Nitin Nitsure
Articles written in Proceedings – Mathematical Sciences
Volume 95 Issue 1 September 1986 pp 61-77
Cohomology of the moduli of parabolic vector bundles
The purpose of this paper is to compute the Betti numbers of the moduli space of
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.
Volume 106 Issue 2 May 1996 pp 133-137
Quasi-parabolic Siegel formula
The result of Siegel that the Tamagawa number of
Volume 107 Issue 2 May 1997 pp 221-222 Erratum
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.
Volume 112 Issue 4 November 2002 pp 539-542
We prove a necessary and sufficient condition for the automorphisms of a coherent sheaf to be representable by a group scheme.
Volume 114 Issue 1 February 2004 pp 7-14
Representability of Hom implies flatness
Let
We prove the converse of the above, in fact, we show that if
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 on
Volume 119 Issue 2 April 2009 pp 179-186
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
Volume 124 Issue 3 August 2014 pp 315-332
Schematic Harder–Narasimhan Stratification for Families of Principal Bundles and 𝛬-modules
Sudarshan Gurjar Nitin Nitsure
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.
Volume 131 All articles Published: 13 February 2021 Article ID 0004 Article
Curvature, torsion and the quadrilateral gaps
For a manifold with an affine connection, we prove formulas which infinitesimally quantify the gap in a certain naturally defined open geodesic quadrilateral associated to a pair of tangent vectors $u$, $v$ at a point of the manifold. We show that the first-order infinitesimal obstruction to the quadrilateral to close is always zero, the second-order infinitesimal obstruction to the quadrilateral to close is $−T (u, v)$, where $T$ is the torsion tensor of the connection, and if $T = 0$, then the third-order infinitesimal obstruction to the quadrilateral to close is $(1/2)R(u, v)(u +v)$ in terms of the curvature tensor of the connection. Consequently, the torsion of the connection, and if the torsion is identically zero, then also the curvature of the connection can be recovered uniquely from knowing all the quadrilateral gaps. In particular, this answers a question of Rajaram Nityananda about the quadrilateral gaps on a curved Riemannian surface. The angles of $3\pi/4$ and $−\pi/4$ radians feature prominently in the answer, along with the value of the Gaussian curvature. This article is essentially self-contained, and written in an expository style.
Volume 132, 2022
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