Quasi-perfect scheme-maps and boundedness of the twisted inverse image functor

扭曲霍奇菱形导致非傅里叶-向井函子的产生

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。

三次扭曲曲线的空间

The space of twisted cubics

本文研究了射影n空间中三次扭曲曲线空间的Cohen-Macaulay紧化，证明其为Hilbert多项式3t+1的CM曲线的精细模概形，并揭示了其在射影3空间中与Hilbert概形分支的同构关系。

扭曲 $\mathcal D$-模族与 Harish-Chandra 模的算术模型

Families of twisted $\mathcal D$-modules and arithmetic models of Harish-Chandra modules

本文发展了一般基概形上的扭曲 $\mathcal D$-模理论及闭 $K$-轨道形式理论，并利用它们几何构造了上同调诱导模的半整模型，进而证明了其投射性和无挠相对李代数上同调等算术良好性质。

到代数栈的扭曲映射的退化

Degenerations of twisted maps to algebraic stacks

本文定义了具有射影好模空间的商栈的扭曲映射，证明所得函子满足真的赋值判据的存在性部分。

Néron models of $Pic^0$ via $Pic^0$

We provide a new description of the Néron model of the Jacobian of a smooth curve $C_K$ with stable reduction $C_R$ on a discrete valuation ring $R$ with field of fractions $K$. Instead of the regular semistable model, our approach uses the regular twisted model, a twisted curve in the sense of Abramovich and Vistoli whose Picard functor contains a larger separated subgroup than the usual Picard functor of $C_R$. In this way, after extracting a suitable $l$th root from the uniformizer of $R$, the pullback of the Néron model of the Jacobian represents a Picard functor $Pic^{0,l}$ of line bundle…

Twisted complexes on a ringed space as a dg-enhancement of perfect complexes

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…

Frobenius扭转的de Rham上同调的自同构

Automorphisms of Frobenius twisted de Rham cohomology

本文证明Frobenius扭转的de Rham上同调函子的自同构模空间为$\mathbb{G}_m$，并利用该作用重新证明了Bhatt-Petrov-Vologodsky的分解定理。

Relations between derived Hochschild functors via twisting

Let $k$ be a regular ring, and let $A,B$ be essentially finite type $k$-algebras. For any functor $F:{D}(A)\times\dots\times{D}(A)\to{D}(B)$ between their derived categories, we define its twist $F^{!}:{D}(A)\times\dots\times{D}(A)\to{D}(B)$ with respect to dualizing complexes, generalizing Grothendieck's construction of $f^{!}$. We show that relations between functors are preserved between their twists, and deduce that various relations hold between derived Hochschild (co)-homology and the $f^{!}$ functor. We also deduce that the set of isomorphism classes of dualizing complexes over a ring (…

Twisted Polytope Sheaves and Coherent-Constructible Correspondence for Toric Varieties

Given a smooth projective toric variety $X_Σ$ of complex dimension $n$, Fang-Liu-Treumann-Zaslow \cite{FLTZ} showed that there is a quasi-embedding of the differential graded (dg) derived category of coherent sheaves $Coh(X_Σ)$ into the dg derived category of constructible sheaves on a torus $Sh(T^n, Λ_Σ)$. Recently, Kuwagaki \cite{Ku2} proved that the quasi-embedding is a quasi-equivalence, and generalized the result to toric stacks. Here we give a different proof in the smooth projective case, using non-characteristic deformation of sheaves to find twisted polytope sheaves that co-represent …

Singular localization and intertwining functors for reductive Lie algebras in prime characteristic

In math.RT/0205144 we observed that, on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of finite characteristic with a given (generalized) regular central character can be identified with coherent sheaves on the formal neighborhood of the corresponding (generalized) Springer fiber. In the present paper we treat singular central characters. The basic step is the Beilinson-Bernstein localization of modules with a fixed (generalized) central character as sheaves on the partial flag variety corresponding to the singularity of the ch…

The twisted inverse image pseudofunctor over commutative DG rings and perfect base change

Let $K$ be a Gorenstein noetherian ring of finite Krull dimension, and consider the category of cohomologically noetherian commutative differential graded rings $A$ over $K$, such that $H^0(A)$ is essentially of finite type over $K$, and $A$ has finite flat dimension over $K$. We extend Grothendieck's twisted inverse image pseudofunctor to this category by generalizing the theory of rigid dualizing complexes to this setup. We prove functoriality results with respect to cohomologically finite and cohomologically essentially smooth maps, and prove a perfect base change result for $f^{!}$ in this…

Derived Azumaya algebras and twisted $K$-theory

We construct a relative version of topological $K$-theory of dg categories over an arbitrary quasi-compact, quasi-separated $\mathbb{C}$-scheme $X$. This has as input a $\text{Perf}(X)$-linear stable $\infty$-category and output a sheaf of spectra on $X(\mathbb{C})$, the space of complex points of $X$. We then characterize the values of this functor on inputs of the form $Mod_{A}^ω$, for $A$ a derived Azumaya algebra over $X$. In such cases we show that this coincides with the $α$-twisted topological $K$-theory of $X(\mathbb{C})$ for some appropriately defined twist of $K$-theory. We use this …

Relation between two twisted inverse image pseudofunctors in duality theory

Grothendieck duality theory assigns to essentially-finite-type maps f of noetherian schemes a pseudofunctor f^\times right-adjoint to Rf_*, and a pseudofunctor f^! agreeing with f^\times when f is proper, but equal to the usual inverse image f^* when f is etale. We define and study a canonical map from the first pseudofunctor to the second. This map behaves well with respect to flat base change, and is taken to an isomorphism by "compactly supported" versions of standard derived functors. Concrete realizations are described, for instance for maps of affine schemes. Applications include proofs …

On mirabolic D-modules

Let an algebraic group G act on X, a connected algebraic manifold, with finitely many orbits. For any Harish-Chandra pair (D,G) where D is a sheaf of twisted differential operators on X, we form a left ideal D.g in D generated by the Lie algebra g, of G. Then, D/D.g is a holonomic D-module and its restriction to a unique Zariski open dense G-orbit in X is a G-equivariant local system. We prove a criterion saying that the D-module D/D.g is isomorphic, under certain (quite restrictive) conditions, to a direct image of that local system to X. We apply this criterion in the special case where the …