arXiv subjects

Christopher Dodd

Publications and source records attributed to Christopher Dodd.

10 recordsLinked to original sources

arXiv2024

Lagrangian Skeleta and Koszul Duality on Bionic Symplectic Varieties

We consider the category of modules over sheaves of Deformation-Quantization (DQ) algebras on bionic symplectic varieties. These spaces are equipped with both an elliptic $\mathbb{G}_m$-action and a Hamiltonian $\mathbb{G}_m$-action, with finitely many fixed points. On these spaces one can consider geometric category $\mathcal{O}$: the category of (holonomic) modules supported on the Lagrangian attracting set of the Hamiltonian action. We show that there exists a local generator in geometric category $\mathcal{O}$ whose dg endomorphism ring, cohomologically supported on the Lagrangian attracting set, is derived equivalent to the category of all DQ-modules. This is a version of Koszul duality generalizing the equivalence between D-modules on a smooth variety and dg-modules over the de Rham complex.

math.AG↗

arXiv2023

Witt Differential Operators

For a smooth scheme $X$ over a perfect field $k$ of positive characteristic, we define (for each $m\in\mathbb{Z}$) a sheaf of rings $\mathcal{\widehat{D}}_{W(X)}^{(m)}$ of differential operators (of level $m$) over the Witt vectors of $X$. If $\mathfrak{X}$ is a lift of $X$ to a smooth formal scheme over $W(k)$, then for $m\geq0$ modules over $\mathcal{\widehat{D}}_{W(X)}^{(m)}$ are closely related to modules over Berthelot's ring $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(m)}$ of differential operators of level $m$ on $\mathfrak{X}$. Our construction therefore gives an description of suitable categories of modules over these algebras, which depends only on the special fibre $X$. There is an embedding of the category of crystals on $X$ (over $W_{r}(k)$) into modules over $\mathcal{\widehat{D}}_{W(X)}^{(0)}/p^{r}$; and so we obtain an alternate description of this category as well. For a map $\varphi:X\to Y$ we develop the formalism of pullback and pushforward of $\mathcal{\widehat{D}}_{W(X)}^{(m)}$-modules and show all of the expected properties. When working mod $p^{r}$, this includes compatibility with the corresponding formalism for crystals, assuming $\varphi$ is smooth. In this case we also show that there is a ``relative de Rham Witt resolution'' (analogous to the usual relative de Rham resolution in $\mathcal{D}$-module theory) and therefore that the pushforward of (a quite general subcategory of) modules over $\mathcal{\widehat{D}}_{W(X)}^{(0)}/p^{r}$ can be computed via the reduction mod $p^{r}$ of Langer-Zink's relative de Rham Witt complex. Finally we explain a generalization of Bloch's theorem relating integrable de Rham-Witt connections to crystals.

math.AG↗

arXiv2022

Differential Operators, Gauges, and Mixed Hodge Modules

The purpose of this paper is to develop a new theory of gauges in mixed characteristic. Namely, let $k$ be a perfect field of characteristic $p>0$ and $W(k)$ the $p$-typical Witt vectors. Making use of Berthelot's arithmetic differential operators, we define for a smooth formal scheme $\mathfrak{X}$ over $W(k)$, a new sheaf of algebras $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$ which can be considered a higher dimensional analogue of the (commutative) Dieudonne ring. Modules over this sheaf of algebras can be considered the analogue (over $\mathfrak{X}$) of the gauges of Ekedahl and Fontain-Jannsen. We show that modules over $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$ admit all of the usual $\mathcal{D}$-module operations, and we prove a robust generalization of Mazur's theorem in this context. Finally, we show that an integral form of a mixed Hodge module of geometric origin admits, after a suitable $p$-adic completion, the structure of a module over $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$. This allows us to prove a version of Mazur's theorem for the intersection cohomology and the ordinary cohomology of an arbitrary quasiprojective variety defined over a number field.

math.AG↗

arXiv2016

Associated graded of Hodge modules and categorical sl_2 actions

One of the most mysterious aspects of Saito's theory of Hodge modules are the Hodge and weight filtrations that accompany the pushforward of a Hodge module under an open embedding. In this paper we consider the open embedding in a product of complementary Grassmannians given by pairs of transverse subspaces. The push-forward of the structure sheaf under this open embedding is an important Hodge module from the viewpoint of geometric representation theory and homological knot invariants. We compute the associated graded of this push-forward with respect to the induced Hodge filtration as well as the resulting weight filtration. The main tool is a categorical $\sl_2$ action on the category of $\D_h$-modules on Grassmannians. Along the way we also clarify the interaction of kernels for $\D_h$-modules with the associated graded functor. Both of these results may be of independent interest.

math.AG↗

arXiv2015

The p-cycle of Holonomic D-modules and Quantization of Exact Algebraic Lagrangians

Let $X=\mathbb{A}^{n}$ be complex affine space, and let $T^{*}X$ be its cotangent bundle. For any exact Lagrangian $L\subset T^{*}X$, we define a new invariant, A, living in $ \text{Div}_{\mathbb{Q}/\mathbb{Z}}(L)$. We call this invariant the monodromy divisor of $L$. We conjecture that the existence of a finite order character of $\pi_{1}(L$) whose monodromy is exactly A defines an obstruction to attaching a holonomic $\mathcal{D}_{X}$-module M associated to L - here, the association goes via positive characteristic and p-supports. In the case where $\mathbb{H}_{dR}^{1}(L)=0$, we prove this conjecture, and then go on the show that the set of such holonomic $\mathcal{D}_{X}$-modules forms a torsor over the group of finite order characters of $\pi_{1}$. This proves a version of a conjecture of Kontsevich. As a consequence, we deduce that the group of Morita autoequivalences of the n-th Weyl algebra is isomorphic to the group of symplectomorphisms of $T^{*}\mathbb{A}^{n}$. This generalizes an old theorem of Dixmier (in the case n=1) and settles a conjecture of Belov-Kanel and Kontsevich in general.

math.AG↗

arXiv2013

Categorical Cell Decomposition of Quantized Symplectic Algebraic Varieties

We prove a new symplectic analogue of Kashiwara's Equivalence from D-module theory. As a consequence, we establish a structure theory for module categories over deformation quantizations that mirrors, at a higher categorical level, the Bialynicki-Birula stratification of a variety with an action of the multiplicative group. The resulting categorical cell decomposition provides an algebro-geometric parallel to the structure of Fukaya categories of Weinstein manifolds. From it, we derive concrete consequences for invariants such as K-theory and Hochschild homology of module categories of interest in geometric representation theory.

math.AG↗

arXiv2011

Equivariant Coherent Sheaves, Soergel Bimodules, and Categorification of Affine Hecke Algebras

We give a description of certain categories of equivariant coherent sheaves on Grothendieck's resolution in terms of the categorical affine Hecke algebra of Soergel. As an application, we deduce a relationship of these coherent sheaf categories to the categories of perverse sheaves considered in the work of Bezrukavnikov-Yun, generalizing results of Arkhipov-Bezrukavnikov. In addition, we deduce that the weak braid group action on sheaves of Riche and Bezrukavnikov-Riche can be upgraded to a strict braid group action.

math.AG↗

arXiv2010

Injectivity of a certain cycle map for finite dimensional W-algebras

We study a certain cycle map defined on finite dimensional modules for the W-algebra with regular integral central character. Via comparison with the theory in postive characteristic, we show that this map injects into the top Borel-Moore homology group of a Springer fibre. This is the first result in a larger program to completely desribe the finite dimensional modules for the W algebras.

math.RT↗

arXiv2009

A Localization Theorem for Finite W-algebras

Following the work of Beilinson-Bernstein and Kashiwara-Rouquier, we give a geometric interpretation of certain categories of modules over the finite W-algebra. As an application we reprove the Skryabin equivalence.

math.RT↗

arXiv2007

Minimal Betti Numbers

We give conditions for determining the extremal behavior for the (graded) Betti numbers of squarefree monomial ideals. For the case of non-unique minima, we give several conditions which we use to produce infinite families, exponentially growing with dimension, of Hilbert functions which have no smallest (graded) Betti numbers among squarefree monomial ideals and all ideals. For the case of unique minima, we give two families of Hilbert functions, one with exponential and one with linear growth as dimension grows, that have unique minimal Betti numbers among squarefree monomial ideals.

math.AC↗