A Dolbeault Isomorphism Theorem in Infinite Dimensions

$L^2$-type Dolbeault isomorphisms and vanishing theorems for logarithmic sheaves twisted by multiplier ideal sheaves

In this article, we first establish an $L^2$-type Dolbeault isomorphism for the sheaf of logarithmic differential forms twisted by the multiplier ideal sheaf. By using this isomorphism and $L^2$-estimates equipped with a singular Hermitian metric, we obtain logarithmic vanishing theorems involving multiplier ideal sheaves on compact Kähler manifolds with simple normal crossing divisors.

de Rham and Dolbeault Cohomology of solvmanifolds with local systems

Let $G$ be a simply connected solvable Lie group with a lattice $Γ$ and the Lie algebra $\g$ and a representation $ρ:G\to GL(V_ρ)$ whose restriction on the nilradical is unipotent. Consider the flat bundle $E_ρ$ given by $ρ$. By using "many" characters $\{α\}$ of $G$ and "many" flat line bundles $\{E_α\}$ over $G/Γ$, we show that an isomorphism \[\bigoplus_{\{α\}} H^{\ast}(\g, V_α\otimes V_ρ)\cong \bigoplus_{\{E_α\}} H^{\ast}(G/Γ, E_α\otimes E_ρ)\] holds. This isomorphism is a generalization of the well-known fact:"If $G$ is nilpotent and $ρ$ is unipotent then, the isomorphism $H^{\ast}(\g, V_…

The equivalence of pseudodifferential operators and their symbols via Čech-Dolbeault cohomology

In this paper we construct the sheaf morphism from the sheaf of pseudodifferential operators to its symbol class. Since the map is hard to construct directly, we realize it with two original ideas as follows. First, to calculate cohomologies we use the theory of Čech-Dolbeault cohomology introduced by Honda, Izawa and Suwa. Secondly we construct a new symbol class, which is called the symbols of $C^\infty$-type. These ideas enable us to construct the sheaf morphism, which is actually an isomorphism of sheaves.

$L^2$-Dolbeault 解析与弱伪凸复空间上 Nadel 消失定理

$L^2$-Dolbeault resolutions and Nadel vanishing on weakly pseudoconvex complex spaces with singular Hermitian metrics

本文研究了弱伪凸复空间上 $L^2$-Dolbeault 解析与 $L^2$-估计，推广了 Nadel 消失定理。

The Dolbeault complex in infinite dimensions. II

We prove that the equation d-bar u = f can be solved on a ball B(R) of radius R in the Banach space l^1 if f is a closed Lipschitz continuous (0,1) form on B(R). We also present examples of closed (0,1) forms f of various regularities on the spaces l^p that are not exact. In particular, in the first result above, it is not enough to assume that f is merely continuous, rather than Lipschitz continuous.

On the annihilator of a Dolbeault group

We show that any Dolbeault cohomology group $H^{p,q}(D)$, $p\ge0$, $q\ge1$, of an open subset $D$ of a closed finite codimensional complex Hilbert submanifold of $\ell_2$ is either zero or infinite dimensional. We also show that any continuous character of the algebra of holomorphic functions of a closed complex Hilbert submanifold $M$ of $\ell_2$ is induced by evaluation at a point of $M$. Lastly, we prove that any closed split infinite dimensional complex Banach submanifold of $\ell_1$ admits a nowhere critical holomorphic function.