关于解析簇的对偶定理

On complex Legendre duality

Complex Legendre duality is a generalization of Legendre transformation from Euclidean spaces to Kahler manifolds, that Berndtsson and collaborators have recently constructed. It is a local isometry of the space of Kahler potentials. We show that the fixed point of such a transformation must correspond to a real analytic Kahler metric.

Explicit Serre duality on complex spaces

In this paper we use recently developed calculus of residue currents together with integral formulas to give a new explicit analytic realization, as well as a new analytic proof of Serre duality on any reduced pure $n$-dimensional paracompact complex space $X$. At the core of the paper is the introduction of concrete fine sheaves $\mathscr{B}_X^{n,q}$ of certain currents on $X$ of bidegree $(n,q)$, such that the Dolbeault complex $(\mathscr{B}_X^{n,\bullet},\,\bar{\partial})$ becomes, in a certain sense, a dualizing complex. In particular, if $X$ is Cohen-Macaulay (e.g., Gorenstein or a comple…

Summability and duality

We formalize the observation that the same summability methods converge in a Banach space $X$ and its dual $X^*$. At the same time we determine conditions under which these methods converge in the weak and weak*-topologies on $X$ and $X^*$ respectively. We also derive a general limitation theorem, which yields a necessary condition for the convergence of a summability method in $X$. These results are then illustrated by applications to a wide variety of function spaces, including spaces of continuous functions, Lebesgue spaces, the disk algebra, Hardy and Bergman spaces, the BMOA space, the Bl…

液体塔恩卡对偶I：经典情况

Liquid Tannaka Duality I: Classical Case

本文通过液体向量空间理论，在复代数簇中重建拓扑基本群及扭曲基本群oid，利用液体局部系统范畴实现塔恩卡对偶在经典情况下的应用。

复勒让德对偶

Complex Legendre duality

本文提出经典勒让德变换的复推广，构造Kähler度量空间上Mabuchi度量的显式局部等距对称性，从而回答了Semmes工作中提出的问题。

分布族与对偶下的Pfaff系统

Families of distributions and Pfaff systems under duality

非奇异簇$X$上的奇异分布可以由切层的子层$D$定义，也可以由1-形式层的子层$D^0$的零点定义。虽然这两个定义在$D$上的温和条件下是等价的，但它们通常会产生不等价的平坦族概念。在这项工作中，我们调查的条件下，这两个概念的平坦的家庭是等价的。在最后一节中，我们重点讨论了分布可积的情况，并利用我们的结果推广了Cukierman和Pereira的一个定理。