On coisotropic deformations of holomorphic submanifolds

Deformations of compact holomorphic Poisson submanifolds

In this paper, we study deformations of compact holomorphic Poisson submanifolds which extend Kodaira's series of papers on semi-regularity (deformations of compact complex submanifolds of codimension 1), deformations of compact complex submanifolds of arbitrary codimensions, and stability of compact complex submanifolds in the context of holomorphic Poisson deformations. We also study simultaneous deformations of holomorphic Poisson structures and holomorphic Poisson submanifolds on a fixed underlying compact complex manifold. In appendices, we present deformations of Poisson closed subscheme…

Chern classes of quantizable coisotropic bundles

Let $M$ be a smooth algebraic variety of dimension $2(p+q)$ with an algebraic symplectic form and a compatible deformation quantization $\mathcal{O}_h$ of the structure sheaf. Consider a smooth coisotropic subvariety $j: Y \to M$ of codimension $q$ and a vector bundle $E$ on $Y$. We show that if $j_* E$ admits a deformation quantization (as a module) then its characteristic class $\widehat{A}(M) exp(-c(\mathcal{O}_h)) ch(j_* E)$ lifts to a cohomology group associated to the null foliation of $Y$. Moreover, it can only be nonzero in degrees $2q, \ldots, 2(p+q)$. For Lagrangian $Y$ this reduces …

可双亚纯收缩的全纯 Lagrangian 子流形的标准形式

Normal form of bimeromorphically contractible holomorphic Lagrangian submanifolds

本文通过到法锥的形变研究可双亚纯收缩至一点的紧全纯 Lagrangian 子流形，证明其必为 $CP^n$，且局部辛双全纯等价于余切丛零截面邻域，建立 Weinstein 定理的全纯版本。

Images of real submanifolds under finite holomorphic mappings

We give some results concerning the smoothness of the image of a real-analytic submanifold in complex space under the action of a finite holomorphic mapping. For instance, if the submanifold is not contained in a proper complex subvariety, we give a necessary and sufficient condition guaranteeing that its image is smooth and the mapping is transversal to the image.

全纯射流沿子流形的最优全纯延拓的渐近性

The asymptotics of the optimal holomorphic extensions of holomorphic jets along submanifolds

本文研究沿子流形的全纯射流的$L^2$-最优延拓算子的渐近公式，通过分析Schwartz核与Bergman投影，证明了射流空间上两个自然范数的渐近等距。

Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions

We define a new family of open Gromov-Witten type invariants based on intersection theory on the moduli space of pseudoholomorphic curves of arbitrary genus with boundary in a Lagrangian submanifold. We assume the Lagrangian submanifold arises as the fixed points of an anti-symplectic involution and has dimension 2 or 3. In the strongly semi-positive genus 0 case, the new invariants coincide with Welschinger's invariant counts of real pseudoholomorphic curves. Furthermore, we calculate the new invariant for the real quintic threefold in genus 0 and degree 1 to be 30.

Extending Nirenberg-Spencer's question on holomorphic embeddings to families of holomorphic embeddings

Nirenberg and Spencer posed the question whether the germ of a compact complex submanifold in a complex manifold is determined by its infinitesimal neighborhood of finite order when the normal bundle is sufficiently positive. To study the problem for a larger class of submanifolds, including free rational curves, we reformulate the question in the setting of families of submanifolds and their infinitesimal neighborhoods. When the submanifolds have no nonzero vector fields, we prove that it is sufficient to consider only first-order neighborhoods to have an affirmative answer to the reformulate…

Holomorphic versus algebraic equivalence for deformations of real-algebraic CR manifolds

We consider (small) algebraic deformations of germs of real-algebraic CR submanifolds in complex space and study the biholomorphic equivalence problem for such deformations. We show that two algebraic deformations of minimal holomorphically nondegenerate real-algebraic CR submanifolds are holomorphically equivalent if and only if they are algebraically equivalent.

Holomorphic correspondences between CR manifolds

It is proved that a germ of a real analytic CR map from a smooth real-analytic minimal CR manifold M to an essentially finite real-algebraic generic submanifold M' of P^N of the same CR-dimension extends as a holomorphic correspondence along M. Applications are given for pseudoconcave submanifolds of P^N.

Analyticity of smooth CR mappings of generic submanifolds

We consider a smooth CR mapping $f$ from a real-analytic generic submanifold $M$ in $\bC^N$ into $\bC^N$. For $M$ of finite type and essentially finite at a point $p\in M$, and $f$ formally finite at $p$, we give a necessary and sufficient condition for $f$ to extend as a holomorphic mapping in some neighborhood of $p$. In a similar vein, we consider a formal holomorphic mapping $H$ and give a necessary and sufficient condition for $H$ to be convergent.

Complex embeddings, Toeplitz operators and transitivity of optimal holomorphic extensions

In a setting of a complex manifold with a fixed positive line bundle and a submanifold, we consider the optimal Ohsawa-Takegoshi extension operator, sending a holomorphic section of the line bundle on the submanifold to the holomorphic extension of it on the ambient manifold with the minimal $L^2$-norm. We show that for a tower of submanifolds in the semiclassical setting, i.e. when we consider a large tensor power of the line bundle, the extension operators satisfy transitivity property modulo some small defect, which can be expressed through Toeplitz type operators. We calculate the first si…

Variations of Hodge Structures of Rank Three k-Higgs Bundles and Moduli Spaces of Holomorphic Triples

There is an isomorphism between the moduli spaces of $σ$-stable holomorphic triples and some of the critical submanifolds of the moduli space of $k$-Higgs bundles of rank three, whose elements $(E,φ^k)$ correspond to variations of Hodge structure, VHS. There are special embeddings on the moduli spaces of $k$-Higgs bundles of rank three. The main objective here is to study the cohomology of the critical submanifolds of such moduli spaces, extending those embeddings to moduli spaces of holomorphic triples.

Expected topology of random real algebraic submanifolds

Let X be a smooth complex projective manifold of dimension n equipped with an ample line bundle L and a rank k holomorphic vector bundle E. We assume that 0< k <=n, that X, E and L are defined over the reals and denote by RX the real locus of X. Then, we estimate from above and below the expected Betti numbers of the vanishing loci in RX of holomorphic real sections of E tensored with L^d, where d is a large enough integer. Moreover, given any closed connected codimension k submanifold S of R^n with trivial normal bundle, we prove that a real section of E tensored with L^d has a positive proba…