This paper contains the details and complete proofs of our earlier announcement in math.AG/9907004 .
We construct a general semiregularity map for algebraic cycles as asked for by S. Bloch in 1972. The existence of such a semiregularity map has well known consequences for the structure of the Hilbert scheme and for the variational Hodge conjecture. Aside from generalizing and extending considerably previously known results in this direction, we give new applications to deformations of modules that encompass, for example, results of Artamkin and Mukai.
The formation of the semiregularity map here involves powers of the cotangent complex, Atiyah classes, and trace maps, and is defined not only for subspaces of manifolds but for perfect complexes on arbitrary complex spaces. It generalizes in particular Illusie's treatment of the Chern character to the analytic context and specializes to Bloch's earlier description of the semiregularity map for locally complete intersections as well as to the infinitesimal Abel-Jacobi map for submanifolds.
In this paper, we study relative deformations of maps into a family of Kähler manifolds whose images are divisors. We show that if the map satisfies a condition called semiregularity, then it allows relative deformations if and only if the cycle class of the image remains Hodge in the family. This gives a refinement of the so-called variational Hodge conjecture. We also show that the semiregularity of maps is related to classical notions such as Cayley-Bacharach conditions and d-semistability.
Let $E^*$ be a finite complex of locally free sheaves on a complex manifold $X$. We prove that to every connection of type $(1,0)$ on $E^*$ it is canonically associated an $L_{\infty}$ morphism $g\colon A^{0, *}_X(\mathcal{H}om^*_{O_X}(E^*,E^*))\to \dfrac{A^{*,*}_X}{A^{\ge 2,*}_X}[2]$ that lifts the 1-component of Buchweitz-Flenner semiregularity map. An application to deformations of coherent sheaves on projective manifolds is given.
We consider matrices with entries in a local ring, Mat(m,n,R). Fix a group action, G on Mat(m,n,R), and a subset of allowed deformations, Σ\subseteq Mat(m,n,R). The standard question of Singularity Theory is the finite-(Σ,G)-determinacy of matrices. Finite determinacy implies algebraizability and is equivalent to a stronger notion: stable algebraizability. In our previous work this determinacy question was reduced to the study of the tangent spaces to Σand to the orbit, T_{(Σ,A)}, T_{(GA,A)} , and their quotient, the tangent module to the miniversal deformation. In particular, the order of det…
In this paper we generalize Artin-Verdier, Esnault and Wunram construction of McKay correspondence to arbitrary Gorenstein surface singularities. The key idea is the definition and a systematic use of a degeneracy module, which is an enhancement of the first Chern class construction via a degeneracy locus. We study also deformation and moduli questions. Among our main result we quote: a full classification of special reflexive MCM modules on normal Gorenstein surface singularities in terms of divisorial valuations centered at the singularity, a first Chern class determination at an adequate re…
We consider matrices with entries in a local ring, Mat(R). Fix a group action, G on Mat(R), and a subset of allowed deformations, Σ. The traditional objects of study in Singularity Theory and Algebraic Geometry are the tangent spaces T_{(Σ,A)}, T_{(GA,A)}, and their quotient, the tangent module to the miniversal deformation, T^1_{(Σ,G,A)}. This module plays the key role in various deformation problems, e.g., deformations of maps, of modules, of (skew-)symmetric forms. In particular, the first question is to determine the support/annihilator of this tangent module. In [Belitski-Kerner.1] we hav…
Blowing up a rational surface singularity in a reflexive module gives a (any) partial resolution dominated by the minimal resolution. The main theorem shows how deformations of the pair (singularity, module) relates to deformations of the corresponding pair of partial resolution and locally free strict transform, and to deformations of the underlying spaces. The results imply some recent conjectures on small resolutions and flops.
Deformations of modules of maximal grade and the Hilbert scheme at determinantal schemes
设R是一个多项式环，M是R生成的分次R-模的极大分次（这意味着M的齐次表示矩阵\cA的极大子式生成的理想I_t（\cA）在R中有极大余维数）.设X：=Proj（R/I_t（\cA））在一个足够大的开子集上是光滑的，且dim X > 0.然后证明了在弱假设下M的局部分次形变函子与X \subset Proj（R）上的局部Hilbert（scheme）函子同构。在此假设下，我们得到了Hilbert格式在（X）上是光滑的，并给出了其局部环维数的显式表达式.作为推论，我们证明了R. M. Miró-Roig和作者指出，在表示矩阵中具有固定度数的标准行列式格式的轨迹的闭包是Hilbert格式的一般光滑分量V。当dim X > 0时，证明了他们关于V的维数的猜想.证明了在Proj（R）中X的正规层的上同调H^ i_{*}（{ \cN }_X）对0 < i < dim X-1为零.最后，对上述结果稍加修改，用多项式环的任意Cohen-Macaulay商代替R，仍为真。
Classifying obstructions to the problem of finding extensions between two fixed modules goes back at least to L. Illusie's thesis. Our approach, following in the footsteps of J. Wise, is to introduce an analogous Grothendieck Topology on the category $A$-mod of modules over a fixed ring $A$ in a topos $E$. The problem of finding extensions becomes a banded gerbe and furnishes a cohomology class on the site $A$-mod. We compare our obstruction and that coming from Illusie's work, giving another construction of the exact sequence Illusie used to obtain his obstruction. Our work circumvents the co…
In a previous article (J. Algebra 367 (2012), 142-165) we established axiomatic parametrised Cohen-Macaulay approximation which in particular was applied to pairs consisting of a finite type flat family of Cohen-Macaulay rings and modules. In this sequel we study the induced maps of deformation functors and deduce properties like smoothness and injectivity under general, mainly cohomological conditions on the module.
Let $k$ be a perfect field of positive characteristic and $Z$ an effective Cartier divisor in the projective line over $k$ with complement $U$. In this note, we establish some results about the formal deformation theory of overconvergent isocrystals on $U$ with fixed "local monodromy" along $Z$. En route, we show that a Hochschild cochain complex governs deformations of a module over an arbitrary associative algebra. We also relate this Hochschild cochain complex to a de Rham complex in order to understand the deformation theory of a differential module over a differential ring.
For a pair (algebra, module) with equidimensional and isolated singularity we establish the existence of a versal henselian deformation. Obstruction theory in terms of an André-Quillen cohomology for pairs is a central ingredient in the Artin theory used. In particular we give a long exact sequence relating the algebra cohomology and the module cohomology with the cohomology of the pair and define a Kodaira-Spencer class for pairs. Cohen-Macaulay approximation induces maps between versal base spaces for pairs and cohomology conditions imply properties like smoothness, isomorphism and linear se…
We identify the type of $\mathbb{C}[[\hbar]]$-linear structure inherent in the $\infty$-categories which arise in the theory of Deformation Quantization modules. Using this structure, we show that the $\infty$-category of quasicoherent cohomologically complete DQ-modules is a deformation of the $\infty$-category of quasicoherent sheaves. We also obtain integral representation results for DQ-modules similar to the ones of Toën and Ben-Zvi-Nadler-Francis, stating that suitably linear functors between $\infty$-categories of DQ-modules are integral transforms.
Abelian and derived deformations in the presence of Z-generating geometric helices
对于Grothendieck范畴C，通过Z-生成序列（O（n））n in Z}等价于伴随Z-代数A上的“拟凝聚模”范畴，我们证明了在适当的上同调条件下，“取拟凝聚模”定义了A的线性变形与C的交换变形之间的等价性.如果（O（n））n in Z}同时是导出范畴中的几何螺旋，我们证明了将（变形）Z-代数限制为对象的“线”定义了与相关矩阵代数的线性变形的进一步等价。
A Dwork family is a one-parameter monomial deformation of a Fermat hypersurface. In this paper we compute algebraically the invariant part of its Gauss-Manin cohomology under the action of certain subgroup of automorphisms. To achieve that goal we use the algebraic theory of $\mathcal{D}$-modules, especially one-dimensional hypergeometric ones.
We show that the deformation theory of a perfect complex and that of its determinant are related by the trace map, in a general setting of sheaves on a site. The key technical step, in passing from the setting of modules over a ring where one has global resolutions to the general setting, is achieved using $K$-theory and higher category theory.
