New classes of domains with explicit Bergman kernel
Guy Roos, Weiping Yin
备注：
22 pages. Version 2: presentation of the polynomial $χ$ has been simplified and does no longer depend on the parity of $a$ and $r$. One reference precised
发表信息：
Science in China Ser.A Mathematics 2004 vol.47(3) 352-371 (English version) - Science in China Ser.A Mathematics 2004, 34(3), 283-303 (Chinese version)
We introduce two classes of "egg type" domains, built on general bounded symmetric domains, for which we compute the Bergmann kernel in explicit form. We use the characterization of bounded symmetric domains through Jordan triple systems. The egg type domains are defined using the generic norm. A generalization of the Hua integral is computed; the result shows the existence of a special polynomial with integer or half-integer coefficients, attached to each irreducible bounded symmetric domain.
We introduce the notion of "virtual Bergman kernel" and apply it to the computation of the Bergman kernel of "domains inflated by Hermitian balls", in particular when the base domain is a bounded symmetric domain.
Herein, the theory of Bergman kernel is developed to the weighted case. A general form of weighted Bergman reproducing kernel is obtained, by which we can calculate concrete Bergman kernel functions for specific weights and domains.
We investigate Lp regularity of weighted Bergman projections and zeros of weighted Bergman kernels for the weights that are radially symmetric and comparable to 1 on the unit disc.
We discuss topics related to zeroes of the Bergman kernels, and present a method for generating Bergman kernels with arbitrarily, but finitely, many zeroes. It is also shown that a Bergman kernel induced by a radial weight on the unit disk cannot have infinitely many zeroes. Similar questions for the Segal-Bargmann spaces of the complex plane are briefly discussed.
We shall give a variational formula of the full Bergman kernels associated to a family of smoothly bounded strongly pseudoconvex domains. An equivalent criterion for the triviality of holomorphic motions of planar domains in terms of the Bergman kernel is given as an application.
The goal of this note is to explore the Bergman projection on forms. In particular, we show that some of most basic facts used to construct the Bergman kernel on functions, such as pointwise evaluation in $L^2_{0,q}(Ω)\cap\ker\bar\partial_q$, fail for $(0,q)$-forms, $q \geq 1$. We do, however, provide a careful construction of the Bergman kernel and explicitly compute the Bergman kernel on $(0,n-1)$-forms. For the ball in $\mathbb{C}^2$, we also show that the size of the Bergman kernel on $(0,1)$-forms is not governed by the control metric, in stark contrast to Bergman kernel on functions.
We solved the problem of the best rational approximation of the Bergman kernels on the unit circle of the complex plane in the quadratic and uniform metrics.
In this paper we consider the zeros of the Bergman kernel of the Fock-Bargmann-Hartogs domain $D_{n,m}$. We describe how the existence of zeros of the Bergman kernel depends on the integers $m$ and $n$ with the help of the interlacing property.
In this article, we consider Bergman kernels with respect to modules at boundary points, and obtain a log-subharmonicity property of the Bergman kernels, which deduces a concavity property related to the Bergman kernels. As applications, we reprove the sharp effectiveness result related to a conjecture posed by Jonsson-Mustcată and the effectiveness result of strong openness property of the modules at boundary points.
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 give a characterization of $L_h^2$-domains of holomorphy with the help of the boundary behavior of the Bergman kernel and geometric properties of the boundary, respectively.
We compute explicitly the Bergman kernels of all two dimensional monomial polyhedra, a class of domains including the Hartogs triangle and some of its generalizations. The kernel is computed from the representation of such domains as quotient of the product of a disc and punctured disc.
We show how to compute the Bergman kernel functions of some special domains in a simple way. As an application of the explicit formulas, we show that the Bergman kernel functions of some convex domains, for instance the domain in C^3 defined by the inequality |z_1|+|z_2|+|z_3|<1, have zeroes.
In this work we provide an asymptotic expansion for the Szegő kernel associated to a suitably defined Hardy space on the the non-smooth worm domain $D'_β$. After describing the singularities of the kernel, we compare it with an asymptotic expansion of the Bergman kernel. In particular, we show that the Bergman kernel has the same singularities of the first derivative of the Szegő kernel with respect to any of the variables. On the side, we prove the boundedness of the Bergman projection operator on Sobolev spaces of integer order.
We use Stokes's theorem to establish an explicit and concrete connection between the Bergman and Szegő projections on the disc, the ball, and on strongly pseudoconvex domains.
We obtain the octonionic Bergman kernel for half space in the octonionic analysis setting by two different methods. As a consequence, we unify the kernel forms in both complex analysis and hyper-complex analysis.
We construct inner products by the Bernstein-Markov inequality on spaces of holomorphic sections of high powers of a line bundle. The corresponding weighted Bergman kernel functions converge to an extremal function. We obtain a uniform convergence speed.
In this paper, we survey some recent results about the asymptotic expansion of Bergman kernel and we give a Bergman kernel proof of Kodaira embedding theorem.
We obtain a quantitative estimate of Bergman distance when $Ω\subset \mathbb{C}^n$ is a bounded domain with log-hyperconvexity index $α_l(Ω)>\frac{n-1+\sqrt{(n-1)(n+3)}}{2}$, as well as the $A^2(\log A)^q$-integrability of the Bergman kernel $K_Ω(\cdot, w)$ when $α_l(Ω)>0$.
Let $\{M_j\}$ be a sequence of complete Riemannian surfaces which converges in the sense of Cheeger-Gromov to a complete Riemannian surface $M$. We study the convergence of the Bergman kernel $K_{M_j}$ of $M_j$ by using isoperimetric inequalities.
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
We give an elementary proof of the existence of an asymptotic expansion in powers of $k$ of the Bergman kernel associated to $L^k$, where $L$ is a positive line bundle. We also give an algorithm for computing the coefficients in the expansion.
We consider a certain Hartogs domain which is related to the Fock-Bargmann space. We give an explicit formula for the Bergman kernel of the domain in terms of the polylogarithm functions. Moreover we solve the Lu Qi-Keng problem of the domain in some cases.
This article is concerned with asymptotics of equivariant Bergman kernels and partial Bergman kernels for polarized projective Kahler manifolds invariant under a Hamiltonian holomorphic $S^1$ action. Asymptotics of partial Bergman kernel are obtained in the allowed region $\mathcal{A}$ resp. forbidden region $\mathcal{F}$, generalizing results of Shiffman-Zelditch, Shiffman-Tate-Zelditch and Pokorny-Singer for toric Kahler manifolds. The main result gives scaling asymptotics of equivariant Bergman kernels and partial Bergman kernels in the transition region around the interface $\partial \math…
In this paper, we study the Bergman kernel $B_φ(x,y)$ of generalized Bargmann-Fock spaces in the setting of Clifford algebra. The versions of $L^2$-estimate method and weighted subharmonic inequality for Clifford algebra are established. Consequently we show the existence of $B_φ(x,y)$ and then give some estimates on and off the diagonal. As a by-product, we also obtain an upper estimate of the weighted harmonic Bergman kernel.
In this paper we investigate the Bergman kernel function for intersection of two complex ellipsoids $\{(z,w_1,w_2) \in \mathbb{C}^{n+2} : |z_1|^2 + \cdots + |z_n|^2 + |w_1|^q < 1, \quad |z_1|^2 + \cdots + |z_n|^2 + |w_2|^r < 1\}.$
By introducing a suitable new definition for the inner product on the octonionic Bergman space, we determine the explicit form of the octonionic Bergman kernel, in the framework of octonionic analysis which is non-commutative and non-associative.
Using the techniques developed in \cite{SunSun}, we give estimations of the Bergman kernel of the punctured disk with the standard complete Poincaré metric. As an application, we improve the result of \cite{AMM} on the Bergman kernels of punctured Riemann surfaces near singularities.
We present a technique for computing explicit, concrete formulas for the weighted Bergman kernel on a planar domain with weight the modulus squared of a meromorphic function in the case that the meromorphic function has a finite number of zeros on the domain and a concrete formula for the unweighted kernel is known. We apply this theory to the study of the Lu Qi-keng Problem.
A natural class of weighted Bergman spaces on the symmetrized polydisc is isometrically embedded as a subspace in the corresponding weighted Bergman space on the polydisc. We find an orthonormal basis for this subspace. It enables us to compute the kernel function for the weighted Bergman spaces on the symmetrized polydisc using the explicit nature of our embedding. This family of kernel functions include the Szegö and the Bergman kernel on the symmetrized polydisc.
The asymptotic analysis of Bergman kernels with respect to exponentially varying measures near emergent interfaces has attracted recent attention. Such interfaces typically occur when the associated limiting Bergman density function vanishes on a portion of the plane, the off-spectral region. This type of behaviour is observed when the metric is negatively curved somewhere, or when we study partial Bergman kernels in the context of positively curved metrics. In this work, we cover these two situations in a unified way, for exponentially varying planar measures on the complex plane. We obtain u…
Inspired by Berndtsson's work on the subharmonicity property of the Bergman kernel, we give a local variation formula of the full Bergman kernels associated to deformations of complex manifolds. In compact case, it follows from the reproducing property of the Bergman kernel and the curvature formula of the 0-th direct image sheaf. In general, following Schumacher's idea, we use the Lie derivative to compute the variation. An equivalent criterion for the triviality of holomorphic motions of planar domains in terms of the Bergman kernel is given as an application.
We present an elementary proof for an approximate expression of the Bergman kernel on homogeneous spaces, and products of them. The error term is exponentially small with respect to the inverse semiclassical parameter.
A Canonical Positive Definite Kernel Associated with the $ξ$-Bergman Kernel
Indian Institute of Technology Jodhpur（印度理工学院焦特普尔分校）；Indian Institute of Science Education and Research, Pune（印度科学教育研究所浦那）；Indian Institute of Technology Bhubaneswar（印度理工学院布巴内斯瓦尔分校）
In this paper we give an asymptotic expansion of the Bergman kernel for certain weakly pseudoconvex tube domains of finite type in C^2. Our asymptotic formula asserts that the singularity of the Bergman kernel at weakly pseudoconvex points is essentially expressed by using two variables; moreover certain real blowing-up is necessary to understand its singularity. The form of the asymptotic expansion with respect to each variable is similar to that in the strictly pseudoconvex case due to C. Fefferman. We also give an analogous result in the case of the Szego kernel.
In this article, we consider Bergman kernels related to modules at boundary points on Stein manifolds, and obtain a log-subharmonicity property of the Bergman kernels. As applications, we obtain a lower estimate of weighted $L^2$ integrals on Stein manifolds, and reprove an effectiveness result of strong openness property of modules at boundary points on Stein manifolds.
Our main result introduces a new way to characterize two-dimensional finite ball quotients by algebraicity of their Bergman kernels. This characterization is particular to dimension two and fails in higher dimensions, as is illustrated by a counterexample in dimension three constructed in this paper. As a corollary of our main theorem, we prove, e.g., that a smoothly bounded strictly pseudoconvex domain G in $\mathbb{C}^2$ has rational Bergman kernel if and only if there is a rational biholomorphism from G to the 2-dimensional unit ball.
In this article, we derive off-diagonal estimates of the Bergman kernel associated to the tensor-powers of the cotangent bundle defined on a hyperbolic Riemann surface of finite volume, when the distance between the points is less than injectivity radius. We then use these estimates to derive estimates of the Bergman kernel along the diagonal.
We study the Bergman kernel of certain domains in $\mathbb{C}^n$, called elementary Reinhardt domains, generalizing the classical Hartogs triangle. For some elementary Reinhardt domains, we explicitly compute the kernel, which is a rational function of the coordinates. For some other such domains, we show that the kernel is not a rational function. For a general elementary Reinhardt domain, we obtain a representation of the kernel as an infinite series.
Let $\{Ω_t:-1<t<1\}$ be a family of bounded pseudoconvex domains and $φ_t\in PSH(Ω_t)$. Let $K_t(z,w)$ denote the Bergman kernel with weight $φ_t$ on $Ω_t$. We study the continuity and Hölder continuity of $K_t(z,w)$ in $t$. Several applications to singularity theory of psh functions are given, including a new proof of the openness theorem.
In this paper we determine explicitly the kernels $\mathbb K_{α,β}$ associated with new Bergman spaces $\mathcal A_{α,β}^2(\mathbb D)$ considered recently by the first author and M. Zaway. Then we study the distribution of the zeros of these kernels essentially when $α\in\mathbb N$ where the zeros are given by the zeros of a real polynomial $Q_{α,β}$. Some numerical results are given throughout the paper.
We give a version of the comparison principle from pluripotential theory where the Monge-Ampère measure is replaced by the Bergman kernel and use it to derive a maximum principle
In this article, we derive off-diagonal estimates of the Bergman kernel associated to tensor- products of the cotangent line bundle defined over a hyperbolic Riemann surface of finite volume.
