arXiv subjects

An Khuong Doan

Publications and source records attributed to An Khuong Doan.

8 recordsLinked to original sources

arXiv2026

Equivariant deformations of isolated singularities and applications

In this paper, we develop an equivariant version of the classical theory of deformations of germs of complex spaces in the presence of complex Lie groups. As a tradition, the main result is the existence of equivariant semi-universal deformations of germs of complex spaces in the reductive case and the non-existence in general in the non-reductive case. The latter is justified by an explicit counterexample. In particular, it generalizes Grauert-Donin's existence theorem to the equivariant settings and Ferrer-Puerta-Slodowy's one to the case of reductive complex Lie groups. Several applications to deformations of pairs, rigidity and actions on Milnor fibers are given.

math.AG↗

arXiv2025

Formal moduli problems with cohomological constraints

We aim to generalize Lurie-Pridham's famous $\infty$-correspondence between formal moduli problems and differential graded Lie algebras to the context where some cohomological conditions are imposed. Specifically, a natural equivalence between formal moduli problems with cohomological constraints and derived cohomology jump functors is provided, thereby answering affirmatively a question posed by N. Budur and B. Wang in \cite{1}.

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arXiv2024

Local structure of the Teichmüller and the Riemann moduli stacks

The goal of this note is to introduce an interesting question proposed by D. Rydh on an analytic version of the local structure of Artin stacks saying that near points with linearly reductive stabilizers, Artin stacks are étale-locally quotient stacks. We give some supporting evidence by verifying it on two fundamental classes of classical analytic moduli spaces: the Teichmüller moduli space and the Riemann moduli space of integrable complex structures whose analytic stack versions have been constructed by a recent work of L. Meersseman.

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arXiv2023

A note on the group extension problem to semi-universal deformation

The aim of this note is twofold. Firstly, we explain in detail Remark 4.1 in \cite{doan-a} by showing that the action of the automorphism group of the second Hirzebruch surface $\mathbb{F}_2$ on itself extends to its formal semi-universal deformation only up to the first order. Secondly, we show that for reductive group actions, the locality of the extended actions on the Kuranishi space constructed in \cite{doan-equivariant} is the best one could expect in general.

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arXiv2023

Semi-prorepresentability of formal moduli problems and equivariant structures

We generalize the notion of semi-universality in the classical deformation problems to the context of derived deformation theories. A criterion for a formal moduli problem to be semi-prorepresentable is produced. This can be seen as an analogue of Schlessinger's conditions for a functor of Artinian rings to have a semi-universal element. We also give a sufficient condition for a semi-prorepresentable formal moduli problem to admit a $G$-equivariant structure in a sense specified below, where $G$ is a linearly reductive group. Finally, by making use of these criteria, we derive many classical results including the existence of ($G$-equivariant) formal semi-universal deformations of algebraic schemes and that of complex compact manifolds.

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arXiv2022

Group actions on local moduli space of holomorphic vector bundles

We prove that actions of complex reductive Lie groups on a holomorphic vector bundle over a complex compact manifold are locally extendable to its local moduli space.

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arXiv2021

Equivariant Kuranishi family of complex compact manifolds

We prove that actions of complex reductive Lie groups on a complex compact manifold are locally extendable to its Kuranishi family. This can be seen as an analogue of Rim's result (see [11]) in the analytic setting.

math.CV↗

arXiv2020

A counter-example to the equivariance structure on semi-universal deformation

If $X$ is a projective variety and $G$ is an algebraic group acting algebraically on $X$, we provide a counter-example to the existence of a $G$-equivariant extension on the formal semi-universal deformation of $X$.

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