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Vasudevan Srinivas

Publications and source records attributed to Vasudevan Srinivas.

17 recordsLinked to original sources
arXiv2026

Bounds on the Mordell-Weil rank of elliptic fibrations

We prove that the Mordell-Weil group of a higher dimensional elliptic fibration naturally embeds into that of a suitable elliptic surface. We give sufficient conditions for the existence of such surfaces. We apply our result to obtain explicit and uniform bounds for the Mordell-Weil rank of elliptic threefolds of Kodaira dimension zero, including Calabi-Yau threefolds, confirming predictions from physics. We prove new explicit bounds for a broad class of elliptic fourfolds. These results suggest a general linear bound for the Mordell-Weil rank in terms of the dimension of the elliptic fibration, which we formulate as a conjecture.

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arXiv2023

Fundamental groups of proper varieties are finitely presented

It was recently proven by Esnault, Shusterman and the second named author, that the étale fundamental group of a connected smooth projective variety over an algebraically closed field $k$ is finitely presented. In this note, we extend this result to all connected proper schemes over $k$.

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arXiv2022

Euler characteristics of homogeneous and weighted-homogeneous hypersurfaces

Let $k$ be a field and let $\text{GW}(k)$ be the Grothendieck-Witt ring of virtual non-degenerate symmetric bilinear forms over $k$. We develop methods for computing the quadratic Euler characteristic $χ(X/k)\in \text{GW}(k)$ for $X$ a smooth hypersurface in a projective space or a weighted projective space. We raise the question of a quadratic refinement of classical conductor formulas and find such a formula for the degeneration of a smooth hypersurface $X$ in $\mathbb{P}^{n+1}$ to the cone over a smooth hyperplane section of $X$; we also find a similar formula in the weighted homogeneous case. We formulate a conjecture that generalizes these computations to similar types of degenerations. Finally, we give an interpretation of the quadratic conductor formulas in terms of Ayoub's nearby cycles functor.

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arXiv2021

Bounding ramification by covers and curves

We prove that $\bar {\mathbb Q}_\ell$-local systems of bounded rank and ramification on a smooth variety $X$ defined over an algebraically closed field $k$ of characteristic $p\neq \ell$ are tamified outside of codimension $2$ by a finite separable cover of bounded degree. In rank one, there is a curve which preserves their monodromy. There is a curve defined over the algebraic closure of a purely transcendental extension of $k$ of finite degree which fulfills the Lefschetz theorem. Last version: minor typos corrected.

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arXiv2021

An obstruction to lifting to characteristic $0$

We introduce a new obstruction to lifting smooth proper varieties in characteristic $p>0$ to characteristic $0$. It is based on Grothendieck's specialization homomorphism and the resulting discrete finiteness properties of étale fundamental groups.

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arXiv2021

Finite presentation of the tame fundamental group

Let $p$ be a prime number, and let $k$ be an algebraically closed field of characteristic $p$. We show that the tame fundamental group of a smooth affine curve over $k$ is a projective profinite group. We prove that the fundamental group of a smooth projective variety over $k$ is finitely presented. More generally we prove that the tame fundamental group of a smooth quasi-projective variety over $k$ which admits a good compactification is finitely presented. v2: references added. Thank you to all for the friendly and fruitful comments.

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arXiv2018

A note on fierce ramification

We show that bounding ramification at infinity bounds fierce ramification. This answers positively a question of Deligne posed to the first named author.

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arXiv2015

Simply connected varieties in characteristic $p>0$

We show that there are no non-trivial stratified bundles over a smooth simply connected quasi-projective variety over the algebraic closure of a finite field, if the variety admits a normal projective compactification with boundary locus of codimension $\ge 2$. In the appendix, various strong forms of the Lefschetz LEF are proven. Final version, to appear in Compositio.

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arXiv2015

Nilpotence of Frobenius action and the Hodge filtration on local cohomology

An $F$-nilpotent local ring is a local ring $(R, \mathfrak{m})$ of prime characteristic defined by the nilpotence of the Frobenius action on its local cohomology modules $H^i_{\mathfrak{m}}(R)$. A singularity in characteristic zero is said to be of $F$-nilpotent type if its modulo $p$ reduction is $F$-nilpotent for almost all $p$. In this paper, we give a Hodge-theoretic interpretation of three-dimensional normal isolated singularities of $F$-nilpotent type. In the graded case, this yields a characterization of these singularities in terms of divisor class groups and Brauer groups.

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arXiv2013

On the Holonomic Rank Problem

A tautological system, introduced in \cite{LSY}\cite{LY}, arises as a regular holonomic system of partial differential equations that govern the period integrals of a family of complete intersections in a complex manifold $X$, equipped with a suitable Lie group action. In this article, we introduce two formulas -- one purely algebraic, the other geometric -- to compute the rank of the solution sheaf of such a system for CY hypersurfaces in a generalized flag variety. The algebraic version gives the local solution space as a Lie algebra homology group, while the geometric one as the middle de Rham cohomology of the complement of a hyperplane section in $X$. We use both formulas to find certain degenerate points for which the rank of the solution sheaf becomes 1. These rank 1 points appear to be good candidates for the so-called large complex structure limits in mirror symmetry. The formulas are also used to prove a conjecture of Hosono, Lian and Yau on the completeness of the extended GKZ system when $X$ is $¶^n$.

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arXiv2012

Abelian varieties and theta functions associated to compact Riemannian manifolds; constructions inspired by superstring theory

We look into a construction of principal abelian varieties attached to certain spin manifolds, due to Witten and Moore-Witten around 2000 and try to place it in a broader framework. This is related to Weil intermediate Jacobians but it also suggests to associate abelian varieties to polarized even weight Hodge structures. The latter construction can also be explained in terms of algebraic groups which might be useful from the point of view of Tannakian categories. The constructions depend on moduli much as in Teichmüller theory although the period maps in general are only real analytic. One of the nice features is how the index for certain differential operators canonically associated to the geometry of the situation (spin structure, complex structure etc.) leads to integrality of skew pairings on the topological K-group (coming from the Index Theorem) which then serves as a polarization for the jacobian.

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arXiv2011

Algebraic versus topological entropy for surfaces over finite fields

We show that, as in de Rham cohomology over the complex numbers, the value of the entropy of an automorphism of the surface over a finite field $\F_q $ is taken on the span of the Néron-Severi group inside of $\ell$-adic cohomology. v2: (some) typos removed, exposition (partly) improved.

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arXiv2011

Ordinary varieties and the comparison between multiplier ideals and test ideals

We consider the following conjecture: if X is a smooth projective variety over a field of characteristic zero, then there is a dense set of reductions X_s of X to positive characteristic such that the action of the Frobenius morphism on the top Zariski cohomology of the structure sheaf of X_s is bijective. We also consider a conjecture relating certain invariants of singularities in characteristic zero (the multiplier ideals) with invariants in positive characteristic (the test ideals). We prove that the former conjecture implies the latter one in the case of ambient nonsingular varieties.

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arXiv2008

Finite Schur filtration dimension for modules over an algebra with Schur filtration

Let G be GL_N or SL_N as reductive linear algebraic group over a field k of positive characteristic p. We prove several results that were previously established only when N < 6 or p > 2^N. Let G act rationally on a finitely generated commutative k-algebra A. Assume that A as a G-module has a good filtration or a Schur filtration. Let M be a noetherian A-module with compatible G action. Then M has finite good/Schur filtration dimension, so that there are at most finitely many nonzero H^i(G,M). Moreover these H^i(G,M) are noetherian modules over the ring of invariants A^G. Our main tool is a resolution involving Schur functors of the ideal of the diagonal in a product of Grassmannians.

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arXiv2007

A local ring such that the map between Grothendieck groups with rational coefficient induced by completion is not injective

In this paper, we construct a local ring $A$ such that the kernel of the map $G_0(A)\subq \to G_0(\hat{A})\subq$ is not zero, where $\hat{A}$ is the comletion of $A$ with respect to the maximal ideal, and $G_0()\subq$ is the Grothendieck group of finitely generated modules with rational coefficient. In our example, $A$ is a two-dimensional local ring which is essentially of finite type over ${\Bbb C}$, but it is not normal.

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arXiv2007

Diagonal Subschemes and Vector Bundles

We study when a smooth variety $X$, embedded diagonally in its Cartesian square, is the zero scheme of a section of a vector bundle of rank $\dim(X)$ on $X\times X$. We call this the diagonal property (D). It was known that it holds for all flag manifolds ${\rm SL}_n/P$. We consider mainly the cases of proper smooth varieties, and the analogous problems for smooth manifolds (the topological case). Our main new observation in the case of proper varieties is a relation between (D) and cohomologically trivial line bundles on $X$, obtained by a variation of Serre's classic argument relating rank 2 vector bundles and codimension 2 subschemes, combined with Serre duality. Based on this, we have several detailed results on surfaces, and some results in higher dimensions. For smooth affine varieties, we observe that for an affine algebraic group over an algebraically closed field, the diagonal is in fact a complete intersection; thus (D) holds, using the trivial bundle. We conjecture the existence of smooth affine complex varieties for which (D) fails; this leads to an interesting question on projective modules. The arguments in the topological case have a different flavour, with arguments from homotopy theory, topological K-theory, index theory etc. There are 3 variants of the diagonal problem, depending on the type of vector bundle we want (arbitrary, oriented or complex). We obtain a homotopy theoretic reformulation of the diagonal property as an extension problem for a certain homotopy class of maps. We also have detailed results in several cases: spheres, odd dimensional complex projective quadric hypersurfaces, and manifolds of even dimension $\leq 6$ with an almost complex structure.

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