• Nitin Nitsure

Articles written in Proceedings – Mathematical Sciences

• Cohomology of the moduli of parabolic vector bundles

The purpose of this paper is to compute the Betti numbers of the moduli space ofparabolic vector bundles on a curve (see Seshadri [7], [8] and Mehta & Seshadri [4]), in the case where every semi-stable parabolic bundle is necessarily stable. We do this by generalizing the method of Atiyah and Bott [1] in the case of moduli of ordinary vector bundles. Recall that (see Seshadri [7]) the underlying topological space of the moduli of parabolic vector bundles is the space of equivalence classes of certain unitary representations of a discrete subgroup Γ which is a lattice in PSL (2,R). (The lattice Γ need not necessarily be co-compact).

While the structure of the proof is essentially the same as that of Atiyah and Bott, there are some difficulties of a technical nature in the parabolic case. For instance the Harder-Narasimhan stratification has to be further refined in order to get the connected strata. These connected strata turn out to have different codimensions even when they are part of the same Harder-Narasimhan strata.

If in addition to ‘stable = semistable’ the rank and degree are coprime, then the moduli space turns out to be torsion-free in its cohomology.

The arrangement of the paper is as follows. In § 1 we prove the necessary basic results about algebraic families of parabolic bundles. These are generalizations of the corresponding results proved by Shatz [9]. Following this, in § 2 we generalize the analytical part of the argument of Atiyah and Bott (§ 14 of [1]). Finally in § 3 we show how to obtain an inductive formula for the Betti numbers of the moduli space. We illustrate our method by computing explicitly the Betti numbers in the special case of rank = 2, and one parabolic point.

• Quasi-parabolic Siegel formula

The result of Siegel that the Tamagawa number ofSLr over a function field is 1 has an expression purely in terms of vector bundles on a curve, which is known as the Siegel formula. We prove an analogous formula for vector bundles with quasi-parabolic structures. This formula can be used to calculate the Betti numbers of the moduli of parabolic vector bundles using the Weil conjuctures

• Erratum to Quasi-parabolic Siegel formula

The main result of the above paper is mistaken, because of a defective lemma. Here we replace the defective lemma, and derive the corrected quasi-parabolic analogue of the Siegel formula.

• Representability ofGLE

We prove a necessary and sufficient condition for the automorphisms of a coherent sheaf to be representable by a group scheme.

• Representability of Hom implies flatness

LetX be a projective scheme over a noetherian base schemeS, and letF be a coherent sheaf onX. For any coherent sheaf ε onX, consider the set-valued contravariant functor Hom(ε,F)S-schemes, defined by Hom(ε,F) (T)= Hom(εT,FT) where εT andFT are the pull-backs of ε andF toXT =X xS T. A basic result of Grothendieck ([EGA], III 7.7.8, 7.7.9) says that ifF is flat over S then Komε,F) is representable for all ε.

We prove the converse of the above, in fact, we show that ifL is a relatively ample line bundle onX over S such that the functor Hom(L-n,F) is representable for infinitely many positive integersn, thenF is flat overS. As a corollary, takingX =S, it follows that ifF is a coherent sheaf on S then the functorTH°(T, Ft) on the category ofS-schemes is representable if and only ifF is locally free onS. This answers a question posed by Angelo Vistoli.

The techniques we use involve the proof of flattening stratification, together with the methods used in proving the author’s earlier result (see [N1]) that the automorphism group functor of a coherent sheaf onS is representable if and only if the sheaf is locally free.

• Sign (di)Lemma for Dimension Shifting

There is a surprising occurrence of some minus signs in the isomorphisms produced in the well-known technique of dimension shifting in calculating derived functors in homological algebra. We explicitly determine these signs. Getting these signs right is important in order to avoid basic contradictions. We illustrate the result – which we call as the sign lemma for dimension shifting – by some de Rham cohomology and Chern class considerations for compact Riemann surfaces.

• Schematic Harder–Narasimhan Stratification for Families of Principal Bundles and 𝛬-modules

Let 𝐺 be a reductive algebraic group over a field 𝑘 of characteristic zero, let $X\to S$ be a smooth projective family of curves over 𝑘, and let 𝐸 be a principal 𝐺 bundle on 𝑋. The main result of this note is that for each Harder–Narasimhan type 𝜏 there exists a locally closed subscheme $S^\tau (E)$ of 𝑆 which satisfies the following universal property. If $f:T\to S$ is any base-change, then 𝑓 factors via $S^\tau (E)$ if and only if the pullback family $f^∗E$ admits a relative canonical reduction of Harder–Narasimhan type 𝜏. As a consequence, all principal bundles of a fixed Harder–Narasimhan type form an Artin stack. We also show the existence of a schematic Harder–Narasimhan stratification for flat families of pure sheaves of 𝛬-modules (in the sense of Simpson) in arbitrary dimensions and in mixed characteristic, generalizing the result for sheaves of $\mathcal{O}$-modules proved earlier by Nitsure. This again has the implication that 𝛬-modules of a fixed Harder–Narasimhan type form an Artin stack.

• # Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 5
November 2019

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019