arXiv subjects

Jaya NN Iyer

Publications and source records attributed to Jaya NN Iyer.

At least 19 recordsLinked to original sources

arXiv2026

Persistent Simple-homotopy invariants via discrete Morse theory

Persistent homology records the evolution of homological features along a filtration, but does not retain finer information related to simple-homotopy theory. In this paper, we develop two approaches to capturing such information for filtered simplicial complexes. We first introduce the Morse complexity profile, which records the minimal number of critical simplices at each filtration level. We study its invariance and stability properties and develop computable approximations using several discrete Morse matchings. We then introduce a persistent version of Whitehead torsion and show that it is invariant under both levelwise homotopy equivalence and interleaving equivalence of filtrations.

math.AT↗

arXiv2026

Torsion of extended Chern-Simons classes for canonical extensions of flat bundles

Let $X$ be a smooth complex projective variety and $D = D_1+\cdots+D_k\subset X$ a divisor with simple normal crossings. Consider Deligne's canonical extension $(F,\nabla)$ of a flat algebraic vector bundle on $X^*:=X\setminus D$ with unipotent monodromy around every component of $D$. We define and compare the various constructions of the extended Chern-Simons classes $$ \mathrm{CS}_p(\nabla^{\mathrm{Del}})\;\in\; H^{2p-1}(X,\mathbb{C}/\mathbb{Z}),\, p\geq 1, $$ attached to $(F,\nabla)$. Our main theorem states that $\mathrm{CS}_p(\nabla^{\mathrm{Del}})$ is torsion in $H^{2p-1}(X,\mathbb{C}/\mathbb{Z})$, for every $p\geq 2$, extending \cite{Reznikov}, \cite{Reznikov2}, \cite{IS-arXiv},\cite{IS-2div}). We treat the case of quasi-unipotent local monodromies via locally abelian parabolic bundles, and deduce the torsion of Chern-Simons classes. We also extend the Deligne-Sullivan theorem \cite{DeSu} on triviality of flat bundle on a finite covering of a smooth manifold, to that of a canonical extension, and provide torsion-bounds on the extended characteristic classes.

math.AG↗

arXiv2026

Regulators of canonical extensions are torsion:the case of two transversally intersecting smooth divisors

This note extends the main result of \cite{IS} 2007 --- torsion of the extended Chern--Simons (regulator) classes of the Deligne canonical extension of a flat bundle with unipotent monodromy at infinity --- from the case of a smooth irreducible boundary divisor to the case of a boundary divisor $D = D_1\cup D_2$ with two smooth irreducible components meeting transversally along a smooth center $Z=D_1\cap D_2$. Let $X$ be a smooth projective variety defined over $\mathbb{C}$, and $U:=X-D$. Given a flat bundle $(E,\nabla)$ on $U$ with unipotent monodromy around the components of $D$ consider Deligne's canonical extension $(\overline{E},\overline{\nabla})$ on $X$. Then the extended Chern-Simons classes $$ c_p(\overline{E},\overline{\nabla})\in H^{2p-1}(X,\mathbb{C}/\mathbb{Z}) $$ are torsion, for $p\geq 2$. These notes were prepared in 2009-2010, and the preprint \cite[2026]{IS2} treats the full normal crossing case via a different approach.

math.AG↗

arXiv2025

A note on the Brill-Noether loci of small codimension in moduli space of stable bundles

Let $X$ be a smooth projective curve of genus $g$ over the field $\mathbb{C}$. Let $M_{X}(2,L)$ denote the moduli space of stable rank $2$ vector bundles on $X$ with fixed determinant $L$ of degree $2g-1$. Consider the Brill-Noether subvariety $W^{1}_{X}(2,L)$ of $M_{X}(2,L)$ which parametrises stable vector bundles having at least two linearly independent global sections. In this article, for generic $X$ and $L$, we show that $W^{1}_{X}(2,L)$ is stably-rational when $g=3$, unirational when $g=4$, and rationally chain connected by Hecke curves, when $g\geq 5$. We also show triviality of low dimensional rational Chow groups of an associated Brill-Noether hypersurface.

math.AG↗

arXiv2025

Tautological algebra of the moduli stack of semistable bundles of rank 2 on a general curve

Our aim is to determine the tautological algebra generated by the cohomology classes of the Brill-Noether loci in the rational cohomology of the moduli stack $\mathcal{U}_C(n,d)$ of semistable bundles of rank $n$ and degree $d$. We show that for a general smooth projective curve $C$ of genus $g\geq 2$, $d=2g-2$, the tautological algebra of $ \mathcal{U}_C(2,2g-2)$ (resp. the moduli stack $\mathcal{SU}_C(2,\mathcal{L})$ of semistable bundles of rank $2$ and determinant $\mathcal{L}$ with $°(\mathcal{L})=2g-2$) is generated by the divisor classes (resp. the class of the Theta divisor $Θ$). This is previously known in rank one situation, called the (classical) Porteous formula.

math.AG↗

arXiv2025

Tropical Geometry Based Edge Detection Using Min-Plus and Max-Plus Algebra

This paper proposes a tropical geometry-based edge detection framework that reformulates convolution and gradient computations using min-plus and max-plus algebra. The tropical formulation emphasizes dominant intensity variations, contributing to sharper and more continuous edge representations. Three variants are explored: an adaptive threshold-based method, a multi-kernel min-plus method, and a max-plus method emphasizing structural continuity. The framework integrates multi-scale processing, Hessian filtering, and wavelet shrinkage to enhance edge transitions while maintaining computational efficiency. Experiments on MATLAB built-in grayscale and color images suggest that tropical formulations integrated with classical operators, such as Canny and LoG, can improve boundary detection in low-contrast and textured regions. Quantitative evaluation using standard edge metrics indicates favorable edge clarity and structural coherence. These results highlight the potential of tropical algebra as a scalable and noise-aware formulation for edge detection in practical image analysis tasks.

math.AG↗

arXiv2021

Push-forwards of Chow groups of smooth ample divisors

We introduce a homological Lefschetz conjecture on (rational) Chow groups, which can be deduced from some well known conjectures, and illustrate it by a series of key examples. We then prove the injectivity of the push-forward morphism on Chow groups, induced by the closed embedding of the Theta divisor in it's Jacobian $J(C)$. Here $C$ is a smooth irreducible complex projective curve.

math.AG↗

arXiv2019

Brauer groups of schemes associated to symmetric powers of smooth projective curves in arbitrary characteristics

In this paper we show that the l^n-torsion part of the cohomological Brauer groups of certain schemes associated to symmetric powers of a projective smooth curve over a separably closed field k are isomorphic, when `l is invertible in k. The schemes considered are the Symmetric powers themselves, then the corresponding Picard schemes and also certain Quot-schemes. We also obtain similar results for Prym varieties associated to certain finite covers of such curves: we prove such results only for curves defined over the field of complex numbers.

math.AG↗

arXiv2016

On the kernel of the push-forward homomorphism between Chow groups

In this note we prove that the kernel of the push-forward homomorphism on $d$-cycles modulo rational equivalence, induced by the closed embedding of an ample divisor linearly equivalent to some multiple of the theta divisor inside the Jacobian variety $J(C)$ is trivial. Here $C$ is a smooth projective curve of genus $g$.

math.AG↗

arXiv2013

Bundles of Verlinde spaces and group actions

A Verlinde space of level $k$ is the space of global sections of the $k$-th power of the determinant line bundle on the moduli space $\cSU_C(r)$ of semi-stable bundles of rank $r$ on a curve $C$. The aim of this note is to make accessible some remarks on the action of the Theta group on the Verlinde spaces of higher level. This gives a decomposition of the bundle of Verlinde spaces over the moduli space of curves and we indicate how to compute the rank of the isotypical components in the decomposition.

math.AG↗

arXiv2013

Tertiary classes for a one-parameter variation of flat connections on a smooth manifold

In this note, we extend the theory of Chern-Cheeger-Simons to construct canonical invariants for a one-parameter family of flat connections on a smooth manifold. These invariants lie in degrees $(2p-2)$-cohomology with $\C/\Z$-cohomology, for $p\geq 2$. Furthermore, they are shown to be rigid in a variation of paths (parametrising flat connections), in degrees at least three.

math.DG↗

arXiv2013

Embedding properties of linear series on hyperelliptic varieties

In this paper, we investigate linear systems on hyperelliptic varieties. We prove analogues of well-known theorems on abelian varieties, like Lefschetz's embedding theorem and higher k-jet embedding theorems. Syzygy or $N_p$-properties are also deduced for appropriate powers of ample line bundles.

math.AG↗

arXiv2012

Tautological ring of the moduli space of generalised parabolic line bundles on a curve

In this paper, we consider the tautological ring containing the extended Brill-Noether algebraic classes on the normalization of the compactified Jacobian of a complex nodal projective curve (with one node). This smallest $\Q$-subalgebra of algebraic classes under algebraic equivalence, stable under extensions of the maps induced by multiplication maps, Pontrayagin product and Fourier transform, is shown to be generated by pullback of the Brill-Noether classes of the Jacobian of the normalized curve and some natural classes.

math.AG↗

arXiv2010

Abel-Jacobi isomorphism for one cycles on Kirwan's resolution of the moduli space SU_C(2,O_C)

In this paper, we consider the moduli space $\cSU_C(r,\cO_C)$ of rank $r$ semistable vector bundles with trivial determinant on a smooth projective curve $C$ of genus $g$. When the rank $r=2$, F. Kirwan constructed a smooth log resolution $\ov{X}\rar \cSU_C(2,\cO_C)$. Based on earlier work of M. Kerr and J. Lewis, Lewis explains in the Appendix the notion of a relative Chow group (w.r.to the normal crossing divisor), and a subsequent Abel-Jacobi map on the relative Chow group of null-homologous one cycles (tensored with $\Q$). This map takes values in the intermediate Jacobian of the compactly supported cohomology of the stable locus. We show that this is an isomorphism and since the intermediate Jacobian is identified with the Jacobian $Jac(C)\otimes \Q$, this can be thought of as a weak-representability result for open smooth varieties. A Hard Lefschetz theorem is also proved for the odd degree bottom weight cohomology of the moduli space $\cSU_C^s(2,\cO_C)$. When the rank $r\geq 2$, we compute the codimension two rational Chow groups of $\cSU_C(r,\cO_C)$.

math.AG↗

arXiv2010

A note on the unirationality of a moduli space of double covers

In this note we look at the moduli space $\cR_{3,2}$ of double covers of genus three curves, branched along 4 distinct points. This space was studied by Bardelli, Ciliberto and Verra. It admits a dominating morphism $\cR_{3,2} \to {\mathcal A}_4$ to Siegel space. We show that there is a birational model of $\cR_{3,2}$ as a group quotient of a product of two Grassmannian varieties. This gives a proof of the unirationality of $\cR_{3,2}$ and hence a new proof for the unirationality of ${\mathcal A}_4$.

math.AG↗

arXiv2010

Absolute Chow-Kuenneth decomposition for rational homogeneous bundles and for log homogeneous varieties

In this paper, we investigate Murre's conjecture on the existence of a Chow--Kuenneth decomposition for a rational homogeneous bundle $Z\to S$ over a smooth variety, defined over complex numbers. Chow-Künneth decomposition is exhibited for $Z$ whenever $S$ has a Chow--Kuenneth decomposition. The same conclusion holds for a class of log homogeneous varieties, studied by M. Brion.

math.AG↗

arXiv2009

Hilbert schemes of fat r-planes and triviality of Chow groups of complete intersections

In this paper, we investigate the question of triviality of the rational Chow groups of complete intersections in projective spaces and obtain improved bounds for this triviality to hold. Along the way, we study the dimension and nonemptiness of some Hilbert schemes of fat $r$-planes contained in a complete intersection $Y$, generalizing well-known results on the Fano varieties of $r$-planes contained in $Y$.

math.AG↗

arXiv2008

A report on "Regulators of canonical extension are torsion; the smooth divisor case"

In this note, we report on a work jointly done with C. Simpson on a generalization of Reznikov's theorem which says that the Chern-Simons classes and in particular the Deligne Chern classes (in degrees $> 1$) are torsion, of a flat vector bundle on a smooth complex projective variety. We consider the case of a smooth quasi--projective variety with an irreducible smooth divisor at infinity. We define the Chern-Simons classes of the Deligne's \textit{canonical extension} of a flat vector bundle with unipotent monodromy at infinity, which lift the Deligne Chern classes and prove that these classes are torsion. The details of the proof can be found in arxiv:0707.0372 [math.AG].

math.AG↗