David E. Barrett
We show that the efficiency of a natural pairing between certain projectively invariant Hardy spaces on dual strongly C-linearly convex real hypersurfaces in complex projective space is measured by the norm of the corresponding Leray transform.
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If a mapping of several complex variables into projective space is holomorphic in each pair of variables, then it is globally holomorphic.
In this paper, we study holomorphic curves satisfying the Fubini-Study derivative $\|f'(z)\|=O(|z|^σ)$ for some $σ>-1$ from the viewpoint of Nevanlinna theory
A theorem of A. Ostrowski describing meromorphic functions f such that the family {f(kz):k in C*} is normal, is generalized to holomorphic maps from $C*$ to a projective space.
We consider local modifications $ω_n+f^*ω_d$ of the Fubini-Study metric (with associated $(1,1)$-form $ω_n$) on an open subset $Ω\subset \bC\bP^n$ induced by a local holomorphic mapping $f\colon Ω\to \bP^d$. Our main result is that there are "gaps" in potential dimensions $m$ such that the modification can be obtained as $h^*ω_m$ for some local holomorphic mapping $h\colon Ω\to \bC\bP^m$. We also consider the case of rational conformal factors.
In this paper, we prove a type of partial rigidity result for proper holomorphic mappings of certain $\ell$-concave domains in projective space into model quadratic $\ell$-concave domains. The main technical result is a degree estimate for proper holomorphic mappings into the model domains, provided that the mappings extend to projective space as rational mappings, and the source domain contains algebraic varieties and has a boundary with low CR complexity.
Avoidance Criteria for Normal Holomorphic Curves on Complex Projective Space
研究全纯曲线族在复射影空间中避开足够多移动超曲面的条件，并探讨共享超平面的全纯曲线族的正常性条件。
A holomorphic map from the complex line to a complex projective space is called normal (a. k. a. Brody curve) if it is uniformly continuous from the Euclidean metric to the Fubini--Study metric. The paper contains a survey of known results about such maps, as well as some new theorems.
Yan and Chen proved a weak Cartan-type second main theorem for holomorphic curves meeting hypersurfaces in projective space that included truncated counting functions. Here we give an explicit estimate for the level of truncation.
Let M be a projective manifold, p:M_{G} --> M a regular covering over M with a free abelian transformation group G. We describe holomorphic functions on M_{G} of an exponential growth with respect to the distance defined by a metric pulled back from M. As a corollary we obtain for such functions Cartwright and Liouville type theorems. Our approach brings together L_{2} cohomology technique for holomorphic vector bundles on complete Kähler manifolds and geometric properties of projective manifolds.
Let $Σ$ be an open Riemann surface and $Hol (Σ)$ be the Lie algebra of holomorphic vector fields on $Σ.$ We fix a projective structure (i.e. a local $SL_2(C)-$structure) on $Σ.$ We calculate the first group of cohomology of $Hol(Σ)$ with coefficients in the space of linear holomorphic operators acting on tensor densities, vanishing on the Lie algebra $SL_2 (C).$ The result is independant on the choice of the projective structure. We give explicit formulae of 1-cocycles generating this cohomology group.
Defect relation for holomorphic maps from complex discs into projective varieties and hypersurfaces
本文针对复圆盘上有限增长指数的全纯映射，建立了与射影簇中任意超曲面族相交的第二主要定理，给出截断亏量之和的上界，并推广改进了先前相关结果。
Wronskian curvature positivity is equivalent to the existence of a holomorphic projective connection
研究复流形中朗斯基曲率正性与全纯射影联络的关系，证明复流形存在满足特定条件光滑联络当且仅当存在全纯射影联络，固定无挠联络时条件等同于射影类全纯性。
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.
Askey-Wilson version of Second Main Theorem for holomorphic curves in projective space
该研究引入Askey-Wilson版Wronskian-Casorati行列式，建立射影空间全纯曲线一般形式第二基本定理的Askey-Wilson版本，改进了已有结果，并推导了超曲面$l$-次一般位置下的截断版本。
We describe a construction of complex geometrical analysis which corresponds to the classical theory of spherical harmonics.
We study the dynamics of compositions of a sequence of holomorphic mappings in projective space. We define ergodicity and mixing for non-autonomous dynamical systems, and we construct totally invariant measures for which our sequence satisfies these properties.
We study the global property of local holomorphic isometric mappings from a class of Kahler manifolds into a product of projective algebraic manifolds with induced Fubini-Study metrics, where isometric factors are allowed to be negative.
Holomorphic mappings of maximal rank into projective spaces
本文针对最大秩全纯映射与一般位置超平面相交的情形，证明了截断至水平n+1-p的Cartan型第二基本定理，加强了Stoll和Vitter的结果，并统一了Cartan与Carlson-Griffiths的工作。
We prove an effective version of a theorem of Dufresnoy: For any set of 2n+1 hyperplanes in general position in n-dimensional complex projective space, we find an explicit constant K such that for every holomorphic map f from the unit disc to the complement of these hyperplanes, the derivative of f at the origin measured with respect to the Fubuni-Study metric is bouned above by K. This result gives an explicit lower bound on the Royden function, i.e., the ratio of the Kobayashi metric on the hyperplane complement to the Fubini-Study metric. Our estimate is based on the potential-theoretic met…
We study codimension one holomorphic foliations on complex projective spaces and compact manifolds under the assumption that the foliation has a projective transverse structure in the complement of some invariant codimension one analytic subset. The basic motivation is the characterization of pull-backs of Riccati foliations on projective spaces. Our techniques apply to give a description of the generic models of codimension one foliations on compact manifolds of dimension $\ge 3$.
Non-existence of a holomorphic embedding of the Sobolev loop space into the projective Hilbert space
本文对比射影希尔伯特空间与黎曼球面Sobolev环路空间上的亚纯映射性质，证明后者无法全纯嵌入甚至不存在到前者的非退化亚纯映射。
In this paper we studied a broader type of generalized balls which are domains on the complex projective with possibly Levi-degenerate boundaries. We proved rigidity theorems for proper holomorphic mappings among them by exploring the structure of the moduli spaces of projective linear subspaces, which generalized some earlier results for the ordinary generalized balls with Levi-nondegenerate boundaries.
Given a compact connected Riemann surface $X$ equipped with an antiholomorphic involution $τ$, we consider the projective structures on $X$ satisfying a compatibility condition with respect to $τ$. For a projective structure $P$ on $X$, there are holomorphic connections and holomorphic differential operators on $X$ that are constructed using $P$. When the projective structure $P$ is compatible with $τ$, the relationships between $τ$ and the holomorphic connections, or the differential operators, associated to $P$ are investigated. The moduli space of projective structures on a compact oriented…
In this paper, we study the regular quantizations of Kähler manifolds by using the first two coefficients of Bergman function expansions. Firstly, we obtain sufficient and necessary conditions for certain Hermitian holomorphic vector bundles and their ball subbundles to be regular quantizations. Secondly, we obtain that some projective bundles over the Fano manifolds $M$ admit regular quantizations if and only if $M$ are biholomorphically isomorphism to the complex projective spaces. Finally, we obtain the balanced metrics on certain Hermitian holomorphic vector bundles and their ball subbundl…
We solve the $\bar{\partial}$-equation for $(p,q)$-forms locally on any reduced pure-dimensional complex space and we prove an explicit version of Serre duality by introducing suitable concrete fine sheaves of certain $(p,q)$-currents. In particular this gives a precise condition for the $\bar{\partial}$-equation to be globally solvable. Our results extend results for $(0,q)$-forms and give information about holomorphic $p$-forms on singular spaces.
In Part I of this paper we introduced the notion of the projective linking number Link(M,Z) of a compact oriented real submanifold M of dimension 2p-1 in complex projective n-space P^n with an algebraic subvariety Z of codimension p in P^n - M. It is shown here that a basic conjecture concerning the projective hull of real curves in P^2 implies the following result: M is the boundary of a positive holomorphic p-chain T of mass M(T) < K in P^n if and only if Link(M,Z) > -K p!deg(Z) for all algebraic subvarieties Z of codimension p in P^n - M. An analogous result is implied in any projective man…
Chern number formulas for holomorphic jet bundles are computed for projective curves and for projective surfaces. These formulas are used to show that certain minimal surfaces of general type (generic hypersurfaces of degree at least 5 in projective 3-spaces in particular) are hyperbolic in the sense of complex hyperbolic geometry.
On Grothendieck type duality for the space of holomorphic functions of several variables
本文描述了多复变量全纯函数空间的强对偶空间，将其等同为满足特定条件的调和函数闭子空间，利用Bochner-Martinelli核将经典Grothendieck型对偶性推广至多维，并证明对偶性成立当且仅当Sobolev全纯函数空间在全纯函数空间中稠密。
We prove that a one-dimensional foliation with generic singularities on a projective space, exhibiting a Lie group transverse structure in the complement of some codimension one algebraic subset is logarithmic, i.e., it is the intersection of codimension one foliations given by closed one-forms with simple poles. If there is only one singularity in a suitable affine space, then the foliation is induced by a linear diagonal vector field.
We prove that any unstable holomorphic 2-bundle over the complex projective space of complex dimension n at least 6 must split into a direct sum of two holomorphic line bundles. The statement with the weaker dimension condition of n at least 4 has been an open conjecture since 1977. One ingredient in our method uses Mathias Peternell's singular variety version of the Barth-Lefschetz theorem which requires the strong dimension condition of n at least 6.
The algebra $H^\infty(D)$ of bounded holomorphic functions on $D\subset\mathbb C$ is projective free for a wide class of infinitely connected domains. In particular, for such $D$ every rectangular left-invertible matrix with entries in $H^\infty(D)$ can be extended in this class of matrices to an invertible square matrix (the generalization of the corona theorem for $H^\infty(D)$). This follows from a new result on the structure of the maximal ideal space of $H^\infty(D)$ asserting that its covering dimension is $2$ and the second Čech cohomology group is trivial.
We introduce the notion of the projective linking number Link(M,Z) of a compact oriented real submanifold M of dimension 2p-1 in complex projective n-space P^n with an algebraic subvariety Z in P^n - M of codimension p. This notion is related to projective winding numbers and quasi-plurisubharmonic functions, and it generalizes directly from P^n to any projective manifold. Part 1 of this paper establishes the following result for the case p=1. Let M be an oriented, stable, real analytic curve in P^n. Then M is the boundary of a positive holomorphic 1-chain T with Mass(T) < K in P^n if and only…
In [Ann. of Math. 169 (2009)], Min Ru proved a second main theorem for algebraically nondegenerate holomorphic curves in complex projective varieties intersecting fixed hypersurface targets. In this paper, by introducing a new proof method for the case of projective varieties, we generalize this result to moving hypersurface targets.
We prove that a transversely product component of the singular set of a holomorphic foliation on $\mathbb P^n$ is necessarily a Kupka component.
We show that every CR-automorphism of the closure of a Levi degenerate hyperquadric in the projective space extends to a holomorphic automorphism of the projective space.
We establish a defect relation of holomorphic curves from a general open Riemann surface into a normal complex projective variety, with Zariski-dense image intersecting effective Cartier divisors.
We investigate the holomorphic differential operators on a Riemann surface $M$. This is done by endowing $M$ with a projective structure. Let $\mathcal L$ be a theta characteristic on $M$. We explicitly describe the jet bundle $J^k(E\otimes {\mathcal L}^{\otimes n})$, where $E$ is a holomorphic vector bundle on $M$ equipped with a holomorphic connection, for all $k$ and $n$. This provides a description of holomorphic differential operators from $E\otimes {\mathcal L}^{\otimes n}$ to another holomorphic vector bundle $F$ using the natural isomorphism $\text{Diff}^k(E\otimes {\mathcal L}^{\otime…
On the Grothendieck duality for the space of holomorphic Sobolev functions
本文研究了全纯 Sobolev 函数空间的强对偶性，将其刻画为无穷远消失的外部全纯函数空间，并以此推广了经典的 Grothendieck-Köthe-Sebastião e Silva 对偶性。
In this paper, we pose several conjectures on structures and images of maximal rationally connected fibrations of smooth projective varieties admitting semi-positive holomorphic sectional curvature. Toward these conjectures, we prove that the canonical bundle of images of such fibrations is not big. Our proof gives a generalization of Yang's solution using RC positivity for Yau's conjecture. As an application, we show that any compact Kähler surface with semi-positive holomorphic sectional curvature is rationally connected, or a complex torus, or a ruled surface over an elliptic curve.
Let V and F be holomorphic bundles over a complex manifold M, and s be a holomorphic section of V. We study the cohomology associated to the Koszul complex induced by s, and prove a generalized Serre duality theorem for them.
Holomorphic foliations of degree four on the complex projective space
研究复射影空间上四次全纯叶状结构的结构分类，并对高阶射流的 $d\geq4$ 次叶状结构证明其必属于横截仿射、横截射影或有理拉回三种情形之一。
In this paper, we establish a structure theorem for a smooth projective variety $X$ with semi-positive holomorphic sectional curvature. Our structure theorem contains the solution for Yau's conjecture and it can be regarded as a natural generalization of the structure theorem proved by Howard-Smyth-Wu and Mok for holomorphic bisectional curvature. Specifically, we prove that $X$ admits a locally trivial morphism $ϕ:X \to Y$ such that the fiber $F$ is rationally connected and the image $Y$ has a finite étale cover $A \to Y$ by an abelian variety $A$, by combining the author's previous work with…
Kähler duality and projective embeddings
受 Hermitian 对称空间对偶理论启发，本文引入并研究了 $\mathbb C^n$ 域之间的 Kähler 对偶概念。
In this note, we establish the following Second Main Theorem type estimate for every entire non-algebraically degenerate holomorphic curve $f\colon\mathbb{C}\rightarrow\mathbb{P}^n(\mathbb{C})$, in present of a {\sl generic} hypersuface $D\subset\mathbb{P}^n(\mathbb{C})$ of sufficiently high degree $d\geq 15(5n+1)n^n$: \[ T_f(r) \leq \,N_f^{[1]}(r,D) + O\big(\log T_f(r) + \log r \big)\parallel, \] where $T_f(r)$ and $N_f^{[1]}(r,D)$ stand for the order function and the $1$-truncated counting function in Nevanlinna theory. This inequality quantifies recent results on the logarithmic Green--Grif…
We study the branched holomorphic projective structures on a compact Riemann surface $X$ with a fixed branching divisor $S\, =\, \sum_{i=1}^d x_i$, where $x_i \,\in\, X$ are distinct points. After defining branched ${\rm SO}(3,{\mathbb C})$--opers, we show that the branched holomorphic projective structures on $X$ are in a natural bijection with the branched ${\rm SO}(3,{\mathbb C})$--opers singular at $S$. It is deduced that the branched holomorphic projective structures on $X$ are also identified with a subset of the space of all logarithmic connections on $J^2((TX)\otimes {\mathcal O}_X(S))…
We prove that the canonical bundle of any holomorphic family of compact complex algebraic manifolds carries a singular Hermitian metric having non-negative curvature current and such that every holomorphic section of the canonical bundle of the central fiber is L^2 with respect to this metric. This result was proved by Siu when the members of the family are general type, and used to establish the deformation invariance of plurigenera in that case. In fact, we prove our result in a more general setting of a smooth hypersurface with not-necessarily normal trivial bundle in a manifold that is Ste…
In this paper we provide sufficient conditions for maps of vector bundles on smooth projective varieties to be uniquely determined by their degeneracy schemes. We then specialize to holomorphic distributions and foliations. In particular, we provide sufficient conditions for distributions of arbitrary rank on projective spaces to be uniquely determined by their singular schemes.
Let S be a compact connected oriented orbifold surface We show that using Bers simultaneous uniformization, the moduli space of projective structure on S can be mapped biholomorphically onto the total space of the holomorphic cotangent bundle of the Teichmüller space for S. The total space of the holomorphic cotangent bundle of the Teichmüller space is equipped with the Liouville symplectic form, and the moduli space of projective structures also has a natural holomorphic symplectic form. The above identification is proved to be compatible with these symplectic structures. Similar results are …
We study codimension one holomorphic distributions on the projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2 with locally free tangent sheaves, and show that codimension one distributions of arbitrary degree with only isolated singularities have stable tangent sheaves. Furthermore, we describe the moduli space of distributions in terms of Grothendieck's Quot-scheme for the tangent bundle. In certain cases, we show that the moduli space of codimension one …
Singular Levi-flats and holomorphic foliations on projective spaces
本文研究射影空间上具有不变奇异Levi-平坦簇的余维1全纯叶状结构，引入两类条件刻画其可积性，证明Darboux型二择性并得到闭有理或对数1-形式的存在性。
Meromorphic maps from ${\Bbb C}^{p}$ into semi-abelian varieties and general projective varieties
通过纤维积分降维法，将全纯曲线的Nevanlinna型结果推广到从${\Bbb C}^{p}$到射影簇的亚纯映射，包括Bloch定理、第二基本定理等。
Earlier we introduced and studied the concept of holomorphic {\it branched Cartan geometry}. We define here a foliated version of this notion; this is done in terms of Atiyah bundle. We show that any complex compact manifold of algebraic dimension $d$ admits, away from a closed analytic subset of positive codimension, a nonsingular holomorphic foliation of complex codimension $d$ endowed with a transversely flat branched complex projective geometry (equivalently, a ${\mathbb C}P^d$-geometry). We also prove that transversely branched holomorphic Cartan geometries on compact complex projective r…
On a geometrically smooth complex algebraic curve X^1 in P^2(C), represented in complex affine coordinates (x,y) as the zero-locus R(x,y) = 0 of some polynomial R of degree d >= k+3, an explicit family of generating independent holomorphic jet differentials J_R^1, ..., J_R^k expressed in terms of R and its partial derivatives is exhibited with its new precious nonlinearity features as a complete explicitation of all holomorphic sections of the Green-Griffiths bundle of m-homogeneous polynomialized order k jets of local holomorphic maps from a complex disc into X^1.
On the structure of compact Kähler manifolds with nonnegative holomorphic sectional curvature
本文研究具有非负全纯截面曲率的紧凯勒流形，通过建立完整和原理的伪有效版本，证明在无真正平坦切向量等条件下流形必为射影且有理连通，从而回答了 Matsumura-Yang 问题并推广了 Yau 猜想，同时分类了非射影三维情形并推广到正实双截面曲率度量。
We strengthen our previous results regarding the moduli spaces of Zoll metrics and Zoll projective structures on S^2. In particular, we describe a concrete, open condition which suffices to guarantee that a totally real embedding of RP^2 in CP_2 arises from a unique Zoll projective structure on the 2-sphere. Our methods ultimately reflect the special role such structures play in the initial value problem for the 3-dimensional Lorentzian Einstein-Weyl equations.
We prove that if in a (smooth) holomorphic family of compact complex manifolds all the fibres, except one, are projective, then the remaining (limit) fibre must be Moishezon. In an earlier work, we proved this result under the extra assumption that the limit fibre carries a strongly Gauduchon metric. In the present paper, we remove the extra assumption by proving that if all the fibres, except one, are $\partial\bar\partial$-manifolds, then the limit fibre carries a strongly Gauduchon metric. The $\partial\bar\partial$-assumption on the generic fibre is much weaker than the projective, Kähler …
Pseudo-Holomorphic Hamiltonian Systems and Kähler Duality of Complex Coadjoint Orbits
本文研究伪全纯哈密顿系统并引入PHHS以在近复流形上描述经典力学，同时证明复余伴随轨道具有超凯勒与全纯凯勒结构的凯勒对偶性，并猜想其源于双余切丛。
In this work we introduce a GSV type index for varieties invariant by holomorphic Pfaff systems (possibly non locally decomposables) on projective manifolds. We prove a non-negativity property for the index. As an application, we prove that the non-negativity of the GSV-index gives us an obstruction to the solution of the Poincaré problem for Pfaff systems on projectives spaces.
We prove that holomorphic normal projective connections on compact complex surfaces are flat. We show that a holomorphic torsion-free affine connection $\nabla$ on a compact complex surface is locally modelled on a translations-invariant affine connection on $\C^2$, except if $\nabla$ is a generic connection on a principal elliptic bundle over a Riemann surface of genus $g \geq 2$, with odd first Betti number. In the last case, the local Killing Lie algebra is of dimension one, generated by the fundamental vector field of the principal fibration.
Foliated affine and projective structures
本文形式化复流形上沿全纯叶状结构叶片的仿射与射影结构，给出奇点局部正规形式和紧致情形下的指标公式，并证明一般型正则叶状结构不承认叶状射影结构，最终分类了紧复曲面上的此类结构。
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