On Grauert's examples of complete Kähler metrics

Weakly complete domains in Grauert type surfaces

The aim of this short note is to investigate the geometry of weakly complete subdomains of Grauert type surfaces, i.e. open connected sets D, sitting inside a Grauert type surface X, which admit a smooth plurisubharmonic exhaustion function. We prove that they are either modifications of Stein spaces or Grauert type surfaces themselves and we apply these results to the special case of Hopf surfaces.

Non-negative divisors and the Grauert metric

Grauert showed that it is possible to construct complete Kähler metrics on the complement of complex analytic sets in a domain of holomorphy. In this note, we study the holomorphic sectional curvatures of such metrics on the complement of a principal divisor in $\mathbb{C}^n$, $n \ge 1$. In addition, we also study how this metric and its holomorphic sectional curvature behaves when the corresponding principal divisors vary continuously.

On the integral homology and counterexamples to the Andreotti-Grauert conjecture

In this paper, we prove by means of a counterexample that there exist pair of integers (n,p) with $n\geq 3$, $2\leq p\leq n-1$, and open sets $D$ in $C^{n}$ which are cohomologically $p$-complete with respect to the structure sheaf of $D$ such that the cohomology group $H_{n+p}(D,Z)$ does not vanish. In particular $D$ is not $p$-complete.

圆2球面的线丛与Grauert管Hardy量子化的交织

Intertwining the line bundle and Grauert-tube Hardy quantizations of the round 2-sphere

本文比较圆2球面单位余切丛上的两种Hardy量子化，计算其直线归一化重叠，推导相关迹级数与渐近，给出等变酉算子及Fredholm行列式等结果。

Sharp Dimension Estimates of Holomorphic Functions and Rigidity

Let $M^n$ be a complete noncompact K$\ddot{a}$hler manifold of complex dimension $n$ with nonnegative holomorphic bisectional curvature. Denote by $\mathcal{O}$$_d(M^n)$ the space of holomorphic functions of polynomial growth of degree at most $d$ on $M^n$. In this paper we prove that $$dim_{\mathbb{C}}{\mathcal{O}}_d(M^n)\leq dim_{\mathbb{C}}{\mathcal{O}}_{[d]}(\mathbb{C}^n),$$ for all $d>0$, with equality for some positive integer $d$ if and only if $M^n$ is holomorphically isometric to $\mathbb{C}^n$. We also obtain sharp improved dimension estimates when its volume growth is not maximal or…

关于带角点的$q$-完备与$q$-凹复流形

On $q$-complete and $q$-concave with corners complex manifolds

本文研究带角点的$q$-完备与$q$-凹复流形，证明了上同调有限性定理，并构造反例表明上同调$q$-完备不等价于$q$-完备，且对任意$(n,q)$存在非上同调$\hat{q}$-完备的带角点$q$-完备开集。