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Ajneet Dhillon

Publications and source records attributed to Ajneet Dhillon.

At least 19 recordsLinked to original sources
arXiv2026

Adelic points and unmramified Brauer approximation for classifying stacks

Let $k$ be a number field and let $G$ be a connected linear algebraic group over $k$. We compare three approaches to strong approximation for the classifying stack $BG$ with respect to its full Brauer group: the homogeneous-space method of \cite{DhillonClassifying}, Kottwitz's local--global sequence, and, for reductive groups, Borovoi's localization theorem. Under the natural identification \[ Br(BG)/Br(k)\simeq Pic(G), \] we identify the Kottwitz and Borovoi obstruction maps with Brauer evaluation. The three approaches therefore give the same exact description of the localization image as the projected full Brauer--Manin set. We then study strong approximation with respect to the ordinary unramified Brauer group. We prove \[ Br^{un}(BG)/ Br(k)\simeq Sha^1_{\mathrm{cyc}}(k,\widehat G), \qquad \widehat G=X^*(G_{\bar{k}}), \] and give an exact local criterion for unramified Brauer approximation off a finite set of places. Examples show that this approximation can fail even when a finite place is omitted, and exhibit a torus for which the unramified Brauer group cuts out the global image as a proper subset of the adelic space.

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arXiv2026

Finite extensions of abelian schemes and duality for commutative group stacks

Let $R$ be a discrete valuation ring. We prove that every proper flat finitely presented commutative $R$-group scheme $P$ fits into an exact sequence \[ 0\longrightarrow E\longrightarrow P\longrightarrow B\longrightarrow0, \] where $E$ is finite flat and $B$ is an abelian scheme. For a quasiabelian model---that is, a proper flat finitely presented $R$-group scheme with abelian generic fibre---we prove a finer structure theorem: its normalization is an abelian scheme, and the model is obtained from it by a pushout involving finite flat group schemes. An analogous abelian quotient exists for proper flat commutative group stacks with finite flat inertia. These structure theorems give explicit quotient presentations for duals. Over an arbitrary base, we show that the dual of a proper flat finitely presented commutative group algebraic space is algebraic, proper, flat, and finitely presented, and biduality holds. When $2$ is invertible, the corresponding result for group stacks proves Brochard's duality conjecture. As an application over a discrete valuation ring, Polishchuk's kernel-algebra Fourier--Mukai argument gives equivalences between the unbounded derived categories of quasi-coherent sheaves on every proper flat finitely presented commutative group scheme and its dual, and between their bounded derived categories of coherent sheaves, with no tameness or residue-characteristic hypothesis. For a quasiabelian model, the dual category is a finite-flat equivariant category on its abelian normalization.

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arXiv2026

Approximation theorems for classifying stacks over number fields

Approximation theorems for algebraic stacks over a number field $k$ are studied in this article. For G a connected linear algebraic group over a number field we prove strong approximation with Brauer-Manin obstruction for the classifying stack $BG$. This result answers a very concrete question, given $G$-torsors $P_v$ over $k_v$, where $v$ ranges over a finite number of places, when can you approximate the $P_v$ by a $G$-torsor $P$ defined over $k$.

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arXiv2026

The étale Brauer-Manin obstruction for classifying stacks

We study the strong approximation for classifying stacks $BG$, where $G$ is a linear algebraic group over a number field $k$. More specifically, we prove that the étale Brauer-Manin obstruction is the only obstruction to strong approximation for $BG$. To prove the result, we formulate the theory of torsors and Galois twists for algebraic stacks.

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arXiv2024

On the essential dimension of symplectic vector bundles over curves

Let $X$ be a smooth geometrically connected projective curve of genus at least 2 over a field of characteristic zero. We compute the essential dimension of the moduli stack of symplectic bundles over $X$. Unlike the case of vector bundles, we are able to precisely compute the essential dimension as the generic gerbe of the moduli stack has period 2 over it's moduli space.

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arXiv2024

Semiorthogonal decompositions for generalized Severi-Brauer schemes

The purpose of this paper is to use conservative descent to study semi-orthogonal decompositions for some homogeneous varieties over general bases. We produce a semi-orthogonal decomposition for the bounded derived category of coherent sheaves on a generalized Severi-Brauer scheme. This extends known results for Sever-Brauer varieties and Grassmanianns. We use our results to construct semi-orthogonal decompositions for flag varieties over arbitrary bases. This generalises a result of Kapranov.

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arXiv2023

Mukai Duality for abelian stacks

An abelian stack is a stacky generalization of an abelian variety that was introduced by Brochard. Just as an abelian variety has a dual, an abelian stack $\mathcal{A}$ has a dual $\mathfrak{D}(\mathcal{A})$ which generalizes the classical dual. In general, $\mathfrak{D}(\mathcal{A})$ is no longer an abelian stack but a commutative group scheme which is an extension of a finite, flat, and finitely presented commutative group scheme by an abelian scheme. We show that Fourier-Mukai duality holds for tame abelian stacks and their duals. Our approach is as follows. Let $QC_\infty(\mathcal{A})$ be the stable infinity category of quasi-coherent sheaves on $\mathcal{A}$. We define a Poincare bundle on $\mathcal{A}\times \mathfrak{D}(\mathcal{A})$ and use this to show that $QC_\infty(\mathcal{A})$ and $QC_\infty(\mathfrak{D}(\mathcal{A}))$ are dual as objects in the infinity category of stable infinity categories. By a result of Ben-Zvi,Francis and Nadler we have that $QC_\infty(\mathcal{A})$ is self dual, giving that $QC_\infty(\mathcal{A})\cong QC_\infty(\mathfrak{D}(\mathcal{A}))$ which gives the statement for the derived categories. In addition we give new examples of tame abelian stacks.

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arXiv2022

A homotopical Skolem--Noether theorem

The classical Skolem--Noether Theorem [Giraud, 71] shows us (1) how we can assign to an Azumaya algebra $A$ on a scheme $X$ a cohomological Brauer class in $H^2(X,\mathbf G_m)$ and (2) how Azumaya algebras correspond to twisted vector bundles. The Derived Skolem--Noether Theorem [Lieblich, 09] generalizes this result to weak algebras in the derived 1-category locally quasi-isomorphic to derived endomorphism algebras of perfect complexes. We show that in general for a co-family $\mathscr C^\otimes$ of presentable monoidal quasi-categories with descent over a quasi-category with a Grothendieck topology, there is a fibre sequence giving in particular the above correspondences. For a totally supported perfect complex $E$ over a quasi-compact and quasi-separated scheme $X$, the long exact sequence on homotopy group sheaves splits giving equalities $π_i(\mathop{\mathrm{Aut}}_{\mathop{\mathrm{Perf}}} E,\mathrm{id}_E)=π_i(\mathop{\mathrm{Aut}}_{\mathop{\mathrm{Alg}}\mathop{\mathrm{Perf}}}\mathop{\mathbf R\mathrm{End} E},\mathrm{id}_{\mathop{\mathbf R\mathrm{End} E}})$ for $i\ge1$. Further applications include complexes in Derived Algebraic Geometry, module spectra in Spectral Algebraic Geometry and ind-coherent sheaves and crystals in Derived Algeraic Geometry in characteristic 0.

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arXiv2021

Twisted forms of perfect complexes and Hilbert 90

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.

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arXiv2021

On the essential (p)-dimension of parabolic bundles on a curve

We study the essential dimension and essential p-dimension of the moduli stack of vector bundles over a smooth orbifold curve containing a rational point. We improve the known bounds on this essential dimension and obtain an equality modulo the famous conjecture of Colliot-Thelene, Karpenko and Merkurjev. In the case of essential p-dimension we obtain an equality.

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arXiv2018

Pushforwards of Tilting Sheaves

We investigate the behaviour of tilting sheaves under pushforward by a finite Galois morphism. We determine conditions under which such a pushforward of a tilting sheaf is a tilting sheaf. We then produce some examples of Severi Brauer flag varieties and arithmetic toric varieties in which our method produces a tilting sheaf, adding to the list of positive results in the literature. We also produce some counterexamples to show that such a pushfoward need not be a tilting sheaf.

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arXiv2018

On the $\infty$-stack of complexes over a scheme

We study fppf descent for enhanced derived categories. We revisit the work of [HS] and [TV08] in a lax context. More precisely, we construct a Cartesian and coCartesian fibration ${}^{\mathrm{op}}\mathscr D^+_S\rightarrow N(\mathop{\mathrm{Sch}}_S)$ whose fibre over an $S$-scheme $T$ is the opposite $\mathscr D^+(T)^{\mathrm{op}}$ of the quasicategory of bounded below complexes of $\mathscr O_T$-modules. We show that this fibration satisfies fppf-descent for schemes.

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arXiv2015

The Period-Index Problem of the Canonical Gerbe of Symplectic and Orthogonal Bundles

We consider regularly stable parabolic symplectic and orthogonal bundles over an irreducible smooth projective curve over an algebraically closed field of characteristic zero. The morphism from the moduli stack of such bundles to its coarse moduli space is a $μ_2$-gerbe. We study the period and index of this gerbe, and solve the corresponding period-index problem.

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arXiv2015

A symplectic analog of the Quot scheme

We construct a symplectic analog of the Quot scheme that parametrizes the torsion quotients of a trivial vector bundle over a compact Riemann surface. Some of its properties are investigated.

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arXiv2015

Brauer Groups of Quot Schemes

Let $X$ be an irreducible smooth complex projective curve. Let ${\mathcal Q}(r,d)$ be the Quot scheme parametrizing all coherent subsheaves of ${\mathcal O}^{\oplus r}_X$ of rank $r$ and degree $-d$. There are natural morphisms ${\mathcal Q}(r,d) \longrightarrow \text{Sym}^d(X)$ and $\text{Sym}^d(X) \longrightarrow \text{Pic}^d(X)$. We prove that both these morphisms induce isomorphism of Brauer groups if $d \geq 2$. Consequently, the Brauer group of ${\mathcal Q}(r,d)$ is identified with the Brauer group of $\text{Pic}^d(X)$ if $d \geq 2$.

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arXiv2014

On the essential dimension of coherent sheaves

We characterize all fields of definition for a given coherent sheaf over a projective scheme in terms of projective modules over a finite-dimensional endomorphism algebra. This yields general results on the essential dimension of such sheaves. Applying them to vector bundles over a smooth projective curve C, we obtain an upper bound for the essential dimension of their moduli stack. The upper bound is sharp if the conjecture of Colliot-Thélène, Karpenko and Merkurjev holds. We find that the genericity property proved for Deligne-Mumford stacks by Brosnan, Reichstein and Vistoli still holds for this Artin stack, unless the curve C is elliptic.

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arXiv2014

A generalized Quot scheme and meromorphic vortices

Let X be a compact connected Riemann surface. Fix a positive integer r and two nonnegative integers d_p and d_z. Consider all pairs of the form (F, f), where F is a holomorphic vector bundle on X of rank r and degree d_z-d_p, and f : {\mathcal O}^{\oplus r}_X \rightarrow F is a meromorphic homomorphism which an isomorphism outside a finite subset of X and has pole (respectively, zero) of total degree d_p (respectively, d_z). Two such pairs $(F_1, f_1) and $(F_2, f_2) are called isomorphic if there is a holomorphic isomorphism of F_1 with F_2 over X that takes f_1 to f_2. We construct a natural compactification of the moduli space equivalence classes pairs of the above type. The Poincaré polynomial of this compactification is computed.

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