An example of noncommutative deformations
Eivind Eriksen
我们计算了第一Weyl代数上模族的非交换变形。这个例子显示了非对易变形理论的一些重要性质，这些性质将它与对易变形理论区分开来。
We compute the noncommutative deformations of a family of modules over the first Weyl algebra. This example shows some important properties of noncommutative deformation theory that separates it from commutative deformation theory.
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We prove that the functor of noncommutative deformations of every flipping or flopping irreducible rational curve in a 3-fold is representable, and hence associate to every such curve a noncommutative deformation algebra. This new invariant extends and unifies known invariants for flopping curves in 3-folds, such as the width of Reid, and the bidegree of the normal bundle. It also applies in the settings of flips and singular schemes. We show that the noncommutative deformation algebra is finite dimensional, and give a new way of obtaining the commutative deformations of the curve, allowing us…
For a general class of contractions of a variety X to a base Y, I discuss recent joint work with M. Wemyss defining a noncommutative enhancement of the locus in Y over which the contraction is not an isomorphism, along with applications to the derived symmetries of X. This note is based on a talk given at the Kinosaki Symposium in 2016.
The Generalized Burnside Theorem in noncommutative deformation theory
设A是域上的结合代数，M是右A-模的有限族.研究非交换变形函子的家庭M导致建设的代数观测和广义伯恩赛德定理，由于劳达尔。在本文中，我们给出了一个概述方面的非交换变形密切相关的广义伯恩赛德定理。
Computing noncommutative deformations of presheaves and sheaves of modules
我们描述了一个非交换变形理论的预层和层的模块，概括了这些全球性的代数结构的交换变形理论，和非交换变形理论的模块代数由于Laudal。本文的第一部分描述了小范畴上模的预层的一个非对易变形函子，并利用整体Hochschild上同调给出了该函子的一个阻塞理论。这个障碍理论的一个重要特点是，它可以在许多有趣的情况下具体计算。在本文的最后一部分，我们描述了环空间$(X, \mathcal{A})$上模的层和拟凝聚层的非交换形变函子。我们证明了对于$X$的任何好的$\mathcal{A}$-仿射开覆盖$\mathsf{U}$，遗忘函子$\mathsf{QCoh}(\mathcal{A}) \to \mathsf{PreSh}(\mathsf{U}, \mathcal{A})$诱导非交换变形函子的同构.当$(X, \mathcal{A}) = (X, \mathcal{O}_X)$是Beilinson和Bernstein意义下的概型或$(X, \mathcal{A}) = (X, \mathcal{D})$是D-概型时，考虑$X$上拟凝聚$\mathcal{A}$-模的非对易变形.在这些情况下，我们可以使用有限交下闭的$X$的任何开仿射覆盖，使用预层方法计算具体的非对易变形。以$X$为椭圆曲线为例，计算了左$\mathcal{D}_X$-模$\mathcal{O}_X$的非交换变形.
An introduction to noncommutative deformations of modules
本文对模的非交换变形作了初步的介绍。这种变形理论的主要结果是由于Laudal。设k是一个代数闭（交换）域，A是一个结合k-代数，M = {M_1，…，M_p}是左A-模的有限族。研究了Laudal引入的非对易变形函子Def（M）：a（p）-> Sets所描述的M族的同时形式变形.特别是，我们证明了变形函子Def（M）有一个pro-representing壳H（M），唯一的非典型同构，并描述了如何计算H（M）使用的Ext群的家庭M和他们的矩阵Massey产品。
Computing noncommutative global deformations of D-modules
设（X，D）是Beilinson和Bernstein意义下的D-概型，由代数簇X和X上环层的态射OX-> D给出.我们考虑X上左D-模的拟凝聚层的非交换变形，并给出如何计算它们的前表示壳。作为应用，我们计算了当X是任意椭圆曲线时左D_X-模O_X的非交换变形.
We show that Braun-Chuang-Lazarev's derived quotient prorepresents a naturally defined noncommutative derived deformation functor. Given a noncommutative partial resolution of a Gorenstein algebra, we show that the associated derived quotient controls part of its dg singularity category. We use a recent result of Hua and Keller to prove a recovery theorem, which can be thought of as providing a solution to a derived enhancement of a conjecture made by Donovan and Wemyss about the birational geometry of threefold flops.
Algebraic deformations of toric varieties II. Noncommutative instantons
我们继续我们的非交换代数和微分几何的一类特殊的变形环面品种的研究，侧重于有关方面的建设和枚举的非交换瞬子对这些品种。我们发展了扭量理论的一个非对易版本，它引入了一个非对易四球的新例子。我们开发了一个编织版本的ADHM建设，并表明，它参数化一定的模空间的框架扭转自由层的非对易射影平面。我们使用这些建设明确建立瞬子规范束与规范的非对易四球，满足适当的反自对偶方程的连接。我们构造了挠自由层的射影模空间，并证明了它们是光滑的。我们定义了这些模空间的等变配分函数，发现它们与C^2上超对称规范理论的通常瞬子配分函数一致。
We study the noncommutative operator graph ${\mathcal L}_{θ}$ depending on complex parameter $θ$ recently introduced by M.E. Shirokov to construct channels with positive quantum zero-error capacity having vanishing n-shot capacity. We define the noncommutative group $G$ and the algebra ${\mathcal A}_{θ}$ which is a quotient of ${\mathbb C}G$ with respect to the special algebraic relation depending on $θ$ such that the matrix representation $ϕ$ of ${\mathcal A}_{θ}$ results in the algebra ${\mathcal M}_{θ}$ generated by ${\mathcal L}_{θ}$. In the case of $θ=\pm 1$ $ϕ$ is degenerated to the fait…
Mirror symmetry for weighted projective planes and their noncommutative deformations
研究了加权射影空间相干层的导出范畴及其非对易变形，以及加权射影空间镜像Landau-Ginzburg模型的拉格朗日消失环的导出范畴.特别地，我们证明了加权射影平面$\CP^2(a,b,c)$上的相干层（B-膜）的导出范畴等价于配备有精确辛形式和超势$W=x+y+z$的仿射超曲面$X=\{x^ay^bz^c=1\}\subset (\C^*)^3$上的消失圈（A-膜）的导出范畴。因此，同调镜像对称猜想对加权射影平面成立。此外，我们还表明，这种镜像之间的对应导出范畴可以扩展到复曲面的非交换变形的$\CP^2(a,b,c)$的B-膜的关注，和他们的镜像对应物，非精确变形的辛结构的$X$的A-膜的关注。我们也得到类似的结果，其他的例子，如加权投影线或Hirzebruch表面。
Algebraic deformations of toric varieties I. General constructions
我们构造和研究非交换变形的环面品种相结合的技术，从环面几何，等谱变形，和非交换几何在编织幺半范畴范畴。我们的方法利用相同的风扇结构的品种，但变形的底层嵌入的代数环面。我们开发了一个层理论使用的技术从非交换代数几何。对投射簇的情形进行了详细的研究，并给出了几个显式的例子，包括Grassmann簇和flag簇的新的非对易变形。我们的构造为深入研究非对易复曲面簇上的瞬子奠定了基础，这将是本文后续部分的主题。
Noncommutative Manifolds the Instanton Algebra and Isospectral Deformations
我们给出了新的例子，非交换流形是不太标准的NC-环面或Moyal变形的$\Rb^n$。它们自然产生于非交换微分拓扑的基本考虑，并具有非平凡的全局特征。新的例子包括瞬子代数和NC-4-球$S^4_θ$。NC-球面上函数的非交换代数$\Ac=C^{\ify} (S^{4}_θ)$是幂等$e \in M_4 (\Ac), e^2 = e, e = e^*$的$\Ac$的循环同调中陈特征标的消失$ {\rm ch}_j (e) = 0, j < 2 $的解。由这个方程定义的泛非交换空间是由非常非平凡的三次关系定义的非交换格拉斯曼空间。这个空间${\rm Gr}$包含悬浮液的NC-3-球密切相关的量子群变形${\rm SU}_q (2)$的${\rm SU} (2)$，但不寻常的值（复数值的模量1）的参数$q$的$q$ -类似物，$q=\exp (2πi \t)$。然后，我们构造的非交换几何的$S_{\t}^4$所给出的谱三元组$(\Ac, \Hc, D)$和检查所有公理的非交换流形。非对易4-球面$S_{\t}^4$上的狄拉克算子$D$给出了定义“体积形式”$ < (e - {1/2}) [D,e]^4 > = \g_5$的基本四次方程的解，其中$<$是$4 \ts 4$矩阵在交换子上的投影。最后，我们表明，任何紧凑的黎曼自旋流形的等距群的秩$r \geq 2$承认非交换几何的等谱变形。
Instantons and vortices on noncommutative toric varieties
我们详细阐述了复曲面品种的量化技术相结合的复曲面几何，isospectral变形和非交换几何编织monoidal类别，并建设瞬子上的非交换代数几何和量化扭量理论相结合的方法。我们对Klein二次曲面的复曲面非对易变形上的实结构进行了分类，并利用它导出了一个新的非对易四球面，它是与非对易扭量对应相容的唯一变形。我们将等变瞬子配分函数的计算推广到具有伴随和基本物质场的非对易规范理论，发现在所有情况下与经典结果一致。从不变瞬子的模出发，构造了非对易涡旋的模空间，并导出了相应的等变配分函数，这些配分函数也与经典极限的配分函数一致。
A noncommutative deformation of a quadric surface is usually described by a three-dimensional cubic Artin-Schelter regular algebra. In this paper we show that for such an algebra its bounded derived category embeds into the bounded derived category of a commutative deformation of the Hilbert scheme of two points on the quadric. This is the second example in support of a conjecture by Orlov. Based on this example, we formulate an infinitesimal version of the conjecture, and provide some evidence in the case of smooth projective surfaces.
Deformation theory of objects in homotopy and derived categories III: abelian categories
这是一系列论文中的第三篇。在第一部分中，我们发展了一个变形理论的同伦和导出范畴的DG范畴的对象。在这里，我们将展示如何使用这个理论来研究变形的同伦和导出范畴的阿贝尔范畴的对象。然后，我们考虑从（非交换）代数几何的例子。特别地，我们研究了非交换Grassmanian空间，它是射影空间中射影子空间的结构层的真非交换模空间。
Deformations of vector bundles on coisotropic subvarieties via the Atiyah class
利用Atiyah类给出了代数Poisson簇$X$的共迷向子簇$Y$上的向量丛允许一阶和二阶非对易变形的一个判别准则.我们还表明，非交换变形的向量丛是由一个弯曲的dg李代数减少到经典的相对Hochschild复杂的泊松结构时，$X$是平凡的。
We construct a period mapping for deformations of a differential graded algebra, that generalizes Griffiths' period mapping. It is constructed as a morphism between differential graded Lie algebras which has a moduli-theoretic interpretation, where the domain of the morphism is the shifted Hochschild cocohain complex. We then use this period mapping to give some applications such as unobstructedness of deformations.
We continue our study on the corresponding noncommutative deformation of the relative $p$-adic Hodge structures of Kedlaya-Liu along our previous work. In this paper, we are going to initiate the study of the corresponding descent of pseudocoherent modules carrying large noncommutative coefficients. And also we are going to more systematically study the corresponding noncommutative geometric aspects of noncommutative deformation of Hodge structures, which will definitely also provide the insights not only for noncommutative Iwasawa theory but also for noncommutative analytic geometry. The nonc…
The symplectic structure on the moduli space of line bundles on a noncommutative Azumaya surface
本文证明了K3-曲面上Azumaya代数上秩为1的无扭模的模空间是等价于K3-曲面上点的Hilbert格式的不可约辛簇变形。
Noncommutative differential calculus, homotopy BV algebras and formality conjectures
定义了强同伦BV代数的概念，并将其应用于形变理论问题。Hochschild链和循环链的形式图解。我们证明了一些部分结果支持这些命题。
The period map from commutative to noncommutative deformations
本文研究了从交换到非交换变形的周期映射，识别了对偶HKR映射，并探讨了导出不变量的性质。
We show that any commutative rationally ruled surface with a choice of anticanonical curve admits a 1-parameter family of noncommutative deformations parametrized by the Jacobian of the anticanonical curve, and show that many standard facts from commutative geometry (blowups commute, Quot schemes are projective, etc.) carry over. The key new tool in studying these deformations is a relatively simple description of their derived categories and the relevant t-structures; this also allows us to establish nontrivial derived equivalences for deformations of elliptic surfaces. We also establish that…
Categories of holomorphic line bundles on higher dimensional noncommutative complex tori
我们构造了高维环面上全纯线丛范畴的显式非交换变形。我们的基本工具是A.施瓦茨我们得到了微分分次范畴作为复非交换环面上Heisenberg模的曲微分分次范畴的全子范畴。同时，我们给出了态射的显式合成公式，它实际上依赖于非交换性。
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