arXiv subjects

Nitin Nitsure

Publications and source records attributed to Nitin Nitsure.

At least 19 recordsLinked to original sources

arXiv2025

The Narasimhan-Seshadri Theorem revisited

Let $X$ be a compact Riemann surface. The famous Narasimhan-Seshadri theorem [13] of 1965 uses the Grothendieck construction [4] of 1956 that associates vector bundles $E(\sigma)$ on $X$ to representations $\sigma$ of a certain Fuchsian group $\pi$. Narasimhan and Seshadri show that by taking the representations $\sigma$ to be irreducible unitary of a certain kind, this exactly gives all stable vector bundles on $X$ of a given rank and degree. In this note we reformulate the correspondence from representations to bundles, which leads to simpler statements and proofs. The Fuchsian group $\pi$ is replaced by the punctured fundamental group $\pi_1(X-x)$ where $x\in X$. The Grothendieck bundles $E(\sigma)$ then become Deligne's logarithmic extensions to $X$ of bundles with connections on $X-x$ associated to representations of $\pi_1(X-x)$ with scalar local monodromy. We also report how some ideas from algebraic geometry (which were all in place by 1970) have simplified some aspects of the original proof over the decades. This simplified approach works equally well for all values of the genus $g$, removing the restriction $g\ge 2$ in the 1965 original. Finally, we comment that such a logarithmic reformulation extends to related kinds of bundles such as principal bundles with reductive structure groups or parabolic bundles.

math.AG↗

arXiv2023

Smooth homomorphisms admit noetherian reductions

We give a short proof that any smooth (means formally smooth and finitely presented) homomorphism of rings can be obtained by base change from a smooth homomorphism of noetherian rings. Together with the elegant short proof by J. Conde-Lago that smooth homomorphisms of noetherian rings are flat, this gives a short and elementary proof of the theorem of Grothendieck that all smooth homomorphisms are flat.

math.AC↗

arXiv2022

On flat pullbacks for Chow groups

It is a fundamental property of the Chow groups of algebraic schemes that they are contra-functorial with respect to flat morphisms between schemes. While the pullback homomorphism is easy to define at the level of algebraic cycles, the crucial step is to show that the pullback of cycles preserves rational equivalence, so that it descends to the Chow groups. The purpose of this note is to give a natural sheaf theoretic proof of the preservation of rational equivalence under flat pullback on cycles.

math.AG↗

arXiv2021

My encounter with Seshadri and with the Narasimhan-Seshadri theorem

C.S. Seshadri passed away in July 2020. The first part of this article contains some reminiscences from the early 1980s, when as a graduate student in the Tata Institute of Fundamental Research (TIFR), Mumbai, I had the good fortune to learn from him. The second part is on the Narasimhan-Seshadri theorem and my encounter with it.

math.HO↗

arXiv2020

Harder-Narasimhan stacks for principal bundles in higher dimensions and arbitrary characteristics

Let $G$ be a split reductive group over a field $k$ of arbitrary characteristic, chosen suitably. Let $X\to S$ be a smooth projective morphism of locally noetherian $k$-schemes, with geometrically connected fibers. We show that for each Harder-Narasimhan type $τ$ for principal $G$-bundles, all pairs consisting of a principal $G$-bundle on a fiber of $X\to S$ together with a given canonical reduction of HN-type $τ$ form an algebraic stack $Bun_{X/S}^τ(G)$ over $S$. The forgetful $1$-morphism $Bun_{X/S}^τ(G) \to Bun_{X/S}(G)$ to the algebraic stack of all principal $G$-bundles on fibers of $X\to S$ is a schematic morphism, which is of finite type, separated, radicial, and induces an isomorphism on residue fields of all points of $Bun_{X/S}^τ(G)$. It factors via an open substack $Bun_{X/S}^{\ngtr τ}(G)$ of $Bun_{X/S}(G)$, inducing a finite morphism $Bun_{X/S}^τ(G) \to Bun_{X/S}^{\ngtr τ}(G)$. This is a closed embedding if the Behrend conjecture is satisfied by $G$. The results of this paper hold in arbitrary characteristic, and in fact it gives better proofs of the results of our earlier papers which had assumed the characteristic to be zero. Along the way we give a new proof of the existence of a canonical reduction over any base field in all dimensions, and we also prove openness of semistability and semicontinuity of canonical type in a family.

math.AG↗

arXiv2019

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 1st order infinitesimal obstruction to the quadrilateral to close is always zero, the 2nd 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 3rd 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 $3π/4$ and $-π/4$ 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.

math.DG↗

arXiv2019

Molding 3D curved structures by selective heating

It is of interest to fabricate curved surfaces in three dimensions from easily available homogeneous material in the form of flat sheets. The aim is not just to obtain a surface $M$ in $\mathbb{R}^3$ which has a desired intrinsic Riemannian metric, but to get the desired embedding $M \subset \mathbb{R}^3$ up to translations and rotations (the Riemannian metric alone need not uniquely determine this). In this paper, we demonstrate three generic methods of molding a flat sheet of thermo-responsive plastic by selective contraction induced by targeted heating. These methods do not involve any cutting and gluing, which is a property they share with origami. The first method is inspired by tailoring, which is the usual method for making garments out of plain pieces of cloth. Unlike usual tailoring, this method produces the desired embedding in $\mathbb{R}^3$, and in particular, we get the desired intrinsic Riemannian metric. The second method just aims to bring about the desired new Riemannian metric via an appropriate pattern of local contractions, without directly controlling the embedding. The third method is based on triangulation, and seeks to induce the desired local distances. This results in getting the desired embedding in $\mathbb{R}^3$, in particular, it also gives us the target Riemannian metric. The second and the third methods, and also the first method for the special case of surfaces of revolution, are algorithmic in nature. We give a theoretical account of these methods, followed by illustrated examples of different shapes that were physically molded by these methods.

cond-mat.soft↗

arXiv2017

On the geometric phenomenology of static friction

In this note we introduce a hierarchy of phase spaces for static friction, which give a graphical way to systematically quantify the directional dependence in static friction via subregions of the phase spaces. We experimentally plot these subregions to obtain phenomenological descriptions for static friction in various examples where the macroscopic shape of the object affects the frictional response. The phase spaces have the universal property that for any experiment in which a given object is put on a substrate fashioned from a chosen material with a specified nature of contact,the frictional behavior can be read off from a uniquely determined classifying map on the control space of the experiment which takes values in the appropriate phase space.

cond-mat.soft↗

arXiv2016

Schematic Harder-Narasimhan stratification for families of principal bundles in higher dimensions

For any family of principal bundles with a reductive structure group G on a family X/S of smooth projective varieties in characteristic zero, it is known that the parameter scheme S has a set theoretic stratification by locally closed subsets which correspond to the Harder-Narasimhan types of the restriction of the principal bundle to the various fibers of X/S. We show that each of these subsets has in fact the structure of a locally closed subscheme of the parameter scheme S, with the following universal property: Under any base change, the pullback family admits a relative Harder-Narasimhan filtration (defined appropriately) with a given Harder-Narasimhan type if and only if the base change factors via the schematic stratum corresponding to that Harder-Narasimhan type. It follows that principal bundles of any given Harder-Narasimhan type on X/S form an Artin algebraic stack over S, and as the Harder-Narasimhan type varies, these stacks define a stratification the stack of all principal G-bundles on X/S by locally closed substacks. This result extends to principal bundles in higher dimensions our earlier similar results which were proved for principal bundles on families of curves. The result is new even for vector bundles, that is, for G = GL(n).

math.AG↗

arXiv2016

Curvilinear polyhedra as dynamical arenas, illustrated by an example of self-organized locomotion

Experiment shows that dumbbells, placed inside a tilted hollow cylindrical drum that rotates slowly around its axis, climb uphill by forming dynamically stable pairs, seemingly against the pull of gravity. Analysis of this experiment shows that the dynamics takes place in an underlying space which is a curvilinear polyhedron inside a six dimensional manifold, carved out by unilateral constraints that arise from the non-interpenetrability of the dumbbells. The energetics over this polyhedron localizes the configuration point within the close proximity of a corner of the polyhedron. This results into a strong entrapment, which provides the configuration of the dumbbells with its observed shape that leads to its functionality -- uphill locomotion. The stability of the configuration is a consequence of the strong entrapment in the corner of the polyhedron.

nlin.AO↗

arXiv2014

How to construct a closed subscheme, or a coherent subsheaf, with prescribed germs

We show that a closed subscheme of a given locally noetherian scheme can be constructed by prescribing it germs at all points of the ambient scheme in a manner consistent with specialization of points, provided the resulting set of all associated points of all the germs is locally finite. More generally, we prove a similar result for constructing a coherent subsheaf of a coherent sheaf by prescribing its stalks at all points in a manner consistent with specializations of points, again provided the set of all associated points of all the corresponding local quotients is locally finite. On any locally noetherian scheme, we show that there exists a unique global section of any coherent sheaf which has a prescribed family of germs which is consistent with specialization of points. It is not clear how to formulate an analogous result for constructing a coherent sheaf in terms of prescribed stalks. Even when the set of all associated points of all the prescribed stalks is locally finite, such a construction need not succeed as we show with an example. This raises the question of how to set up effective descent data for coherent sheaves purely in terms of germs and their specializations.

math.AG↗

arXiv2013

Quadric invariants and degeneration in smooth-etale cohomology

For a regular pair $(X,Y)$ of schemes of pure codimension 1 on which 2 is invertible, we consider quadric bundles on $X$ which are nondegenerate on $X-Y$, but are minimally degenerate on $Y$. We give a formula for the behaviour of the cohomological invariants (characteristic classes) of the nondegenerate quadric bundle on $X-Y$ under the Gysin boundary map to the etale cohomology of $Y$ with mod 2 coefficients. The results here are the algebro-geometric analogs of topological results for complex bundles proved earlier by Holla and Nitsure, continuing further the algebraization program which was commenced with a recent paper by Bhaumik. We use algebraic stacks and their smooth-etale cohomologies, $A^1$-homotopies and Gabber's absolute purity theorem as algebraic replacements for the topological methods used earlier, such as CW complexes, real homotopies, Riemannian metrics and tubular neighbourhoods. Our results also hold in smooth-etale cohomology for quadric bundles over algebraic stacks on which 2 is invertible.

math.AG↗

arXiv2012

Schematic HN stratification for families of principal bundles and lambda modules

For a family of principal bundles with a reductive structure group on a family of curves in characteristic zero, it is known that the Harder Narasimhan type of its restriction to each fiber varies semicontinuously over the parameter scheme of the family. This defines a stratification of the parameter scheme by locally closed subsets, known as the Harder-Narasimhan stratification. In this note, we show how to endow each Harder-Narasimhan stratum with the structure of a locally closed subscheme of the parameter scheme, which enjoys the universal property that under any base change the pullback family admits a relative Harder-Narasimhan reduction with a given Harder-Narasimhan type if and only if the base change factors through the schematic stratum corresponding to that Harder-Narasimhan type. This has the consequence that principal bundles of a given Harder Narasimhan type form an Artin stack. We also prove a similar result showing the existence of a schematic Harder-Narasimhan filtration 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.

math.AG↗

arXiv2011

Moduli stacks and moduli schemes for rank 2 unstable bundles

Let X be a geometrically irreducible smooth projective curve over a field k. We describe the algebra of endomorphisms of indecomposable unstable vector bundles over X of rank 2 and degree d. Fixing some numerical invariants, namely the Harder-Narasimhan type and the dimension of the algebra of endomorphisms, we construct algebraic stacks and moduli schemes for such bundles.

math.AG↗

arXiv2009

Schematic Harder-Narasimhan Stratification

For any flat family of pure-dimensional coherent sheaves on a family of projective schemes, the Harder-Narasimhan type (in the sense of Gieseker semistability) of its restriction to each fiber is known to vary semicontinuously on the parameter scheme of the family. This defines a stratification of the parameter scheme by locally closed subsets, known as the Harder-Narasimhan stratification. In this note, we show how to endow each Harder-Narasimhan stratum with the structure of a locally closed subscheme of the parameter scheme, which enjoys the universal property that under any base change the pullback family admits a relative Harder-Narasimhan filtration with a given Harder-Narasimhan type if and only if the base change factors through the schematic stratum corresponding to that Harder-Narasimhan type. The above schematic stratification induces a stacky stratification on the algebraic stack of pure-dimensional coherent sheaves. We deduce that coherent sheaves of a fixed Harder-Narasimhan type form an algebraic stack in the sense of Artin.

math.AG↗

arXiv2007

Sign 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 lemma by some de Rham cohomology and Chern class considerations for compact Riemann surfaces.

math.AG↗

arXiv2005

Construction of Hilbert and Quot Schemes

This is an expository account of Grothendieck's construction of Hilbert and Quot Schemes, following his talk `Techniques de construction et theoremes d'existence en geometrie algebriques IV : les schemas de Hilbert', Seminaire Bourbaki 221 (1960/61), together with further developments by Mumford and by Altman and Kleiman. Hilbert and Quot schemes are fundamental to modern Algebraic Geometry, in particular, for deformation theory and moduli constructions. These notes are based on a series of six lectures in the summer school `Advanced Basic Algebraic Geometry', held at the Abdus Salam International Centre for Theoretical Physics, Trieste, in July 2003.

math.AG↗

arXiv2003

Representability of Hom implies flatness

Let $X$ be a projective scheme over a noetherian base scheme $S$, and let $F$ be a coherent sheaf on $X$. For any coherent sheaf $E$ on $X$, consider the set-valued contravariant functor $Hom_{E,F}$ on $S$-schemes, defined by $Hom_{E,F}(T) = Hom(E_T,F_T)$ where $E_T$ and $F_T$ are the pull-backs of $E$ and $F$ to $X_T = X\times_S T$. A basic result of Grothendieck ([EGA] III 7.7.8, 7.7.9) says that if $F$ is flat over $S$ then $Hom_{E,F}$ is representable for all $E$. We prove the converse of the above, in fact, we show that if $L$ is a relatively ample line bundle on $X$ over $S$ such that the functor $Hom_{L^{-n},F}$ is representable for infinitely many positive integers $n$, then $F$ is flat over $S$. As a corollary, taking $X=S$, it follows that if $F$ is a coherent sheaf on $S$ then the functor $T\mapsto H^0(T, F_T)$ on the category of $S$-schemes is representable if and only if $F$ is locally free on $S$. 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 arXiv.org/abs/math.AG/0204047) that the automorphism group functor of a coherent sheaf on $S$ is representable if and only if the sheaf is locally free.

math.AG↗