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Ajneet Dhillon

Publications and source records attributed to Ajneet Dhillon.

29 records · Page 2Linked to original sources
arXiv2012

Automorphisms of the Quot schemes associated to compact Riemann surfaces

Let X be a compact connected Riemann surface of genus at least two. Fix positive integers r and d. Let Q denote the Quot scheme that parametrizes the torsion quotients of {\mathcal O}^{\oplus r}_X of degree d. This Q is also the moduli space of vortices for the standard action of U(r) on {\mathbb C}^r. The group \text{PGL}(r, {\mathbb C}) acts on Q via the action of $\text{GL}(r, {\mathbb C})$ on ${\mathcal O}^{\oplus r}_X$. We prove that this subgroup $\text{PGL}(r, {\mathbb C})$ is the connected component, containing the identity element, of the holomorphic automorphism group Aut(\mathcal Q). As an application of it, we prove that the isomorphism class of the complex manifold Q uniquely determines the isomorphism class of the Riemann surface X.

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arXiv2012

A construction of Chern classes of parabolic vector bundles

Given a parabolic vector bundle, we construct for it a projectivization and tautological line bundle. These are analogs of the projectivization and tautological line bundle for an usual vector bundle. Using these we give a construction of the parabolic Chern classes.

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arXiv2012

The Essential Dimension of Stacks of Parabolic Vector Bundles over Curves

We find upper bounds on the essential dimension of the moduli stack of parabolic vector bundles over a curve. When there is no parabolic structure, we improve the known upper bound on the essential dimension of the usual moduli stack. Our calculations also give lower bounds on the essential dimension of the semistable locus inside the moduli stack of vector bundles of rank $r$ and degree $d$ without parabolic structure.

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arXiv2011

Semistability criterion for parabolic vector bundles on curves

We give a cohomological criterion for a parabolic vector bundle on a curve to be semistable. It says that a parabolic vector bundle $E$ with rational parabolic weights is semistable if and only if there is another parabolic vector bundle $F$ with rational parabolic weights such that the cohomologies of the vector bundle underlying the para- bolic tensor product $E \otimes F$ vanish. This criterion generalizes the known semistability criterion of Faltings for vector bundles on curves and significantly improves the result in [Bis07].

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arXiv2010

Pullback of parabolic bundles and covers of ${\mathbb P}^1\setminus\{0,1,\infty\}$

We work over an algebraically closed ground field of characteristic zero. A $G$-cover of ${\mathbb P}^1$ ramified at three points allows one to assign to each finite dimensional representation $V$ of $G$ a vector bundle $\oplus \mathscr{O}(s_i)$ on ${\mathbb P}^1$ with parabolic structure at the ramification points. This produces a tensor functor from representation of $G$ to vector bundles with parabolic structure that characterises the original cover. This work attempts to describe this tensor functor in terms of group theoretic data. More precisely, we construct a pullback functor on vector bundles with parabolic structure and describe the parabolic pullback of the previously described tensor functor.

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arXiv2010

On Nori's Obstruction to Universal Bundles

Let $G$ be $Sl_n, Sp(2n)$ or SO(2n). We consider the moduli space $M$ of semistable principal $G$-bundles over a curve $X$. Our main result is that if $U$ is a Zariski open subset of $M$ then there is no universal bundle on $U\times X$.

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arXiv2006

The motive of the moduli stack of G-bundles over the universal curve

An abelian category of relative pure motives is constructed along the lines of André (over a field of characteristic 0). An algebraic stack is shown to possess a motive in this sense. This motive is studied for the moduli stack of G-bundles over the universal curve for appropriate G, and a motivic Atiyah-Bott formula is obtained. By taking Hodge realization, a description of the corresponding variations of Hodge structures over M_g is given. In particular, it follows that these lie in the tensor category generated by the variation of Hodge structure associated to the curve itself. Similar results are also given for low degree variations of Hodge structures associated to the moduli space of stable G-bundles.

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arXiv2005

On the Motive of the Stack of Bundles

Let $G$ be a split connected semisimple group over a field. We give a conjectural formula for the motive of the stack of $G$-bundles over a curve $C$, in terms of special values of the motivic zeta function of $C$. The formula is true if $C=\pp^1$ or $G=\sln$. If $k=\cc$, upon applying the Poincaré or Serre characteristic, the formula reduces to results of Teleman and Atiyah-Bott on the gauge group. If $k=\ffq$, upon applying the counting measure, it reduces to the fact that the Tamagawa number of $G$ over the function field of $C$ is $|π_1(G)|$.

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arXiv2003

On the Cohomology of Moduli of Vector Bundles

We compute some Hodge and Betti numbers of the moduli space of stable rank $r$ degree $d$ vector bundles on a smooth projective curve. We do not assume $r$ and $d$ are coprime. In the process we equip the cohomology of an arbitrary algebraic stack with a functorial mixed Hodge structure. This Hodge structure is computed in the case of the moduli stack of rank $r$, degree $d$ vector bundles on a curve. Our methods also yield a formula for the Poincare polynomial of the moduli stack that is valid over any ground field.

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arXiv2000

A Generalized Torelli Theorem

Given a smooth projective curve $C$ of genus $g$ over the complex numbers, Torelli's thoerem asserts that the pair $(J(C),W^{g-1})$ determines $C$, where $W^{g-1}$ is an image of the $g-1$st symmetric power of $C$ inside the Jacobian under an Abel-Jacobi map. We show that the theorem holds with $g-1$ replaced by an integer $d$ in the range $1\le d\le g-1$.

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