arXiv subjects
Publications and source records attributed to Dario Beraldo.
We categorify the localized intersection product on arithmetic schemes defined by Kato--Saito in \cite{katosaito04}. As an application, we prove a generalization of Bloch conductor conjecture.
Let $p:X \rightarrow S$ be a flat, proper and regular scheme over a strictly henselian discrete valuation ring. We prove that the singularity category of the special fiber with its natural two-periodic structure allows to recover the $\ell$-adic vanishing cohomology of $p$. Along the way, we compute homotopy-invariant non-connective algebraic K-theory with compact support of certain embeddings $X_t \hookrightarrow X_T$ in terms of the motivic realization of the dg category of relatively perfect complexes.
We show that the category of ind-coherent sheaves on a quasi-smooth scheme is naturally tensored over the category of sheared D-modules on its shifted cotangent bundle, commuting with its natural action of categorified Hoschschild cochains. We prove that it defines a Morita equivalence as such. We then extend these results to quasi-smooth Artin stacks. As a consequence of our formalism, we are able to articulate a precise sense in which the space of unramified automorphic functions over a function field localizes over the stack of arithmetic Arthur parameters.
We apply methods of derived and non-commutative algebraic geometry to understand intersection theoretic phenomena on arithmetic schemes. Specifically, we categorify Bloch's intersection number (in the formulation provided by Kato--Saito). Combining this with Toën--Vezzosi's non-commutative Chern character, we obtain a generalization of Bloch conductor conjecture in several new cases, including the unipotent Deligne--Milnor formula.
We apply methods of derived and non-commutative algebraic geometry to understand ramification phenomena on arithmetic schemes. As an application, we prove the Deligne-Milnor conjecture and, in the pure characteristic case, a generalization of Bloch conductor conjecture.
Let $G$ be a reductive group and $X$ a smooth projective curve. We prove that, for $G$ classical and $σ$ an arbitrary $G$-local system on $X$, the space $\overline{\operatorname{Op}}^{gen}_{G,σ}$ of generic extended oper structures on $σ$ is homologically contractible. This contractibility result is crucial for the proof of the geometric Langlands conjecture.
We prove a gluing theorem on the automorphic side of the geometric Langlands correspondence: roughly speaking, we show that the difference between $\mathrm{DMod}(\mathrm{Bun}_G)$ and its full subcategory $\mathrm{DMod}(\mathrm{Bun}_G)^\mathrm{temp}$ of tempered objects is compensated by the categories $\mathrm{DMod}(\mathrm{Bun}_M)^\mathrm{temp}$ for all standard Levi subgroups $M \subsetneq G$. This theorem is designed to match exactly with the spectral gluing theorem, an analogous result occurring on the other side of the geometric Langlands conjecture. Along the way, we state and prove several facts that might be of independent interest. For instance, for any parabolic $P \subseteq G$, we show that the functors $\mathrm{CT}_{P,*}:\mathrm{DMod}(\mathrm{Bun}_G) \to \mathrm{DMod}(\mathrm{Bun}_M)$ and $\mathrm{Eis}_{P,*}: \mathrm{DMod}(\mathrm{Bun}_M) \to \mathrm{DMod}(\mathrm{Bun}_G)$ preserve tempered objects, whereas the standard Eisenstein functor $\mathrm{Eis}_{P,!}$ preserves anti-tempered objects.
In this paper we prove two results pertaining to the (unramified and global) geometric Langlands program. The first result is an analogue of the Ramanujan conjecture: any cuspidal D-module on Bun_G is tempered. We actually prove a more general statement: any D-module that is *-extended from a quasi-compact open substack of Bun_G is tempered. Then the assertion about cuspidal objects is an immediate consequence of a theorem of Drinfeld-Gaitsgory. Building up on this, we prove our second main result, the automorphic gluing theorem for the group SL_2: it states that any D-module on Bun_{SL_2} is determined by its tempered part and its constant term. This theorem (vaguely speaking, an analogue of Langlands' classification for the group SL_2(R)) corresponds under geometric Langlands to the spectral gluing theorem of Arinkin-Gaitsgory and the author.
In the setting of the geometric Langlands conjecture, we argue that the phenomenon of divergence at infinity on Bun_G (that is, the difference between $!$-extensions and $*$-extensions) is controlled, Langlands-dually, by the locus of semisimple $\check{G}$-local systems. To see this, we first rephrase the question in terms of Deligne-Lusztig duality and then study the Deligne-Lusztig functor DL_G^\spec acting on the spectral Langlands DG category IndCoh_N(LS_G). We prove that DL_G^\spec is the projection IndCoh_N(LS_G) \to QCoh(LS_G), followed by the action of a coherent D-module St_G which we call the {Steinberg} D-module. We argue that St_G might be regarded as the dualizing sheaf of the locus of semisimple $G$-local systems. We also show that DL_G^\spec, while far from being conservative, is fully faithful on the subcategory of compact objects.
We characterize the tempered part of the automorphic Langlands category D-mod(Bun_G) using the geometry of the big cell in the affine Grassmannian. We deduce that, for $G$ non-abelian, tempered D-modules have no de Rham cohomology with compact supports. The latter fact boils down to a concrete statement, which we prove using the Ran space and some explicit t-structure estimates: for $G$ non-abelian and $Σ$ a smooth affine curve, the Borel-Moore homology of the indscheme $Maps(Σ,G)$ vanishes.
Let \Y be a derived algebraic stack satisfying some mild conditions. The purpose of this paper is three-fold. First, we introduce and study H(\Y), a monoidal DG category that might be regarded as a categorification of the ring of differential operators on \Y. When \Y = \LS_G is the derived stack of G-local systems on a smooth projective curve, we expect H(\LS_G) to act on both sides of the geometric Langlands correspondence, compatibly with the conjectural Langlands functor. Second, we construct a novel theory of D-modules on derived algebraic stacks. Contrarily to usual D-modules, this new theory, to be denoted by D^{der}, is sensitive to the derived structure. Third, we identify the Drinfeld center of H(\Y) with D^{der}(L\Y), the DG category of D^{der}-modules on the loop stack of \Y.
We strengthen the gluing theorem occurring on the spectral side of the geometric Langlands conjecture. While the latter embeds $IndCoh_N(LS_G)$ into a category glued out of 'Fourier coefficients' parametrized by standard parabolics, our refinement explicitly identifies the essential image of such embedding.
The notion of Hochschild cochains induces an assignment from $Aff$, affine DG schemes, to monoidal DG categories. We show that this assignment extends, under some appropriate finiteness conditions, to a functor $\mathbb H: Aff \to AlgBimod(DGCat)$, where the latter denotes the category of monoidal DG categories and bimodules. Now, any functor $\mathbb A: Aff \to AlgBimod(DGCat)$ gives rise, by taking modules, to a theory of sheaves of categories $ShvCat^{\mathbb A}$. In this paper, we study $ShvCat^{\mathbb H}$. Vaguely speaking, this theory categorifies the theory of D-modules, in the same way as Gaitsgory's original $ShvCat$ categorifies the theory of quasi-coherent sheaves. We develop the functoriality of $ShvCat^{\mathbb H}$, its descent properties and, most importantly, the notion of $\mathbb H$-affineness. We then prove the $\mathbb H$-affineness of algebraic stacks: for $Y$ a stack satisfying some mild conditions, the $\infty$-category $ShvCat^{\mathbb H}(Y)$ is equivalent to the $\infty$-category of modules for $\mathbb H(Y)$, the monoidal DG category defined in arXiv:1709.07867. As an application, consider a quasi-smooth stack $Y$ and a DG category $C$ with an action of $\mathbb H(Y)$. Then $C$ admits a theory of singular support in $Sing(Y)$, where $Sing(Y)$ is the space of singularities of $Y$.
Let $G$ be a connected reductive group, with connected center, and $X$ a smooth complete curve, both defined over an algebraically closed field of characteristic zero. Let $\operatorname{Bun}_G$ denote the stack of $G$-bundles on $X$. In analogy with the classical theory of Whittaker coefficients for automorphic functions, we construct a "Fourier transform" functor, called $\mathsf{coeff}_{G,\mathsf{ext}}$, from the DG category of $\mathfrak{D}$-modules on $\operatorname{Bun}_G$ to a certain DG category $\mathcal{W}h(G,\mathsf{ext})$, called the \emph{extended Whittaker category}. Combined with work in progress by other mathematicians and the author, this construction allows to formulate the compatibility of the Langlands duality functor $\mathbb{L}_G: \operatorname{IndCoh}_{\mathcal N}(\operatorname{LocSys}_{\check{G}}) \to \mathfrak{D}(\operatorname{Bun}_G)$ with the Whittaker model. For $G=GL_n$ and $G=PGL_n$, we prove that $\mathsf{coeff}_{G,\mathsf{ext}}$ is fully faithful. This result guarantees that, for those groups, $\mathbb{L}_G$ is unique (if it exists) and necessarily fully faithful.
We consider the spherical DG category $Sph_G$ attached to an affine algebraic group $G$. By definition, $Sph_G := IndCoh(LS_G(S^2))$ consists of ind-coherent sheaves of the stack of $G$-local systems on the $2$-sphere $S^2$. The $3$-dimensional version of the pair of pants endows $Sph_G$ with an $E_3$-monoidal structure. More generally, for an algebraic stack $Y$ (satisfying some mild conditions) and $n \geq -1$, we can look at the $E_{n+1}$-monoidal DG category $Sph(Y,n) := IndCoh_0((Y^{S^n})^\wedge_Y)$, where $IndCoh_0$ is the sheaf theory introduced in [AG2] and [centerH]. % The case of $Sph_G$ is recovered by setting $Y =BG$ and $n=2$. The cobordism hypothesis associates to $Sph(Y,n)$ an $(n+1)$-dimensional TQFT, whose value of a manifold $M^d$ of dimension $d \leq n+1$ (possibly with boundary) is given by the {topological chiral homology} $\int_{M^d} Sph(Y,n)$. % In this paper, we compute such homology (in virtually all cases): we have the Stokes style formula $$ \int_{M^d} Sph(Y,n) \simeq IndCoh_0 ( (Y^{\partial(M^d \times D^{n+1-d})})^\wedge_{Y^M} ) , $$ where the formal completion is constructed using the obvious projection $\partial(M^d \times D^{n+1-d}) \to M^d$. The most interesting instance of this formula is for $Sph_G \simeq Sph(BG,2)$, the original spherical category, and $X$ a Riemann surface. In this case, we obtain a monoidal equivalence $\int_X Sph_G \simeq H(LS_G^{Betti}(X))$, where $LS_G^{Betti}(X)$ is the stack of $G$-local systems on the topological space underlying $X$ and $H$ is the sheaf theory introduced in [centerH].
We develop some aspects of the theory of $D$-modules on ind-schemes of pro-finite type. These notions are used to define $D$-modules on (algebraic) loop groups and, consequently, actions of loop groups on DG categories. Let $N$ be the maximal unipotent subgroup of a reductive group $G$. For a non-degenerate character $χ: N(\!(t)\!) \to \mathbb{G}_a$ and a category $\mathcal{C}$ acted upon by $N(\!(t)\!) $, we define the category $\mathcal{C}^{N(\!(t)\!), χ}$ of $(N(\!(t)\!), χ)$-invariant objects, along with the coinvariant category $\mathcal{C}_{N(\!(t)\!), χ}$. These are the Whittaker categories of $\mathcal{C}$, which are in general not equivalent. However, there is always a family of functors $Θ_k: \mathcal{C}_{N(\!(t)\!), χ} \to \mathcal{C}^{N(\!(t)\!), χ}$, parametrized by $k \in \mathbb{Z}$. We conjecture that each $Θ_k$ is an equivalence, provided that the $N(\!(t)\!)$-action on $\mathcal{C}$ extends to a $G(\!(t)\!)$-action. Using the Fourier-Deligne transform (adapted to Tate vector spaces), we prove this conjecture for $G= GL_n$ and show that the Whittaker categories can be obtained by taking invariants of $\mathcal{C}$ with respect to a very explicit pro-unipotent group subscheme (not ind-scheme) of $G(\!(t)\!)$.