Yue Lin L. Tong, I-Hsun Tsai
We use the theory of twisted resolutions and twisted complexes to give a proof of Kontsevich's claim that Yoneda product corresponds to cup product in a canonical isomorphism from the Ext groups of the product space with coefficients in the diagonal sheaf to the cohomology with coefficients in exterior powers of the tangent sheaf.
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For a scheme X, we construct a sheaf C of complexes on X such that for every quasi-compact open subset U of X, C(U) is quasi-isomorphic to the Hochschild complex of the scheme U. Since C is moreover acyclic for taking sections on quasi-compact opens, we obtain a local to global spectral sequence for Hochschild cohomology if X is quasi-compact.
We develop a formalism involving Atiyah classes of sheaves on a smooth manifold, Hochschild chain and cochain complexes. As an application we prove a version of the Riemann--Roch theorem.
In this paper we study twisted complexes on a ringed space and prove that it gives a new dg-enhancement of the derived category of perfect complexes on that space. A twisted complex is a collection of locally defined sheaves together with the homotopic gluing data. In this paper we construct a dg-functor from twisted complexes to perfect complexes, which turns out to be a dg-enhancement. This new enhancement has the advantage of being completely geometric and it comes directly from the definition of perfect complex . In addition we will talk about some applications and further topics around tw…
Over a field of characteristic zero, we introduce two motivic operations on additive higher Chow cycles: analogues of the Connes boundary $B$ operator and the shuffle product on Hochschild complexes. The former allows us to apply the formalism of mixed complexes to additive Chow complexes building a bridge between additive higher Chow theory and additive $K$-theory. The latter induces a wedge product on additive Chow groups for which we show that the Connes operator is a graded derivation for the wedge product using a variation of a Totaro's cycle. Hence, the additive higher Chow groups with t…
The paper generalizes a result of Behrend-Fantechi and Baranovsky-Ginzburg to the 1-shifted cotangent bundle $T^*X[1]$ of a smooth scheme $X$ over the field of complex numbers. We show how one can obtain twisted cotangent bundles as derived intersections of Lagrangians in $T^*X[1]$, moreover and we show that the derived intersection of the quantized Lagrangians coincide with the canonical quantization of the twisted cotangent bundles.
We prove that the Hilbert scheme of degeneracy loci of pairs of global sections of Omega(2), the twisted cotangent bundle on P^(n-1), is unirational and dominated by the Grassmannian of lines in the projective space of skew-symmetric forms over a vector space of dimension n. We provide a constructive method to find the fibers of the dominant map. In classical terminology, this amounts to giving a method to realize all the pencils of linear line complexes having a prescribed set of centers. In particular, we show that the previous map is birational when n=4.
Automorphisms of a perfect complex naturally have the structure of an $\infty$-group: the 1-morphisms are quasi-isomorphisms, the 2-morphisms are homotopies, etc. This article starts by proving some basic properties of this $\infty$-group. We go on to study the deformation theory of this stack of $\infty$-groups and give a criterion for this stack to be formally smooth. The classifying stack of this $\infty$-group classifies forms of a complex. We discuss a version of Hilbert 90 for perfect complexes.
We study functors underlying derived Hochschild cohomology, also called Shukla cohomology, of a commutative algebra S essentially of finite type and of finite flat dimension over a commutative noetherian ring K. We construct a complex of S-modules D, and natural reduction isomorphisms Ext^*_{S\otimes^L_{K}S}(S|K;M\otimes^L_{K}N) ~ Ext^*_S(RHom_S(M,D),N) for all complexes of S-modules N and all complexes M of finite flat dimension over K whose homology H(M) is finitely generated over S; such isomorphisms determine D up to derived isomorphism. Using Grothendieck duality theory we establish analo…
We give counterexamples to the degeneration of the HKR spectral sequence in characteristic $p$, both in the untwisted and twisted settings. We also prove that the de Rham--$\mathrm{HP}$ and crystalline--$\mathrm{TP}$ spectral sequences need not degenerate.
Hochschild (Co)homology of D-modules on rigid analytic spaces I
本文提出D-帽模在刚性解析空间上的Hochschild (co)同调理论，通过显式计算和同构证明，建立了Hodge-de Rham谱序列及与de Rham复形的关联。
We show that much of the structure of the 2-sphere as a complex curve survives the q-deformation and has natural generalizations to the quantum 2-sphere - which, with additional structures, we identify with the quantum projective line. Notably among these is the identification of a quantum homogeneous coordinate ring with the coordinate ring of the quantum plane. In parallel with the fact that positive Hochschild cocycles on the algebra of smooth functions on a compact oriented 2-dimensional manifold encode the information for complex structures on the surface, we formulate a notion of twisted…
We investigate the hypercohomologies of truncated twisted holomorphic de Rham complexes on (not necessarily compact) complex manifolds. In particular, we generalize Leray-Hirsch, Künneth and Poincaré-Serre duality theorems on them. At last, a blowup formula is given, which affirmatively answers a question posed by Chen, Y. and Yang, S. in \cite{CY}.
We construct a comparison map from the topological K-theory of the dg-category of twisted perfect complexes on certain global quotient stacks to twisted equivariant K-theory, generalizing constructions of Halpern-Leistner-Pomerleano and Moulinos. We prove that this map is an equivalence if a version of the projective bundle theorem holds for twisted equivariant K-theory. Along the way, we give a new proof of a theorem of Moulinos that the comparison map is an equivalence in the non-equivariant case.
We define unbounded twisted complexes and bicomplexes generalising the notion of a (bounded) twisted complex over a DG category [BK90]. These need to be considered relative to another DG category $B$ admitting countable direct sums and shifts. The resulting DG category of unbounded twisted complexes has a fully faithful convolution functor into Mod-$B$ which filters through $B$ if the latter admits change of differential. As an application, we rewrite definitions of $A_{\infty}$-structures in terms of twisted complexes to make them work in an arbitrary monoidal DG category or a DG bicategory.
In this paper we consider the dg-category of twisted complexes over simplicial ringed spaces. It is clear that a simplicial map $f: (\mathcal{U},\mathcal{R})\to (\mathcal{V}, \mathcal{S})$ between simplicial ringed spaces induces a dg-functor $f^*: \text{Tw}(\mathcal{V}, \mathcal{S})\to \text{Tw}(\mathcal{U}, \mathcal{R})$ where $\text{Tw}(\mathcal{U}, \mathcal{R})$ denotes the dg-category of twisted complexes on $(\mathcal{U},\mathcal{R})$. In this paper we prove that for simplicial homotopic maps $f$ and $g$, there exists an $A_{\infty}$-natural transformation $Φ: f^*\Rightarrow g^*$ between…
We develop an approach to calculating the cup and cap products on Hochschild cohomology and homology of curved algebras associated with polynomials and their finite abelian symmetry groups. For polynomials with isolated critical points, the approach yields a complete description of the products. We also reformulate the result for the corresponding categories of equivariant matrix factorizations. In an Appendix written jointly with Alexey Basalaev, we apply the formulas to calculate the Hochschild cohomology of a simple but non-trivial class of so-called invertible LG orbifold models. The resul…
We define the Hochschild (co)homology of a ringed space relative to a locally free Lie algebroid. Our definitions mimic those of Swan and Caldararu for an algebraic variety. We show that our (co)homology groups can be computed using suitable standard complexes. Our formulae depend on certain natural structures on jetbundles over Lie algebroids. In an appendix we explain this by showing that such jetbundles are formal groupoids which serve as the formal exponentiation of the Lie algebroid.
Hochschild cohomology of the second kind: Koszul duality and Morita invariance
本文研究了第二种Hochschild上同调的定义、性质及其在Koszul对偶和Morita等价下的不变性，展示了其在不同范畴中的应用。
We prove new Skoda-type division, or ideal membership, theorems. We work in a geometric setting of line bundles over Kahler manifolds that are Stein away from an analytic subvariety. (This includes complex projective manifolds.) Our approach is to combine the twisted Bochner-Kodaira Identity, used in the Ohsawa-Takegoshi Theorem, with Skoda's basic estimate for the division problem. Techniques developed by McNeal and the author are then used to provide many examples of new division theorems. Among other applications, we give a modification of a recent result of Siu regarding effective finite g…
We show that when a torus $T$ acts on a smooth variety $X$, the twisted HKR isomorphism is equivariant. The main consequence is that the Bezrukavnikov- Lachowska isomorphism, relating the Hochschild cohomology of the principal block of the small quantum group to certain sheaf cohomology groups on the Springer resolution $\widetilde{\mathcal{N}}$ , can be upgraded to a ring isomorphism by a twist.
In this paper we study the dg-category of twisted perfect complexes on a ringed space with soft structure sheaf. We prove that this dg-category is quasi-equivalent to the dg-category of complexes of vector bundles on that space. This result could be considered as a dg-enhancement of the classic result on soft sheaves in SGA6.
We show that the equivariant chain complex associated to a minimal CW-structure X on the complement M(A) of a hyperplane arrangement A, is independent of X. When A is a sufficiently general linear section of an aspheric arrangement, we explain a new way for computing the twisted homology of M(A).
We generalize the decomposition theorem of Hochschild, Kostant and Rosenberg for Hochschild (co-)homology to arbitrary morphisms between complex spaces or schemes over a field of characteristic zero. To be precise, we show that for each such morphism, the Hochschild complex, as introduced in math.AG/0606593, decomposes naturally in the derived category into the direct sum of the derived symmetric powers of the shifted cotangent complex, a result due to Quillen in the affine case. Even in the affine case, our proof is new and provides further information. It shows that the decomposition is give…
We give a formula for the Chern character on the DG-category of global matrix factorizations on a smooth scheme $X$ with superpotential $w\in Γ(\mathcal{O}_X)$. Our formula takes values in a Cech model for Hochschild homology. Our methods may also be adapted to get an explicit formula for the Chern character for perfect complexes of sheaves on $X$ taking values in right derived global sections of the De-Rham algebra. Along the way we prove that the DG version of the Chern character coincides with the classical one for perfect complexes.
In this paper we study the homotopy limits of cosimplicial diagrams of dg-categories. We first give an explicit construction of the totalization of such a diagram and then show that the totalization agrees with the homotopy limit in the following two cases: (1) the complexes of sheaves of $\mathcal O$-modules on the Čech nerve of an open cover of a ringed space $(X, \mathcal O)$; (2) the complexes of sheaves on the simplicial nerve of a discrete group $G$ acting on a space. The explicit models we obtain in this way are twisted complexes as well as their $D$-module and $G$-equivariant versions.…
We prove that the Fano variety of lines of a generic cubic fourfold containing a plane is isomorphic to a moduli space of twisted stable complexes on a K3 surface. On the other hand, we show that the Fano varieties are always birational to moduli spaces of twisted stable coherent sheaves on a K3 surface. The moduli spaces of complexes and of sheaves are related by wall-crossing in the derived category of twisted sheaves on the corresponding K3 surface.
Hochschild cohomology and lifts of endomorphisms
研究代数自同射何时能提升为一阶平坦提升，通过关联一阶平坦提升与自同射的霍奇-希尔德上同调类，证明自同射提升当且仅当中心上的诱导自同射保持提升后的泊松结构。
By a theorem of Bernhard Keller the de Rham cohomology of a smooth variety is isomorphic to the periodic cyclic homology of the differential graded category of perfect complexes on the variety. Both the de Rham cohomology and the cyclic homology can be twisted by the exponential of a regular function on the variety. We explain that the isomorphism holds true in the twisted setting and draw some conclusions on derived invariance of the algebraic Gauss-Manin systems associated with regular functions.
Twisted Hodge diamonds give rise to non-Fourier-Mukai functors
本文利用扭曲霍奇菱形计算霍赫希尔德维数，构造并验证了奇数维光滑二次超曲面上大量非傅里叶-向井函子，并证明每个奇数维光滑二次超曲面至少存在一个这样的函子。
We show that, given a projective regular function f on a smooth quasi-projective variety over C, the corresponding cohomology groups of the algebraic de Rham complex with twisted differential d-df and of the complex of algebraic forms with differential df have the same dimension (a result announced by Barannikov and Kontsevitch). We generalize the result to de Rham complexes with coefficients in a mixed Hodge Module.
We introduce exponential complexes of sheaves on manifolds. They are resolutions of the (Tate twisted) constant sheaves of the rational numbers, generalising the short exact exponential sequence. There are canonical maps from the exponential complexes to the de Rham complex. Using this, we introduce new complexes calculating the rational Deligne cohomology. We call them exponential Deligne complexes. Their advantage is that, at least at the generic point of a complex variety, one can define Beilinson's regulator map to the rational Deligne cohomology on the level of complexes. Namely, we defin…
We study twisted cohomologies with paracompactifying families of supports. The Kunneth theorems, Leray-Hirsch theorems and self-intersection formulae are established. Based on these results, we eventually give explicit expressions of complex blow-up formulae for twisted Dolbeault cohomology on arbitrary complex manifolds and the ones of generalized blow-ups formulae for twisted de Rham cohomology on arbitrary oriented smooth manifolds. These expressions are induced by the morphisms of (simple or double) complexes of spaces of forms and currents rather than just the maps between cohomologies, w…
Rigid Dualizing Complexes over Commutative Rings and their Functorial Properties
本文通过刚性对偶复形处理noetherian环的Grothendieck Duality，证明了正则基环上本质有限型环具有唯一的刚性对偶复形，并利用其函子性质构造了扭曲诱导伪函子，为后续在Deligne-Mumford stacks上建立全局对偶性奠定基础。
Twisted logarithmic complexes of positively weighted homogeneous divisors
本文研究正加权齐次除子的扭曲对数复形的拟同构条件，证明LCT与Bernstein-Sato多项式根及一阶算子生成零化子等价，并给出超平面排列情形的简单证明。
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