arXiv subjects
Publications and source records attributed to Vadim Vologodsky.
V. Drinfeld and E. Lau introduced a ``decompletion'' of the ring of $p$-typical Witt vectors, following earlier work of T. Zink. The goal of this paper is to offer an exposition of this construction, which we call the sheared Witt vectors, on the category of rings $R$ whose reduction is a perfect $\mathbb{F}_p$-algebra.
We prove that after inverting the Planck constant $h$ the Bezrukavnikov-Kaledin quantization $(X, \mathcal{O}_h)$ of symplectic variety $X$ in characteristic $p$ is Morita equivalent to a certain central reduction of the algebra of differential operators on $X$.
Let $k$ be a perfect field of characteristic $p$ and $W(k)$ its ring of Witt vectors. We construct an equivalence of categories between the full subcategory of the derived category of quasi-coherent sheaves on the syntomification of $W(k)$ spanned by objects whose Hodge-Tate weights are between $[0,p-2]$ and an appropriate derived category of Fontaine-Laffaille modules.
The KZ equations are differential equations satisfied by the correlation functions (on the Riemann sphere) of two-dimensional conformal field theories associated with an affine Lie algebra at a fixed level. They form a system of complex partial differential equations with regular singular points satisfied by the $n$-point functions of affine primary fields. In [SV1] the KZ equations were identified with equations for flat sections of suitable Gauss-Manin connections, and solutions of the KZ equations were constructed in the form of multidimensional hypergeometric integrals. In [SV2] the KZ equations were considered modulo a prime number $p$, and, for rational levels, polynomial solutions of the KZ equations modulo $p$ were constructed by an elementary procedure as suitable $p$-approximations of the hypergeometric integrals. In this paper we study in detail the first nontrivial example of the KZ equations in characteristic $p$. In particular, if the level is irrational, we prove a version of the steepest descent result that relates the KZ local system to the space of functions on the critical locus of the master function. We use this result to prove the generic irreducibility of the KZ local system at any irrational level. If the level is rational, we describe all solutions of the KZ equations in characteristic $p$ by demonstrating that they all stem from the $p$-hypergeometric solutions. Finally, we prove a Lagrangian property of the subbundle of the KZ bundle spanned by the $p$-hypergeometric sections.
We prove that singular cohomology of the underlying space of Berkovich's analytification of a scheme $X$ locally of finite type over a trivially-valued field $k$ of characteristic $0$ is isomorphic to cdh-cohomology with integer coefficients which is also isomorphic to the weight zero motivic cohomology $H^*(X, \mathbb{Z})$. Using this isomorphism, we demonstrate the vanishing of $RHom_{Sh_{Nis}(cor_k)}(\underline{G},\mathbb{Z})$, where $\underline{G}$ denotes the Nisnevich sheaf with transfers associated with a commutative algebraic group $G$ over $k$. For abelian $k$-varieties $A$ and $B$, we prove that $RHom_{\mathcal{PS}h_{tr}}(\underline{A},\underline{B})$ is isomorphic to $Hom_{\mathbf{Ab_k}}(A,B)$.
We consider the KZ connection associated with a family of hyperelliptic curves of genus $g$ over the ring of $p$-adic integers $\mathbb{Z}_p$. Then the dual connection is the Gauss-Manin connection of that family. We observe that the Gauss-Manin connection has an ordinary $F$-crystal structure and its unit root subcrystal is of rank $g$. We prove that all local flat sections of the KZ connection annihilate the unite root subcrystal, and the space of all local flat sections of the KZ connection is a free $\mathbb{Z}_p$-module of rank $g$. We also consider the reduction modulo $p^s$ of the unit root subcrystal for any $s\geq 1$. We prove that its annihilator is generated by the so-called $p^s$-hypergeometric flat sections of the KZ connection. In particular, that means that the reduction modulo $p^s$ of an arbitrary local flat section of the KZ connection over $\mathbb{Z}_p$ is a linear combination of the $p^s$-hypergeometric flat sections.
We prove that the $p$-adically completed periodic topological cyclic homology of a DG category over a perfect field $k$ of characteristic $p>2$ is isomorphic to the ($p$-adically completed) periodic cyclic homology of a lifting of the DG category over the Witt vectors $W(k)$.
Let $X$ be a smooth symplectic variety over a field $k$ of characteristic $p>2$ equipped with a restricted structure, which is a class $[η] \in H^0(X, Ω^1_X/d\mathcal O_X)$ whose de Rham differential equals the symplectic form. In this paper we construct a functorial in $(X, [η])$ formal quantization of the category $\mathrm{QCoh}(X)$ of quasi-coherent sheaves on $X$. We also construct its natural extension to a quasi-coherent sheaf of categories $\mathrm{QCoh}_h$ on the product $X^{(1)} \times {\mathbb S}$ of the Frobenius twist of $X$ and the projective line ${\mathbb S}=\mathbb P^1$, viewed as the one-point compactification of $\mathrm{Spec}\ \! k[h]$. Its global sections over $X^{(1)} \times \{0\}$ is the category of quasi-coherent sheaves on $X$. If $X$ is affine, $\mathrm{QCoh}_h$, restricted to $X^{(1)}\times \mathrm{Spf} \ \! k[[h]]$, is equivalent to the category of modules over the distinguished "Frobenius-constant" quantization of $(X,[η])$ defined by Bezrukavnikov and Kaledin.
We show the boundedness of finite subgroups in any anisotropic reductive algebraic group over a perfect field that contains all roots of 1. Also, we provide explicit bounds for orders of finite subgroups of automorphism groups of Severi-Brauer varieties and quadrics over such fields.
For a geometrically rational surface X over an arbitrary field of characteristic different from 2 and 3 that contains all roots of 1, we show that either X is birational to a product of a projective line and a conic, or the group of birational automorphisms of X has bounded finite subgroups. As a key step in the proof, we show boundedness of finite subgroups in any anisotropic reductive algebraic group over a perfect field that contains all roots of 1. Also, we provide applications to Jordan property for groups of birational automorphisms.
It is expected that the periodic cyclic homology of a DG algebra over the field of complex numbers (and, more generally, the periodic cyclic homology of a DG category) carries a lot of additional structure similar to the mixed Hodge structure on the de Rham cohomology of algebraic varieties. Whereas a construction of such a structure seems to be out of reach at the moment its counterpart in finite characteristic is much better understood thanks to recent groundbreaking works of Kaledin. In particular, it is proven by Kaledin that under some assumptions on a DG algebra $A$ over a perfect field $k$ of characteristic $p$, a lifting of $A$ over the ring of second Witt vectors $W_2(k)$ specifies the structure of a Fontaine-Laffaille module on the periodic cyclic homology of $A$. The purpose of this paper is to develop a relative version of Kaledin's theory for DG algebras over a base $k$-algebra $R$ incorporating in the picture the Gauss-Manin connection on the relative periodic cyclic homology constructed by Getzler. Our main result asserts that, under some assumptions on $A$, the Gauss-Manin connection on its periodic cyclic homology can be recovered from the Hochschild homology of $A$ equipped with the action of the Kodaira-Spencer operator as the inverse Cartier transform (in the sense of Ogus-Vologodsky). As an application, we prove, using the reduction modulo $p$ technique, that, for a smooth and proper DG algebra over a complex punctured disk, the monodromy of the Gauss-Manin connection on its periodic cyclic homology is quasi-unipotent.
We say that a formal power series $\sum a_n z^n$ with rational coefficients is a 2-function if the numerator of the fraction $a_{n/p}-p^2 a_n$ is divisible by $p^2$ for every prime number $p$. One can prove that 2-functions with rational coefficients appear as building block of BPS generating functions in topological string theory. Using the Frobenius map we define 2-functions with coefficients in algebraic number fields. We establish two results pertaining to these functions. First, we show that the class of 2-functions is closed under the so-called framing operation (related to compositional inverse of power series). Second, we show that 2-functions arise naturally in geometry as $q$-expansion of the truncated normal function associated with an algebraic cycle extending a degenerating family of Calabi-Yau 3-folds.
Recently, Rizzardo and Van den Bergh constructed an example of a triangulated functor between the derived categories of coherent sheaves on smooth projective varieties over a field $k$ of characteristic $0$ which is not of the Fourier-Mukai type. The purpose of this note is to show that if $char \, k =p$ then there are very simple examples of such functors. Namely, for a smooth projective $Y$ over $\mathbb Z_p$ with the special fiber $i: X\hookrightarrow Y$, we consider the functor $L i^* \circ i_*: D^b(X) \to D^b(X)$ from the derived categories of coherent sheaves on $X$ to itself. We show that if $Y$ is a flag variety which is not isomorphic to $\mathbb P^1$ then $L i^* \circ i_*$ is not of the Fourier-Mukai type. Note that by a theorem of Toen (\cite{t}, Theorem 8.15) the latter assertion is equivalent to saying that $L i^* \circ i_*$ does not admit a lifting to a $\mathbb F_p$-linear DG quasi-functor $D^b_{dg}(X) \to D^b_{dg}(X)$, where $D^b_{dg}(X)$ is a (unique) DG enhancement of $D^b(X)$. However, essentially by definition, $L i^* \circ i_*$ lifts to a $\mathbb Z_p$-linear DG quasi-functor.
Motivated by their role for integrality and integrability in topological string theory, we introduce the general mathematical notion of "s-functions" as integral linear combinations of poly-logarithms. 2-functions arise as disk amplitudes in Calabi-Yau D-brane backgrounds and form the simplest and most important special class. We describe s-functions in terms of the action of the Frobenius endomorphism on formal power series and use this description to characterize 2-functions in terms of algebraic K-theory of the completed power series ring. This characterization leads to a general proof of integrality of the framing transformation, via a certain orthogonality relation in K-theory. We comment on a variety of possible applications. We here consider only power series with rational coefficients; the general situation when the coefficients belong to an arbitrary algebraic number field is treated in a companion paper.
We prove a formula expressing the motivic integral (\cite{ls}) of a K3 surface over $\bC((t))$ with semi-stable reduction in terms of the associated limit Hodge structure. Secondly, for every smooth variety over a non-archimedean field we define an analogue of the monodromy pairing, constructed by Grothendieck in the case of Abelian varieties, and prove that our monodromy pairing is a birational invariant of the variety. Finally, we propose a conjectural formula for the motivic integral of maximally degenerated K3 surfaces over an arbitrary non-archimedean field and prove this conjecture for Kummer K3 surfaces.
We construct a functor from the triangulated category of Voevodsky motives to a certain derived category of mixed Hodge structures enriched with integral weight filtration. We use this construction to prove a strong integral version of the Deligne conjecture on 1-motives which was known previously only up to isogeny.
We prove that the embedding of the derived category of 1-motives into the triangulated category of effective Voevodsky motives, as well as its left adjoint functor $LAlb$, commute with the Hodge realization.
We compute the center of the ring of PD differential operators on a smooth variety over $\bZ/p^n\bZ$ confirming a conjecture of Kaledin. More generally, given an associative algebra $A_0$ over $\bF_p$ and its flat deformation $A_n$ over $\bZ/p^{n+1}\bZ$ we prove that under a certain non-degeneracy condition the center of $A_n$ is isomorphic to the ring of length $n+1$ Witt vectors over the center of $A_0$.