Eugen Hellmann
The coefficient space is a kind of resolution of singularities of the universal flat deformation space for a given Galois representation of some local field. It parameterizes (in some sense) the finite flat models for the Galois representation. The aim of this note is to determine the image of the coefficient space in the universal deformation space.
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Let k denote a perfect field of characteristic 5. We show that the versal deformation ring of an element of order 5 and Hasse conductor 2 as automorphism of a ring of formal power series k[[t]] computed by Bertin and Mezard, is in fact universal. This provides the first example of a non trivial universal deformation ring in higher conductor.
Consider a formal (mixed-characteristic) deformation functor D of a representation of a finite group G as automorphisms of a power series ring k[[t]] over a perfect field k of positive characteristic. Assume that the action of G is weakly ramified, i.e., the second ramification group is trivial. Examples of such representations are provided by a group action on an ordinary curve: the action of a ramification group on the completed local ring of any point on such a curve is weakly ramified. We prove that the only such D that are not pro-representable occur if k has characteristic two and G is o…
We classify the nilpotent orbits in a simple Lie algebra for which the restriction of the adjoint quotient map to a Slodowy slice is the universal Poisson deformation of its central fibre. This generalises work of Brieskorn and Slodowy on subregular orbits. In particular, we find in this way new singular symplectic hypersurfaces of dimension 4 and 6.
We study deformation of spherical $CR$ circle bundles over Riemann surfaces of genus > 1. There is a one to one correspondence between such deformation space and the so-called universal Picard variety. Our differential-geometric proof of the structure and dimension of the unramified universal Picard variety has its own interest, and our theory has its counterpart in the Teichmuller theory.
We show that every compact complex analytic space endowed with a fine logarithmic structure and every morphism between such spaces admit a semi-universal deformation. These results generalize the analogous results in complex analytic geometry first independently proved by A. Douady and H. Grauert in the '70. We follow Douady's two steps process approach consisting of an infinite-dimensional construction of the semi-universal deformation space followed by a finite-dimensional reduction.
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 $X$.
We study deformations of plane curve singularities from an analytic point of view and obtain some new concrete results. We show some rather unexpected properties of Puiseux coefficients treated as functions on a suitably defined parameter space. The methods used in paper are very elementary.
Universal Deformations of a Curve and a Differential
本文为紧复曲线与亚纯1-形式构成的局部平面对构造万有局部形变（Kuranishi族），并应用于KdV方程、sinh-Gordon方程等可积系统的谱数据，奠定可积系统谱曲线形变理论的基础。
In this paper, we represent the Hodge metric in terms of the Weil-Petersson metric and its Ricci curvature on the moduli spaces of polarized Calabi-Yau threefolds.
Infinite-Dimensional Geometry of the Universal Deformation of the Complex Disk
从无限维几何的观点研究了复杂圆盘的普遍变形。给出了次对称空间在泛变形下的结构。次对称镜的普遍变形的叶理示出，以确定一个真正的偏振。本文的主题可能感兴趣的专家在代数几何和表示论，以及研究人员处理数学问题的现代量子场论。内容. I. Virasoro-Bott群的旗流形的无限维几何（复圆盘的普遍变形的基础）。二. Virasoro-Bott群旗流形骨架的无限维几何。三.复杂圆盘的普遍变形的无限维几何。
A note on the group extension problem to semi-universal deformation
本文详细解释了第二个Hirzebruch曲面自同构群的作用仅扩展至形式半泛形变的一阶，并证明对于约化群作用，Kuranishi空间上扩展作用的局部性是通常情况下的最佳结果。
Fix a simple complex Lie group G and a principal sl(2,C) subalgebra of Lie(G). Then the moduli space of semi-stable, topologically trivial G-Higgs bundles on a hyperbolic, spin Riemann surface acquires a marked point. This is the unique C*-fixed point on the Hitchin section. We describe a universal analytic family of deformations which provides holomorphic Darboux coordinates in a neighbourhood of the section. This is a special case of a more general deformation-theoretic construction in the spirit of Kuranishi theory. As a toy example of the latter we consider the tautological family of centr…
A p-divisible group over a complete local domain determines a Galois representation on the Tate module of its generic fibre. We determine the image of this representation for the universal deformation in mixed characteristic of a bi-infinitesimal group and for the p-rank strata of the universal deformation in positive characteristic of an infinitesimal group. The method is a reduction to the known case of one-dimensional groups by a deformation argument based on properties of the stratification by Newton polygons.
This paper gives a canonical construction, in terms of additive cohomological functors, of the universal formal deformation of a compact complex manifold without vector fields (more generally of a faithful $g$-module, where $g$ is a sheaf of Lie algebras without sections). The construction is based on a certain (multivariate) Jacobi complex $J(g)$ associatd to $g$: indeed ${\mathbb C}\oplus {\mathbb H}^0(J(g))^*$ is precisely the base ring of the universal deformation.
Let k be an algebraically closed field of characteristic $p>0$, and $G_0$ be a Barsotti-Tate group (or $p$-divisible group) over k. We denote by $S$ the "algebraic" local moduli in characteristic p of $G_0$, by $G$ the universal deformation of $G_0$ over $S$, and by $U\subset S$ the ordinary locus of $G$. The etale part of $G$ over $U$ gives rise to a monodromy representation $ρ$ of the fundamental group of $U$ on the Tate module of $G$. Motivated by a famous theorem of Igusa, we prove in this article that $ρ$ is surjective if $G_0$ is connected and HW-cyclic. This latter condition is equivale…
Minkowski decompositions and universal equivariant deformations of toric pairs
本文构造仿射正规环面簇的显式平坦代数族，其完备化对所有等变变形万有，并证明约化基的不可约分支与切片多面体的极大格友好Minkowski分解双射对应，推广了已有定理。
We introduce an $A_\infty$-algebra structure on the Hochschild cohomology of the endomorphism bimodule of a finite-dimensional representation of an associative algebra. We prove that this structure determines a presentation for non-commutative deformations of the representation. From this, we deduce presentations of universal deformation rings of Galois representations. In turn, we apply these presentations in order to deduce universal deformation rings of Galois pseudorepresentations, supplying a a tangent and obstruction theory for pseudorepresentations. This generalizes the broadly used tan…
For the universal isomonodromic deformation of an irreducible logarithmic rank two connection over a smooth complex projective curve of genus at least two, consider the family of holomorphic vector bundles over curves underlying this universal deformation. In a previous work we proved that the vector bundle corresponding to a general parameter of this family is stable. Here we prove that the vector bundle corresponding to a general parameter is in fact very stable (it does not admit any nonzero nilpotent Higgs field).
On the universality of integrable deformations of solutions of degenerate Riemann-Hilbert-Birkhoff problems
本文研究退化 RHB 问题解的可积形变分类，证明 Sabbah 形变虽非严格普适，却具有相对普适性，并刻画了可局部全纯 Jordan 化的矩阵值映射。
In this paper, we study and describe the universal Poisson deformation space of hypertoric varieties concretely. In the first application, we show that affine hypertoric varieties as conical symplectic varieties are classified by the associated regular matroids (this is a partial generalization of the result by Arbo and Proudfoot). As a corollary, we obtain a criterion when two quiver varieties whose dimension vector have all coordinates equal to one are isomorphic to each other. Then we describe all 4- and 6-dimensional affine hypertoric varieties as quiver varieties and give some examples of…
Equivariant deformation theory for nilpotent slices in symplectic Lie algebras
针对辛李代数中双块若尔当形幂零轨道的Slodowy切片非通用泊松形变问题，证明其幂零Slodowy簇与D型同构，C型切片为$\u2124_2$-等变通用泊松形变且有非交换对应。
Stable cohomology of the mapping class group with symplectic coefficients and of the universal Abel-Jacobi map
这一替换纠正了主要结果的陈述和证明。此外，还增加了关于普适Abel-Jacobi映射的一节。
A projective variety has a large fundamental group if its universal covering has no compact subvarieties. In this paper we investigate whether the property of a variety having a large fundamental groups is stable under Kähler deformations.
Deformation theory and Lie algebra homology
给出了函数环在多模问题的一种通用形式变形的基础上的一种描述。答案是给出了一个同调组的某些dg李代数规范（到一个基本上唯一的准同构）与一个问题。
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