An explicit d-bar-integration formula for weighted homogeneous varieties

An explicit d-bar-integration formula for weighted homogeneous varieties II: forms of higher degree

Let Y be a weighted homogeneous (singular) subvariety of C^n. The main objective of this paper is to present a class of explicit integral formulae for solving the d-bar-equation $ω=\dbarλ$ on the regular part of Y, where $ω$ is a d-bar-closed (0,q)-form with compact support and degree q>=1. Particular cases of these formulae yield L^p-bounded solution operators for $1<=p<=\infty$ if Y is a homogeneous and pure dimensional subvariety with an arbitrary singular locus.

The Hartogs phenomenon in almost homogeneous varieties

We study the Hartogs extension phenomenon in noncompact almost homogeneous algebraic varieties and we prove the cohomological and weight criteria for the Hartogs phenomenon. In the case of spherical varieties, we prove a criterion for the Hartogs phenomenon in terms of colored fans.

The d-bar-equation on homogeneous varieties with an isolated singularity

Let X be a regular irreducible variety in CP^{n-1}, Y the associated homogeneous variety in C^n, and N the restriction of the universal bundle of CP^{n-1} to X. In the present paper, we compute the obstructions to solving the d-bar-equation in the L^p-sense on Y for 1<=p<=\infty in terms of cohomology groups H^q(X,O(N^m)). That allows to identify obstructions explicitly if X is specified more precisely, for example if it is equivalent to CP^1 or an elliptic curve.

Some L2 results for d-bar on projective varieties with general singularities

Let X be an irreducible n-dimensional projective variety in CP^N with arbitrary singular locus. We prove that the L2-(p,1)-d-bar cohomology groups (with respect to the Fubini-Study metric) of the regular part of X are finite dimensional.

有理齐性空间中舒伯特簇的同调刚性与舒尔刚性

Homological rigidity and Schur rigidity of Schubert varieties in rational homogeneous spaces

研究高皮卡数有理齐性空间中舒伯特簇的同调刚性与舒尔刚性问题，证明长根情形下光滑舒伯特簇有同调刚性，给出子图型舒伯特簇同调刚性列表，还表明长根情形下子图型舒伯特簇除非有纤维丛结构否则有舒尔刚性。