Priska Jahnke, Ivo Radloff
The authors give a complete classification of projective threefolds admitting a holomorphic normal projective connection. Moreover, they prove a general structure theorem on complex projective manifolds admitting a holomorphic normal projective connection, saying in particular, that any such manifold is either the projective space or minimal in the sense of Mori.
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We prove Chern class equalities for abelian families with a holomorphic normal projective connection.
Holomorphic projective connections on compact complex threefolds
本文分类复射影三维流形上的全纯射影联络，证明其若非平坦则必位于阿贝尔三维流形上，并排除平凡典范线丛的单连通紧复三维流形存在此类联络。
The authors give a complete classification of projective threefolds admitting a holomorphic conformal structure. A Corollary is the complete list of projective threefolds, whose tangent bundle is a symmetric square.
Locally Homogeneous Holomorphic Geometric Structures on Projective Varieties
本文研究带齐次代数簇模型的光滑射影簇上的全纯局部齐次结构，并刻画所有能展开到模型的子簇。
We introduce the concept of a branched holomorphic Cartan geometry. It generalizes to higher dimension the definition of branched (flat) complex projective structure on a Riemann surface introduced by Mandelbaum. This new framework is much more flexible than that of the usual holomorphic Cartan geometries. We show that all compact complex projective manifolds admit branched flat holomorphic projective structure. We also give an example of a non-flat branched holomorphic normal projective structure on a compact complex surface. It is known that no compact complex surface admits such a structure…
Let $(M,ω)$ be a compact Kähler manifold with negative holomorphic sectional curvature. It was proved by Wu-Yau and Tosatti-Yang that $M$ is necessarily projective and has ample canonical bundle. In this paper, we show that any irreducible subvariety of $M$ is of general type. Moreover, we can extend the theorem to the quasi-negative curvature case building on earlier results of Diverio-Trapani. Finally, we investigate the more general setting of a quasi-projective manifold $X^{\circ}$ endowed with a Kähler metric with negative holomorphic sectional curvature and we prove that such a manifold …
We study projective manifolds M admitting a (flat) holomorphic normal projective connection and show that the Iitaka fibration (up to etale coverings) defines a smooth abelian group scheme structure on M.
Let $\mathcal{O}(D)$ be an equivariant line bundle which is big and nef on a complex projective nonsingular toric variety $X$. Given a continuous toric metric $\|\cdot\|$ on $\mathcal{O}(D)$, we define the energy at equilibrium of $(X,ϕ_{\bar{D}})$ where $ϕ_{\bar{D}}$ is the weight of the metrized toric divisor $\bar{D}=(D,\|\cdot\|)$. We show that this energy describes the asymptotic behaviour as $k\rightarrow \infty$ of the volume of the $L^2$-norm unit ball induced by $(X,kϕ_{\bar{D}})$ on the space of global holomorphic sections $H^0(X,\mathcal{O}(kD))$.
We prove that the only Calabi--Yau projective manifolds which bear holomorphic Cartan geometries are precisely the abelian varieties. (Nous démontrons que les seules variétés projectives de Calabi--Yau qui possèdent des géométrie holomorphes de Cartan sont les variétés abéliennes.)
This manuscript is an introduction to the theory of holomorphic foliations on the complex projective plane. Historically the subject has emerged from the theory of ODEs in the complex domain and various attempts to solve Hilbert's 16th Problem, but with the introduction of complex algebraic geometry, foliation theory and dynamical systems, it has now become an interesting subject of its own.
Answering a problem raised by Lazarsfeld, Hwang and Mok proved that a surjective holomorphic map from a rational homogeneous space of Picard number 1 onto projective manifold different from projective space must be a biholomorphism. THe aim of this paper is to generalized this result to irreducible rational homogeneous space of higher Picard number.
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.
The topology of spaces of holomorphic maps to projective space
本文证明了从光滑复射影簇到射影空间的全纯映射与连续映射在依赖次数的范围内同调相同。
We extend the holomorphic analytic torsion classes of Bismut and Köhler to arbitrary projective morphisms between smooth algebraic complex varieties. To this end, we propose an axiomatic definition and give a classification of the theories of generalized holomorphic analytic torsion classes for arbitrary projective morphisms.
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…
The mirror of a projective toric manifold $X_Σ$ is given by a Landau-Ginzburg model $(Y,W)$. We introduce a class of Lagrangian submanifolds in $(Y,W)$ and show that, under the SYZ mirror transformation, they can be transformed to torus-invariant hermitian metrics on holomorphic line bundles over $X_Σ$. Through this geometric correspondence, we also identify the mirrors of Hermitian-Einstein metrics, which are given by distinguished Lagrangian sections whose potentials satisfy certain Laplace-type equations.
We associate to an SU(2) hyperbolic monopole a holomorphic sphere embedded in projective space and use this to uncover various features of the monopole.
We show that a nonsingular complex projective variety admitting a holomorphic vector field with nonempty isolated zeroes, is rational using a key technique by Harvey-Lawson on finite volume flows. This statement was conjectured by J. Carrell. By the same technique, we obtain a uniform upper bound of Betti numbers of nonsingular complex projective variety admitting a holomorphic vector field with exact one zero point. Such an upper bound depends only on the dimension of the variety, which is a stronger version of a result of Akyildiz and Carrell.
We prove rigidity of various types of holomorphic parabolic geometry on smooth complex projective varieties.
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…
Non-standard Holomorphic Structures on Line Bundles over the Quantum Projective Line
本文探讨量子射影线上的线丛非标准全纯结构，证明存在无限多个非规范等价结构，反驳了2011年提出的疑问。
In this note we show that if a projective manifold admits a Kähler metric with negative holomorphic sectional curvature then the canonical bundle of the manifold is ample. This confirms a conjecture of the second author.
Fixed points and Holomorphic Structures on Line Bundles over the Quantum Projective Line
研究量子射影线上的线丛全纯结构不唯一性，通过不动点理论分析平坦$\bar\partial$连接，并将可逆解的存在性与特定非线性映射在适当Banach空间中的不动点存在性联系起来。
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 …
We introduce a weak concept of Morita equivalence, in the birational context, for Poisson modules on complex normal Poisson projective varieties. We show that Poisson modules, on projective varieties with mild singularities, are either rationally Morita equivalent to a flat partial holomorphic sheaf, or a sheaf with a meromorphic flat connection or a co-Higgs sheaf. As an application, we study the geometry of rank two meromorphic rank two $\mathfrak{sl}_2$-Poisson modules which can be interpreted as a Poisson analogous to transversally projective structures for codimension one holomorphic foli…
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.
Holomorphic Bisectional Curvature and Applications to Deformations and Rigidity for Variations of Mixed Hodge Structure
本文证明了容许分级极化混合 Hodge 结构周期映射的刚性准则，并在拟射影曲线族、具有普通双点的射影曲线及正规函数等多种情形下建立了刚性。
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。
Geometry of Holomorphic One-forms on Smooth Projective Varieties
本文证明光滑射影簇到简单阿贝尔簇的态射光滑当且仅当存在无零点的拉回全纯一次形式，并研究了有零点的全纯一次形式空间的结构。
We prove that if a Calabi--Yau manifold $M$ admits a holomorphic Cartan geometry, then $M$ is covered by a complex torus. This is done by establishing the Bogomolov inequality for semistable sheaves on compact Kähler manifolds. We also classify all holomorphic Cartan geometries on rationally connected complex projective manifolds.
This article studies codimension one foliations on projective man-ifolds having a compact leaf (free of singularities). It explores the interplay between Ueda theory (order of flatness of the normal bundle) and the holo-nomy representation (dynamics of the foliation in the transverse direction). We address in particular the following problems: existence of foliation having as a leaf a given hypersurface with topologically torsion normal bundle, global structure of foliations having a compact leaf whose holonomy is abelian (resp. solvable), and factorization results.
An unobstructedness theorem is proved for deformations of compact holomorphic Poisson manifolds and applied to a class of examples. These include certain rational surfaces and Hilbert schemes of points on Poisson surfaces. We study in particular the Hilbert schemes of the projective plane and show that a generic deformation is determined by two parameters -- an elliptic curve and a translation on it.
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…
Varieties admitting a holomorphic symplectic form: LLV algebras and derived equivalences
利用Beauville-Bogomolov分解计算具有全纯辛形式的复光滑射影簇的LLV代数，推广超凯勒簇和阿贝尔簇的已知结果，并证明Orlov猜想对许多此类簇成立，特别对所有四维情形成立。
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。
We classify projective manifolds with flat holomorphic conformal structures.
In this paper we study the topology of the spaces Hol(M,P{n},k) of (basepoint preserving) holomorphic maps of a given degree k from a Riemann surface M of genus g>0 into the n-th complex projective space P{n}, n>0. Using symmetric products of the surface as well as the geometry of the associated Abel-Jacobi maps, we construct an explicit topological compactification of Hol(M,P{n},k) which allow us to construct spectral sequences with known E_1-terms that can be used to determine the homology groups of Hol(M,P{n},k). Complete calculations are given for the elliptic case (g=1). Complete rational…
Numerically flat foliations and holomorphic Poisson geometry
本文研究了紧Kähler流形上具有数值平坦切丛的光滑全纯叶状结构，证明了其诱导切丛分解等性质，并由此导出了射影流形上秩为2的Poisson结构的全局Weinstein分裂定理及零点存在性的新Hodge理论条件。
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.
We show that much of the structure of the 2-sphere as a complex curve survives the q-deformation and has natural generalizations to the quantum 2-sphere - which, with additional structures, we identify with the quantum projective line. Notably among these is the identification of a quantum homogeneous coordinate ring with the coordinate ring of the quantum plane. In parallel with the fact that positive Hochschild cocycles on the algebra of smooth functions on a compact oriented 2-dimensional manifold encode the information for complex structures on the surface, we formulate a notion of twisted…
Positive holomorphic sectional curvature on rational surfaces
本文证明每个从射影环流形经有限次点爆破得到的射影流形都允许具有正全纯截面曲率的Kähler度量，从而完成了Hitchin关于有理曲面刻画的结果，并解决了Yau提出的一个复曲面情形问题。
In a previous paper, we proved that a projective Kähler manifold of positive total scalar curvature is uniruled. At the other end of the spectrum, it is a well-known theorem of Campana and Kollár-Miyaoka-Mori that a projective Kähler manifold of positive Ricci curvature is rationally connected. In the present work, we investigate the intermediate notion of $k$-positive Ricci curvature and prove that for a projective $n$-dimensional Kähler manifold of $k$-positive Ricci curvature the MRC fibration has generic fibers of dimension at least $n-k+1$. We also establish an analogous result for projec…
Two conjectures relating the Kodaira dimension of a smooth projective variety and existence of number of nowhere vanishing 1-forms on the variety are proposed and verified in dimension 3.
Fundamental groups of compact Kähler manifolds with semi-positive holomorphic sectional curvature
本文证明具有半正全纯截面曲率的紧Kähler流形容许局部平凡纤维化，其纤维为有理连通射影流形，底空间为环面有限étale商，从而将射影流形结构定理推广至紧Kähler流形。
The goal of this note is to exhibit the integrability properties (in the sense of the Frobenius theorem) of holomorphic p-forms with values in certain line bundles with seminegative curvature on a compact Kaehler manifold. There are in fact very strong restrictions, both on the holomorphic form and on the curvature of the seminegative line bundle. As a consequence, these observations provide interesting information on the structure of projective manifolds which admit a contact structure: either they are Fano manifolds or, thanks to results of Kebekus-Peternell-Sommese-Wisniewski, they are biho…
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。
In this paper, with the aim of establishing a structure theorem for a compact Kähler manifold $X$ with semi-positive holomorphic sectional curvature, we study a morphism $ϕ: X \to Y$ to a compact Kähler manifold $Y$ with pseudo-effective canonical bundle. We prove that the morphism $ϕ$ is always smooth (that is, a submersion), the image $Y$ admits a finite etale cover $T \to Y$ by a complex torus $T$, and further that all the fibers are isomorphic when $X$ is projective. Moreover, by applying a modified method to maximal rationally connected fibrations, we show that $X$ is rationally connected…
We give a complete description of the Bridgeland stability manifold for the bounded derived category of holomorphic triples over a smooth projective curve of genus 1 as a connected, four dimensional complex manifold.
The paper studies the deformation theory of a holomorphic surjective map from a normal compact complex space to a compact Kaehler manifold and describes the component of the space of holomorphic maps, generalizing results in the projective case by Kebekus and the authors.
In this note we derive from deep results due to Clozel-Ullmo the density of Noether-Lefschetz loci inside the moduli space of marked (polarized) irreducible holomorphic symplectic (IHS) varieties. In particular we obtain the density of Hilbert schemes of points on projective $K3$ surfaces and of projective generalized Kummer varieties in their moduli spaces. We present applications to the existence of rational curves on projective deformations of such varieties, to the study of relevant cones of divisors, and a refinement of Hassett's result on cubic fourfolds whose Fano variety of lines is is…
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.
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 show that every holomorphic one-form on a smooth complex projective variety of general type must vanish at some point. The proof uses generic vanishing theory for Hodge D-modules on abelian varieties.
A generically generated vector bundle on a smooth projective variety yields a rational map to a Grassmannian, called Kodaira map. We answer a previous question, raised by the asymptotic behaviour of such maps, giving rise to a birational characterization of abelian varieties. In particular we prove that, under the conjectures of the Minimal Model Program, a smooth projective variety is birational to an abelian variety if and only if it has Kodaira dimension 0 and some symmetric power of its cotangent sheaf is generically generated by its global sections.
This paper is motivated by the real symplectic isotopy problem : does there exists a nonsingular real pseudoholomorphic curve not isotopic in the projective plane to any real algebraic curve of the same degree? Here, we focus our study on symmetric real curves on the projective plane. We give a classification of real schemes (resp. complex schemes) realizable by symmetric real curves of degree 7 with respect to the type of the curve (resp. M-symmetric real curves of degree 7). In particular, we exhibit two real schemes which are realizable by real symmetric dividing pseudoholomorphic curves of…
We construct quasi-projective moduli spaces of $K$-general lattice polarized irreducible holomorphic symplectic manifolds. Moreover, we study their Baily--Borel compactification and investigate a relation between one-dimensional boundary components and equivalence classes of rational Lagrangian fibrations defined on mirror manifolds.
We prove that a generic holomorphic foliation on a weighted projective plane has no algebraic solutions when the degree is big enough. We also prove an analogous result for foliations on Hirzebruch surfaces.
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