Linearization of transition functions of a semi-positive line bundle along a certain submanifold

与负向量丛相关的奇点的不可约消歧的唯一性

Uniqueness of irreducible desingularization of singularities associated to negative vector bundles

本文证明紧复流形上负全纯向量丛经 Grauert 向下爆破所得奇点的不可约消歧唯一，并据此刻画负线丛同构及子流形到超曲面的唯一修改方式。

Embeddings of submanifolds and normal bundles

This paper is devoted to the study of the embeddings of a complex submanifold $S$ inside a larger complex manifold $M$; in particular, we are interested in comparing the embedding of $S$ in $M$ with the embedding of $S$ as the zero section in the total space of the normal bundle $N_S$ of $S$ in $M$. We explicitely describe some cohomological classes allowing to measure the difference between the two embeddings, in the spirit of the work by Grauert, Griffiths, and Camacho-Movasati-Sad; we are also able to explain the geometrical meaning of the separate vanishing of these classes. Our results ho…

Positivity of metrics on conic neighbourhoods of $1$-convex submanifolds

We consider positivity properties of vector bundles over conic neighbourhoods of sections of submersions over $1$-convex manifolds.

Characterization on projective submanifolds of codimensions 2 and 3

In this article we give a necessary and sufficient condition to characterize projective submanifolds in ${\mathbb P}^N$ with codimensions 2 and 3. The conditions involve the Chern classes of the manifold and a very ample line bundle on the manifold. This generalizes our earlier characterization for hypersurfaces. The higher codimensional cases are proposed as a general question.

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。

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…

Bohr-Sommerfeld拉格朗日子流形切丛上全纯截面的极化和收敛

Polarizations and Convergences of holomorphic sections on the tangent bundle of a Bohr-Sommerfeld Lagrangian submanifold

本文研究 Kähler 流形上预量子线丛的 Bohr-Sommerfeld 拉格朗日子流形附近全纯截面的渐近行为，证明了截面 $L^2$-范数的下界以及通过缩放映射拉回后截面收敛到纤维上常数函数。

半经典Ohsawa-Takegoshi延拓定理与正交Bergman核的渐近性

Semiclassical Ohsawa-Takegoshi extension theorem and asymptotics of the orthogonal Bergman kernel

该研究针对复流形中固定复子流形，推导了正线丛高张量幂下Ohsawa-Takegoshi延拓算子的显式渐近展开，同时得到正交Bergman投影算子的类似渐近结果。

Curvature and $L^p$ Bergman spaces on complex submanifolds in ${\mathbb C}^N$

Let $M$ be a closed complex submanifold in ${\mathbb C}^N$ with the complete Kähler metric induced by the Euclidean metric. Several finiteness theorems on the $L^p$ Bergman space of holomorphic sections of a given Hermitian line bundle $L$ over $M$ and the associated $L^2$ cohomology groups are obtained. Some infiniteness theorems are also given in order to test the accuracy of finiteness theorems. As applications we obtain some rigidity results concerning growth of curvatures.

Total reality of conormal bundles of hypersurfaces in almost complex manifolds

A generalization to the almost complex setting of a well-known result by S. Webster is given. Namely, we prove that if $Γ$ is a strongly pseudoconvex hypersurface in an almost complex manifold $(M, J)$, then the conormal bundle of $Γ$ is a totally real submanifold of $(T^*M, \J)$, where $\J$ is the lifted almost complex structure on $T^*M$ defined by Ishihara and Yano.