On the complement of a hypersurface with flat normal bundle which corresponds to a semipositive line bundle

Effective algebraic degeneracy of entire curves in complements of smooth projective hypersurfaces

In this work, it is established that for a generic projective hypersurface $H\subset\mathbb{P}^n(\mathbb{C})$ of degree $d\geq(5n)^2\,n^{n}$, any holomorphic entire curve $f\colon\mathbb{C}\to\mathbb{P}^n(\mathbb{C})\setminus H$ has its image contained in a proper algebraic subvariety $Z\subsetneq\mathbb{P}^{n}(\mathbb{C})$, that does not depend on the curve $f$. Here generic means that the coefficients of the defining equation of $H$ have to lie outside of a proper algebraic subvariety of the projective space of coefficients of homogeneous polynomials of degree $d$ (that parametrizes the alge…

On homotopy types of complements of analytic sets and Milnor fibres

We prove that for any germ of complex analytic set in $\CC^n$ there exists a hypersurface singularity whose Milnor fibration has trivial geometric monodromy and fibre homotopic to the complement of the germ of complex analytic set. As an application we show an example of a quasi-homogeneous hypersurface singularity, with trivial geometric monodromy and simply connected and non-formal Milnor fibre.

On the order map for hypersurface coamoebas

Given a hypersurface coamoeba of a Laurent polynomial f, it is an open problem to describe the structure of its set of connected complement components. In this paper we approach this problem by introducing the lopsided coamoeba. We show that the closed lopsided coamoeba comes naturally equipped with an order map, i.e. a map v from its set of connected complement components to a translated lattice inside the zonotope of a Gale dual of the point configuration supp(f). Under a natural assumption, the map v is a bijection. Finally we use this map to obtain new results concerning coamoebas of polyn…

Levi-flat hypersurfaces and their complement in complex surfaces

In this work we study analytic Levi-flat hypersurfaces in complex algebraic surfaces. First, we show that if this foliation admits chaotic dynamics (i.e. if it does not admit a transverse invariant measure), then the connected components of the complement of the hypersurface are modifications of Stein domains. This allows us to extend the CR foliation to a singular algebraic foliation on the ambient complex surface. We apply this result to prove, by contradiction, that analytic Levi-flat hypersurfaces admitting a transverse affine structure in a complex algebraic surface have a transverse inva…

Local systems on complements of arrangements of smooth, complex algebraic hypersurfaces

We consider smooth, complex quasi-projective varieties $U$ which admit a compactification with a boundary which is an arrangement of smooth algebraic hypersurfaces. If the hypersurfaces intersect locally like hyperplanes, and the relative interiors of the hypersurfaces are Stein manifolds, we prove that the cohomology of certain local systems on $U$ vanishes. As an application, we show that complements of linear, toric, and elliptic arrangements are both duality and abelian duality spaces.

Complex K3 surfaces containing Levi-flat hypersurfaces

We show the existence of a complex K3 surface $X$ which is not a Kummer surface and has a one-parameter family of Levi-flat hypersurfaces in which all the leaves are dense. We construct such $X$ by patching two open complex surfaces obtained as the complements of tubular neighborhoods of elliptic curves embedded in blow-ups of the projective planes at general nine points.

Vanishing homology of projective hypersurfaces with 1-dimensional singularities

We introduce and study the vanishing homology of singular projective hypersurfaces. We prove its concentration in two levels in case of 1-dimensional singular locus $Σ$, and moreover determine the ranks of the nontrivial homology groups. These two groups depend on the monodromy at special points of $Σ$ and on the effect of the monodromy of the local system over its complement.

Weak analytic hyperbolicity of complements of generic surfaces of high degree in projective 3-space

In this article we prove that every entire curve in the complement of a generic hypersurface of degree $d\geq 586$ in $\mathbb{P}_{\mathbb{C}}^{3}$ is algebraically degenerate i.e there exists a proper subvariety which contains the entire curve.

Equations differentielles sur les hypersurfaces de l'espace projectif complexe de dimension 4

The main goal of this work is to prove that every entire curve in a smooth hypersurface of degree greater than or equal to 97 in the complex projective space of dimension 4 must satisfy an algebraic differential equation of order 3. A logarithmic version of this result is given proving that every entire curve in the complement of a smooth surface of degree greater than or equal to 92 in the complex projective space of dimension 3 must satisfy an algebraic differential equation of order 3.

On Levi flat hypersurfaces with transversely affine foliation

We prove the non-existence of real analytic Levi flat hypersurface whose complement is 1-convex and Levi foliation is transversely affine in compact Kahler surfaces.

Homotopy groups of complements and non-isolated singularities

We obtain sufficient conditions for the vanishing of higher homotopy groups of the complements to hypersurfaces in ${\mathbb C}^n$ in terms of the behavior at infinity and relate the monodromy of non isolated singularities to the position of singularities in generic plane sections.

Stein complements in compact Kähler manifolds

Given a projective or compact Kähler manifold X and a (smooth) hypersurface Y, we study conditions under which $X \setminus Y$ could be Stein. We apply this in particular to the case when X is the projectivization of the so-called canonical extension of the tangent bundle $T_M$ of a projective manifold M with Y being the projectivization of $T_M$ itself.

紧曲面上半正反正则除子补集的$\overline{\partial}$上同调

$\overline{\partial}$ cohomology of the complement of a semi-positive anticanonical divisor of a compact surface

本文研究具有半正曲率反正则线丛的紧复曲面上，其反正则除子补集的$\overline{\partial}$上同调群$H^1(M,\mathcal{O}_M)$的结构与性质。

Analytic Continuation of Holomorphic Mappings From Non-minimal Hypersurfaces

We study the analytic continuation problem for a germ of a biholomorphic mapping from a non-minimal real hypersurface $M\subset\CC{n}$ into a real hyperquadric $\mathcal Q\subset\CP{n}$ and prove that under certain non-degeneracy conditions any such germ extends locally biholomorphically along any path lying in the complement $U\setminus X$ of the complex hypersurface $X$ contained in $M$ for an appropriate neighborhood $U\supset X$. Using the monodromy representation for the multiple-valued mapping obtained by the analytic continuation we establish a connection between nonminimal real hypersu…

On Chern-Moser normal forms of strongly pseudoconvex hypersurfaces with high-dimensional stability group

We explicitly describe germs of strongly pseudoconvex non-spherical real-analytic hypersurfaces $M$ at the origin in $\CC^{n+1}$ for which the group of local CR-automorphisms preserving the origin has dimension $d_0(M)$ equal to either $n^2-2n+1$ with $n\ge 2$, or $n^2-2n$ with $n\ge 3$. The description is given in terms of equations defining hypersurfaces near the origin, written in the Chern-Moser normal form. These results are motivated by the classification of locally homogeneous Levi non-degenerate hypersurfaces in $\CC^3$ with $d_0(M)=1,2$ due to A. Loboda, and complement earlier joint w…

Curvature restrictions for Levi-flat real hypersurfaces in complex projective planes

We study curvature restrictions of Levi-flat real hypersurfaces in complex projective planes, whose existence is in question. We focus on its totally real Ricci curvature, the Ricci curvature of the real hypersurface in the direction of the Reeb vector field, and show that it cannot be greater than -4 along a Levi-flat real hypersurface. We rely on a finiteness theorem for the space of square integrable holomorphic 2-forms on the complement of the Levi-flat real hypersurface, where the curvature plays the role of the size of the infinitesimal holonomy of its Levi foliation.

On the logarithmic Kobayashi conjecture

We study the hyperbolicity of the log variety $(\mathbb{P}^n, X)$, where $X$ is a very general hypersurface of degree $d\geq 2n+1$ (which is the bound predicted by the Kobayashi conjecture). Using a positivity result for the sheaf of (twisted) logarithmic vector fields, which may be of independent interest, we show that any log-subvariety of $(\mathbb{P}^n, X)$ is of log-general type, give a new proof of the algebraic hyperbolicity of $(\mathbb{P}^n, X)$, and exclude the existence of maximal rank families of entire curves in the complement of the universal degree $d$ hypersurface. Moreover, we…

Hyperbolicity of Generic High-Degree Hypersurfaces in Complex Projective Space

We use two ingredients to prove the hyperbolicity of generic hypersurfaces of sufficiently high degree and of their complements in the complex projective space. One is the pullbacks of appropriate low pole order meromorphic jet differentials on the complex projective space to a hypersurface. The other is slanted vector fields of low vertical pole order on the vertical jet space of the universal hypersurface. We also present a number of related results, obtained by the same methods, such as: (i) a Big-Picard-Theorem type statement concerning extendibility, across the puncture, of holomorphic ma…

Homotopical variations and high-dimensional Zariski-van Kampen theorems

In 1933, van Kampen described the fundamental groups of the complements of plane complex projective algebraic curves. Recently, Chéniot-Libgober proved an analogue of this result for higher homotopy groups of the complements of complex projective hypersurfaces with isolated singularities. Their description is in terms of some "homotopical variation operators". We generalize here the notion of "homotopical variation" to (singular) quasi-projective varieties. This is a first step for further generalizations of van Kampen's theorem. A conjecture, with a first approach, is stated in the special ca…

On superheight conditions for the affineness of open subsets

In this paper we consider the open complement U of a hypersurface Y=V(a) in an affine scheme X. We study the relations between the affineness of U, the intersection of Y with closed subschemes, the property that every closed surface in U is affine, the property that every analytic closed surface is Stein and the superheight of the defining ideal a.

Zariski-van Kampen theorem for higher homotopy groups

This paper gives an extension of the classical Zariski-van Kampen theorem describing the fundamental groups of the complements of plane singular curves by generators and relations. It provides a procedure for computation of the first non-trivial higher homotopy groups of the complements of singular projective hypersurfaces in terms of the homotopy variation operators introduced here.

全纯线丛中的Oka管

Oka tubes in holomorphic line bundles

本文证明了在半正定全纯线丛上，当补集具有密度性质时，单位圆盘丛是Oka流形而外部区域是Kobayashi双曲域，并将该结果推广到有理齐性流形，揭示了Oka性质与度量正性的联系。

Effective Cartan-Tanaka connections on C6-smooth strongly pseudoconvex hypersurfaces M3 of C2

In a recent expository article (Notices of the AMS, 58 (2011), no. 1, 20-27), Ezhov, McLaughlin and Schmalz showed how to perform in an effective way Tanaka's prolongation procedure valid generally for filtered structures of constant type when the distribution is equipped with an integrable complex structure, so as to derive the principal curvature invariants and (co)frame(s) associated to strongly pseudoconvex real hypersurfaces M3 in C2, an approach which is alternative and complementary to the `hyperspherical' connection of Elie Cartan, and to the so-called calculi of Fefferman, of Chern-Mo…