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Loring W. Tu

Publications and source records attributed to Loring W. Tu.

14 recordsLinked to original sources
arXiv2023

Equivariant Characteristic Classes

This is a commentary on Raoul Bott and Loring Tu's joint article "Equivariant characteristic classes in the Cartan model," which appeared in "Geometry, Analysis, and Applications (Varanasi, 2000)," World Scientif Publishing, River Edge, NJ, 3--20. The article is also included in "Raoul Bott: Collected Papers," Vol. 5. The commentary discusses the genesis of the article, its influence, and the current state of the problem concerning equivariant characteristic classes.

math.AT↗
arXiv2023

Computing Topological Invariants Using Fixed Points

When a torus acts on a compact oriented manifold with isolated fixed points, the equivariant localization formula of Atiyah--Bott and Berline--Vergne converts the integral of an equivariantly closed form into a finite sum over the fixed points of the action, thus providing a powerful tool for computing integrals on a manifold. An integral can also be viewed as a pushforward map from a manifold to a point, and in this guise it is intimately related to the Gysin homomorphism. This article highlights two applications of the equivariant localization formula. We show how to use it to compute characteristic numbers of a homogeneous space and to derive a formula for the Gysin map of a fiber bundle.

math.AT↗
arXiv2023

Gysin formulas and equivariant cohomology

Under Poincaré duality, a smooth map of compact oriented manifolds induces a pushforward map in cohomology, called the "Gysin map." It plays an important role in enumerative geometry. Using the equivariant localization formula, the author gave in 2017 a general formula for the Gysin map of a fiber bundle with equivariantly formal fibers. Equivariantly formal manifolds include all manifolds with cohomology in only even degrees such as complex projective spaces, Grassmannians, and flag manifolds as well as $G/H$, where $G$ is a compact Lie group and $H$ is a closed subgroup of maximal rank. This article is a simplified exposition using the example of a projective bundle to illustrate the algorithm.

math.AT↗
arXiv2022

Lefschetz fixed point theorems for correspondences

The classical Lefschetz fixed point theorem states that the number of fixed points, counted with multiplicity $\pm 1$, of a smooth map $f$ from a manifold $M$ to itself can be calculated as the alternating sum $\sum (-1)^k \textrm{ tr } f^*|_{H^k(M)}$ of the trace of the induced homomorphism in cohomology. In 1964, at a conference in Woods Hole, Shimura conjectured a Lefschetz fixed point theorem for a holomorphic map, which Atiyah and Bott proved and generalized into a fixed point theorem for elliptic complexes. However, in Shimura's recollection, he had conjectured more than the holomorphic Lefschetz fixed point theorem. He said he had made a conjecture for a holomorphic correspondence, but he could not remember the statement. This paper is an exploration of Shimura's forgotten conjecture, first for a smooth correspondence, then for a holomorphic correspondence in the form of two conjectures and finally in the form of an open problem involving an extension to holomorphic vector bundles over two varieties and the calculation of the trace of a Hecke correspondence.

math.AT↗
arXiv2022

Introduction to Sheaf Cohomology

This article aims to introduce to the uninitiated, in just four lectures of 26 pages, the wonderful techniques of sheaf cohomology, hypercohomology, and spectral sequences.

math.AT↗
arXiv2020

Basis-Free Analysis of Singular Tuples and Eigenpairs of Tensors

A tensor in applied mathematics is usually defined as a multidimensional array of numbers. This presumes a choice of basis in $\mathbb{R}^n$ or in some other vector space, and tensorial concepts are defined accordingly. In this article we define eigenvalues, eigenvectors, singular values, and singular vectors of a tensor intrinsically, without reference to a basis. The basis-free approach has several advantages. First, it shows more clearly the relationship between tensor analysis and areas of pure mathematics such as abstract algebra, differential topology, and algebraic geometry. Second, it obviates the need to prove that a concept defined in terms of coordinates is independent of the choice of basis. Third, an intrinsic definition is usually conceptually simpler. As illustrations we show how Morse theory from differential topology can be used to analyze eigenvalues and eigenvectors of a symmetric tensor. We also reprove a few results that are obvious in the basis-free approach, but not otherwise.

math.RA↗
arXiv2015

Computing the Gysin map using fixed points

The Gysin map of a map between compact oriented manifolds is the map in cohomology induced by the push-forward map in homology. In enumerative algebraic geometry, formulas for the Gysin map of a flag bundle play a vital role. These formulas are usually proven by algebraic or combinatorial means. This article shows how the localization formula in equivariant cohomology provides a systematic method for calculating the Gysin homomorphism in the ordinary cohomology of a fiber bundle. As examples, we recover classical pushforward formulas for generalized flag bundles. Our method extends the classical formulas to fiber bundles with equivariantly formal fibers.

math.AT↗
arXiv2014

From Sheaf Cohomology to the Algebraic de Rham Theorem

Let X be a smooth complex algebraic variety with the Zariski topology, and let Y be the underlying complex manifold with the complex topology. Grothendieck's algebraic de Rham theorem asserts that the singular cohomology of Y with complex coefficients can be computed from the complex of sheaves of algebraic differential forms on X. This article gives an elementary proof of Grothendieck's algebraic de Rham theorem, elementary in the sense that we use only tools from standard textbooks as well as Serre's FAC and GAGA papers.

math.AG↗
arXiv2013

What is ... Equivariant Cohomology?

When a torus acts on a compact oriented manifold with isolated fixed points, the equivariant localization formula of Atiyah--Bott--Berline--Vergne converts the integral of an equivariantly closed form to a finite sum over the fixed points, providing a powerful tool for computing integrals on a manifold. This article seeks to give an accessible exposition of the equivariant localization formula.

math.AT↗
arXiv2013

Computing characteristic numbers using fixed points

Let G be a compact connected Lie group with maximal torus T, and H a closed subgroup containing T . We work out the Atiyah--Bott--Berline--Vergne localization formula for the homogeneous space G/H under the natural action of the maximal torus T. The computation gives explicit formulas for the ordinary and equivariant characteristic numbers of a homogeneous space.

math.AG↗
arXiv1993

Theta Functions for $\SL(n)$ versus $\GL(n)$

Over a smooth complex projective curve $C$ of genus $g$ let $\M (n,d)$ be the moduli space of semistable bundles of rank $n$ and degree $d$ on $C$, and $\SM (n,L)$, the moduli space of those bundles whose determinant is isomorphic to a fixed line bundle $L$ over $C$. Let $θ_F$ and $θ$ be theta bundles over these two moduli spaces. We prove a simple formula relating their spaces of sections: if $h=\gcd (n,d)$ is the greatest common divisor of $n$ and $d$, and $L\in \Pic ^d(C)$, then $$\dim H^0(\SM (n,L), θ^k) \cdot k^g=\dim H^0(\M(n,d),θ_F^k)\cdot h^g.$$ We also formulate a conjectural duality between these two types of spaces of sections.

alg-geom↗