标题： 退化厄米特度量与典范丛的谱几何 显示英文标题

标题： Degenerating Hermitian metrics and spectral geometry of the canonical bundle

摘要： 设$(X,h)$为一个紧致且不可约的埃尔米特复空间，复维数为$m$。 在本文中，我们关注的是作用在$L^2$截面空间上的Dolbeault算子，这些截面属于$reg(X)$的典范丛，其中$X$是其正则部分。 More precisely let $\overline{\mathfrak{d}}_{m,0}:L^2\Omega^{m,0}(reg(X),h)\rightarrow L^2\Omega^{m,1}(reg(X),h)$ be an arbitrarily fixed closed extension of $\overline{\partial}_{m,0}:L^2\Omega^{m,0}(reg(X),h)\rightarrow L^2\Omega^{m,1}(reg(X),h)$ where the domain of the latter operator is $\Omega_c^{m,0}(reg(X))$. 我们建立各种性质，如$\overline{\mathfrak{d}}_{m,0}$的闭合值域，包含关系$\mathcal{D}(\overline{\mathfrak{d}}_{m,0})\hookrightarrow L^2\Omega^{m,0}(reg(X),h)$的紧性，其中$\mathcal{D}(\overline{\mathfrak{d}}_{m,0})$是$\overline{\mathfrak{d}}_{m,0}$的定义域，并赋予相应的图范数，以及相关 Hodge-Kodaira 拉普拉斯算子$\overline{\mathfrak{d}}_{m,0}^*\circ \overline{\mathfrak{d}}_{m,0}$的谱离散性，并给出其特征值增长的估计。 得出若干推论，如与$\overline{\mathfrak{d}}_{m,0}^*\circ \overline{\mathfrak{d}}_{m,0}$相关的热算子的迹类性质，并给出其迹的估计。 最后在最后一部分，我们提供了在具有孤立奇点的紧致不可约厄米特复空间和复射影曲面的设定下，Hodge-Kodaira 拉普拉斯算子的几个应用。 显示英文摘要

摘要： Let $(X,h)$ be a compact and irreducible Hermitian complex space of complex dimension $m$. In this paper we are interested in the Dolbeault operator acting on the space of $L^2$ sections of the canonical bundle of $reg(X)$, the regular part of $X$. More precisely let $\overline{\mathfrak{d}}_{m,0}:L^2\Omega^{m,0}(reg(X),h)\rightarrow L^2\Omega^{m,1}(reg(X),h)$ be an arbitrarily fixed closed extension of $\overline{\partial}_{m,0}:L^2\Omega^{m,0}(reg(X),h)\rightarrow L^2\Omega^{m,1}(reg(X),h)$ where the domain of the latter operator is $\Omega_c^{m,0}(reg(X))$. We establish various properties such as closed range of $\overline{\mathfrak{d}}_{m,0}$, compactness of the inclusion $\mathcal{D}(\overline{\mathfrak{d}}_{m,0})\hookrightarrow L^2\Omega^{m,0}(reg(X),h)$ where $\mathcal{D}(\overline{\mathfrak{d}}_{m,0})$, the domain of $\overline{\mathfrak{d}}_{m,0}$, is endowed with the corresponding graph norm, and discreteness of the spectrum of the associated Hodge-Kodaira Laplacian $\overline{\mathfrak{d}}_{m,0}^*\circ \overline{\mathfrak{d}}_{m,0}$ with an estimate for the growth of its eigenvalues. Several corollaries such as trace class property for the heat operator associated to $\overline{\mathfrak{d}}_{m,0}^*\circ \overline{\mathfrak{d}}_{m,0}$, with an estimate for its trace, are derived. Finally in the last part we provide several applications to the Hodge-Kodaira Laplacian in the setting of both compact irreducible Hermitian complex spaces with isolated singularities and complex projective surfaces.