Jim Agler, John E. McCarthy
A basic result in the theory of holomorphic functions of several complex variables is the following special case of the work of H. Cartan on the sheaf cohomology on Stein domains ([10], or see [14] or [16] for more modern treatments).
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This paper studies the determining sets for analytic functions from the symmetrized bidisk into the open unit disk in $\mathbb C$. It relates the idea to the uniqueness of the solutions of a Nevanlinna-Pick interpolation problem. It also investigates when certain thin sets can be determining.
There are three new things in this paper about the open symmetrized bidisk $\mathbb G = \{(z_1+z_2, z_1z_2) : |z_1|, |z_2| < 1\}$. They are motivated in the Introduction. In this Abstract, we mention them in the order in which they will be proved. \begin{enumerate} \item The Realization Theorem: A realization formula is demonstrated for every $f$ in the norm unit ball of $H^\infty(\mathbb G)$. \item The Interpolation Theorem: Nevanlinna-Pick interpolation theorem is proved for data from the symmetrized bidisk and a specific formula is obtained for the interpolating function. \item The Extensio…
Indefinite Schwarz-Pick inequalities for holomorphic self-maps of the bidisk are given as application of the spectral theory on Hilbert modules.
Suppose that Y is a complex manifold with the property that any holomorphic map from a compact convex set in a complex Euclidean space C^n (for any n) to Y is a uniform limit of entire maps from C^n to Y. We prove that a holomorphic map from a closed complex subvariety X_0 in a Stein manifold X to the manifold Y extends to a holomorphic map of X to Y provided that it extends to a continuous map. We then establish the equivalence of four Oka-type properties of a complex manifold. We also generalize a theorem of Siu and Demailly on the existence of open Stein neighborhoods of Stein subvarieties …
Let X be a Stein manifold and let Y be a complex manifold which admits a spray in the sense of Gromov (Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2, pp. 851-897 (1989)). We prove that for every closed complex subvariety X_0 of X and for every continuous map f_0 from X to Y whose restriction to X_0 is holomorphic there exists a homotopy of maps f_t from X to Y whose restrictions to X_0 agree with f_0 and such that the map f_1 is holomorphic on X. We obtain analogous results for sections of holomorphic submersions with sprays over Stein manifolds or Stein s…
We give a formula for the Carathéodory distance on the Neil parabola, the variety ${z^2=w^3}$ restricted to the bidisk; thus making it the first variety with a singularity to have its Carathéodory distance explicitly computed. In addition, we relate this to a mixed Carathéodory-Pick interpolation problem for which known interpolation theorems do not apply. Finally, we prove a bounded holomorphic function extension result from the Neil parabola to the bidisk.
We show that for every connected analytic subvariety $V$ there is a pseudoconvex set $Ω$ such that every bounded matrix-valued holomorphic function on $V$ extends isometrically to $Ω$. We prove that if $V$ is two analytic disks intersecting at one point, if every bounded scalar valued holomorphic function extends isometrically to $Ω$, then so does every matrix-valued function. In the special case that $Ω$ is the symmetrized bidisk, we show that this cannot be done by finding a linear isometric extension from the functions that vanish at one point.
Bloch's principle for holomorphic maps into subvarieties of semi-abelian varieties
本文将半阿贝尔簇子簇上整曲线的值分布基本定理推广到单位圆盘全纯映射序列，证明了对数一般型子簇的伪小林双曲性，并改进了 Cartan 关于满足 Borel 恒等式的无处消失全纯函数组的经典定理。
The goal of this contribution is to investigate L${}^2$ extension properties for holomorphic sections of vector bundles satisfying weak semi-positivity properties. Using techniques borrowed from recent proofs of the Ohsawa-Takegoshi extension theorem, we obtain several new surjectivity results for the restriction morphism to a non necessarily reduced subvariety, provided the latter is defined as the zero variety of a multiplier ideal sheaf. These extension results come with precise L${}^2$ estimates and (probably) optimal curvature conditions.
In this paper, we obtain two extension theorems for cohomology classes and holomorphic sections defined on analytic subvarieties, which are defined as the supports of the quotient sheaves of multiplier ideal sheaves of quasi-plurisubharmonic functions with arbitrary singularities. The first result gives a positive answer to a question posed by Cao-Demailly-Matsumura, and unifies a few well-known injectivity theorems. The second result generalizes and optimizes a general $L^2$ extension theorem obtained by Demailly.
Let $Ω\subset \mathbb{C}^n$ be a bounded domain and let $\mathcal{A} \subset \mathcal{C}(\barΩ)$ be a uniform algebra generated by a set $F$ of holomorphic and pluriharmonic functions. Under natural assumptions on $Ω$ and $F$ we show that the only obstruction to $\mathcal{A} = \mathcal{C}(\barΩ)$ is that there is a holomorphic disk $D \subset \barΩ$ such that all functions in $F$ are holomorphic on $D$, i.e., the only obstruction is the obvious one. This generalizes work by A. Izzo. We also have a generalization of Wermer's maximality theorem to the (distinguished boundary of the) bidisk.
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.
We study the holomorphic/meromorphic function theory and the fundamental group of Euclidean open neighborhoods of compact subvarieties in homogeneous spaces; building on results of Hironaka, Hartshorne, Napier and Ramachandran in the ample normal bundle case.
In this paper (a sequel to B. Drinovec Drnovsek and F. Forstneric, Holomorphic curves in complex spaces, Duke Math. J. 139 (2007), 203-253) we obtain existence and approximation results for closed complex subvarieties that are normalized by strongly pseudoconvex Stein domains. Our sufficient condition for the existence of such subvarieties in a complex manifold is expressed in terms of the Morse indices and the number of positive Levi eigenvalues of an exhaustion function on the manifold. Examples show that our condition cannot be weakened in general. Optimal results are obtained for subvariet…
Taubes established fundamental properties of $J-$holomorphic subvarieties in dimension 4 in \cite{T1}. In this paper, we further investigate properties of reducible $J-$holomorphic subvarieties. We offer an upper bound of the total genus of a subvariety when the class of the subvariety is $J-$nef. For a spherical class, it has particularly strong consequences. It is shown that, for any tamed $J$, each irreducible component is a smooth rational curve. We also completely classify configurations of maximal dimension. To prove these results we treat subvarieties as weighted graphs and introduce se…
We prove a refined Agler decomposition for bounded analytic functions on the bidisk and show how it can be used to reprove an interesting result of Guo et al. related to extending holomorphic functions without increasing their norm. In addition, we give a new treatment of Heath and Suffridge's characterization of holomorphic retracts on the polydisk.
We define the characteristic numbers of a holomorphic k-distribution of any dimension on $mathbb P^n$ and obtain relations between these numbers and the characteristic numbers of an invariant subvariety. As an application we bound the degree of a smooth invariant hypersurface.
The paper is related to the author's talk at the Hayama Symposium in Complex Analysis in December 2000. In section 1 we survey results on the Oka principle for sections of holomorphic submersions over Stein manifolds. In section 2 we apply these results to the lifting problem for holomorphic mappings from Stein manifolds, and in section 3 we apply them to the problem of removability of intersections of holomorphic maps from Stein manifolds with closed complex subvarieties of the target space.
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…
We show that the complex Radon transform realizes an isomorphism between the space of residual $\bar\partial$-cohomologies of a locally complete intersection subvariety in a linearly concave domain of ${\C}P^n$ and the space of holomorphic solutions of the associated homogeneous system of differential equations withconstant coefficients in the dual domain in $({\C}P^n)^*$.
This is a survey on the homotopy principle in complex analysis on Stein manifolds, also called the Oka principle in this context. We concentrate on the following topics: the Oka-Grauert principle (classification of holomorphic vector bundles); the homotopy principle for holomorphic mappings from Stein manifolds and, more generally, for sections of holomorphic submersions with sprays; question on removability of intersections of holomorphic mappings with complex subvarieties; embeddings and immersions of Stein manifolds in affine spaces of minimal dimension; embeddings of open Riemann surfaces …
Restricted spaces of holomorphic sections vanishing along subvarieties
该研究在紧正规复空间上的全纯线丛框架下，给出沿子簇消失的全纯截面限制空间维数的渐近增长条件，证明其Fubini-Study流的收敛性，并应用于随机截面零点的等分布研究。
The Oeljeklaus-Toma (OT-) manifolds are complex manifolds constructed by Oeljeklaus and Toma from certain number fields, and generalizing the Inoue surfaces $S_m$. On each OT-manifold we construct a holomorphic line bundle with semipositive curvature form and trivial Chern class. Using this form, we prove that the OT-manifolds admitting a locally conformally Kahler structure have no non-trivial complex subvarieties. The proof is based on the Strong Approximation theorem for number fields, which implies that any leaf of the null-foliation of w is Zariski dense.
Let X and Y be complex manifolds. One says that maps from X to Y satisfy the Oka principle if the inclusion of the space of holomorphic maps from X to Y into the space of continuous maps is a weak homotopy equivalence. In 1957 H. Grauert proved the Oka principle for maps from Stein manifolds to complex Lie groups and homogeneous spaces, as well as for sections of fiber bundles with homogeneous fibers over a Stein base. In 1989 M. Gromov extended Grauert's result to sections of submersions over a Stein base which admit dominating sprays over small open sets in the base; for proof see [F. Forstn…
Spectral theory of the invariant Laplacian on the disk and the sphere -- a complex analysis approach
本文通过复分析方法，研究复化单位圆补集上不变拉普拉斯算子的全纯谱理论，给出本征空间闭Möbius不变子空间的内在分类，统一了双曲与球面拉普拉斯算子的本征值理论。
Segre nondegenerate totally real subvarieties
本文研究复空间中不可约实解析簇的芽，在 Segre 非退化条件下定义推广 Moser-Webster 对合的平均算子，并利用其研究全纯函数限制及奇点平坦化条件。
A note on subvarieties of powers of OT-manifolds
该研究证明OT-流形上有限对有限全纯对应空间离散，结合模型论结果推出无真无限复解析子集的OT-流形的笛卡尔幂无有意义复解析子簇族，证明借助OT-流形的有限非分歧覆盖完成。
We give the following positive answer to Gromov's question (in "Oka's principle for holomorphic sections of elliptic bundles", J. Amer. Math. Soc. 2, 851-897 (1989), 3.4.(D), page 881). THEOREM: If every holomorphic map from a compact convex set in a complex Euclidean space C^n to a certain complex manifold Y is a uniform limit of entire maps of C^n to Y, then Y enjoys the parametric Oka property. In particular, for any reduced Stein space X the inclusion of the space of holomorphic maps of X to Y into the space of continuous maps is a weak homotopy equivalence. This shows that all Oka type pr…
Let $\mathcal{F}$ be a singular codimension one holomorphic foliation on a compact complex manifold $X$ of dimension at least three such that its singular set has codimension at least two. In this paper, we determine Lehmann-Suwa residues of $\mathcal{F}$ as multiples of complex numbers by integration currents along irreducible complex subvarieties of $X$. We then prove a formula that determines the Baum-Bott residue of simple almost Liouvillian foliations of codimension one, in terms of Lehmann-Suwa residues, generalizing a result of Marco Brunella. As an application, we give sufficient condi…
The Bruce-Roberts Tjurina number of holomorphic 1-forms along complex analytic varieties
本文定义了全纯1-形式沿复解析簇对的Bruce-Roberts Tjurina数，证明其与多个已知不变量之间的关系，并应用于复二维全纯叶层芽的拟齐性问题。
On geometric properties of holomorphic isometries between bounded symmetric domains
本文研究有界对称域间关于Bergman度量及典范Kähler度量的全纯等距映射，证明仿射线性截面的像与仿射线性子空间相交，并构造满全纯浸没使等距映射成为其整体全纯截面。
Motivated by recent work in the mathematics and engineering literature, we study integrability and non-tangential regularity on the two-torus for rational functions that are holomorphic on the bidisk. One way to study such rational functions is to fix the denominator and look at the ideal of polynomials in the numerator such that the rational function is square integrable. A concrete list of generators is given for this ideal as well as a precise count of the dimension of the subspace of numerators with a specified bound on bidegree. The dimension count is accomplished by constructing a natura…
On the zero set of holomorphic sectional curvature
本文针对具有丰沛典范线丛的紧复 Kähler 流形，研究了全纯截面曲率零点子空间的维数上界，通过联系实代数簇的复子簇理论与多项式平方差分解，提出了基于全纯截面曲率平方分解长度不变量的上界估计。
The multivariate Herglotz-Nevanlinna class: Rational approximation
本文研究多元Herglotz-Nevanlinna类中有界全纯函数的有理逼近问题，基于Takagi-Pfister斜特征向量构造Pade型逼近格式，证明Montessus de Ballore型收敛定理，并揭示Agler函数Taylor系数集的半代数性质。
Projective subvarieties of Bogomolov-Guan manifolds and quasi-diagonals in products of elliptic curves
研究非凯勒全纯辛流形 Bogomolov-Guan 中的复子簇，通过拟对角线分类射影子簇，证明一般情形下射影子簇属于Lagrangian纤维的纤维。
Residue functions and Extension problems
针对Demailly定性延拓定理缺乏延拓估计的问题，研究利用残差函数获取一般情形下全纯延拓的$L^2$估计的可能性，并证明指标1残差函数定义的1-lc测度等于Ohsawa测度。
With a view to proving the conjecture of "dlt extension" related to the abundance conjecture, a sequence of potential candidates for replacing the Ohsawa measure in the Ohsawa-Takegoshi $L^2$ extension theorem, called the "lc-measures", which hopefully could provide the $L^2$ estimate of a holomorphic extension of any suitable holomorphic section on a subvariety with singular locus, are introduced in the first half of the paper. Based on the version of $L^2$ extension theorem proved by Demailly, a proof is provided to show that the lc-measure can replace the Ohsawa measure in the case where th…
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