的亚纯Bergman空间，证明其基本性质及对偶性质，并将Hardy-Littlewood和Fejér-Riesz不等式推广至该空间，应用于Toeplitz算子。
The main result of this note is that, for each $n\in \{1,2,3,\ldots\}$, there exists a Hodge metric on the $n$-th Hirzebruch surface whose positive holomorphic sectional curvature is $\frac{1}{(1+2n)^2}$-pinched. The type of metric under consideration was first studied by Hitchin in this context. In order to address the case $n=0$, we prove a general result on the pinching of the holomorphic sectional curvature of the product metric on the product of two Hermitian manifolds $M$ and $N$ of positive holomorphic sectional curvature.
On horn-shaped Reinhardt domains
本文在n≥2的复空间ℂⁿ中引入一类新的无界Reinhardt域（角形域），研究其Bergman空间维数、Bergman度量全纯截面曲率及全纯等价问题。
In an earlier work, we investigated some consequences of the existence of a Kähler metric of negative holomorphic sectional curvature on a projective manifold. In the present work, we extend our results to the case of semi-negative (i.e., non-positive) holomorphic sectional curvature. In doing so, we define a new invariant that records the largest codimension of maximal subspaces in the tangent spaces on which the holomorphic sectional curvature vanishes. Using this invariant, we establish lower bounds for the nef dimension and, under certain additional assumptions, for the Kodaira dimension o…
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