arXiv subjects

Martin A. Guest

Publications and source records attributed to Martin A. Guest.

24 records · Page 2Linked to original sources

arXiv2001

Introduction to homological geometry: part II

Although this article can be read independently, it is a continuation of the introduction to integrable systems aspects of quantum cohomology given in part 1 (math.DG/0104274). In the same elementary style, i.e. assuming basic properties of quantum cohomology and concentrating on the simplest nontrivial examples, the quantum differential equations of Givental are studied in some detail. The solutions are described first as generating functions for certain Gromov-Witten invariants, then as generalizations of hypergeometric functions, as predicted by the Mirror Theorem.

math.DG↗

arXiv2001

Introduction to homological geometry: part I

This is an introduction to some of the analytic (or integrable systems) aspects of quantum cohomology which have attracted much attention during the last few years. The small quantum cohomology algebra, regarded as an example of a Frobenius manifold, is described in the original naive manner, without going into the technicalities of a rigorous definition. Then three well known analytic phenomena related to quantum cohomology are reviewed: the Landau-Ginzburg description of cohomology, the phase space of the Toda lattice, and the volume functional. The emphasis is on concrete examples, with the intention of alerting a wider audience to the interesting potential of this area. In part 2, the quantum differential equations will be studied in the same way.

math.DG↗

arXiv2000

Quantum cohomology and the periodic Toda lattice

We describe a relation between the periodic one-dimensional Toda lattice and the quantum cohomology of the periodic flag manifold (an infinite-dimensional Kaehler manifold). This generalizes a result of Givental and Kim relating the open Toda lattice and the quantum cohomology of the finite-dimensional flag manifold. We derive a simple and explicit "differential operator formula" for the necessary quantum products, which applies both to the finite-dimensional and to the infinite-dimensional situations.

math.QA↗

arXiv1998

Pseudo vector bundles and quasifibrations

We prove a topological result concerning the kernel of a morphism d : E --> F of holomorphic vector bundles over a complex analytic space. As a consequence, we show that the projectivization P(ker d) is a quasifibration up to some dimension. We give an application to the Abel-Jacobi map of a Riemann surface, and to the space of rational curves in the symmetric product of a Riemann surface.

math.AG↗

arXiv1994

Configuration spaces and the space of rational curves on a toric variety

The space of holomorphic maps from $S^2$ to a complex algebraic variety $X$, i.e. the space of parametrized rational curves on $X$, arises in several areas of geometry. It is a well known problem to determine an integer $n(D)$ such that the inclusion of this space in the corresponding space of continuous maps induces isomorphisms of homotopy groups up to dimension $n(D)$, where $D$ denotes the homotopy class of the maps. The solution to this problem is known for an important but special class of varieties, the generalized flag manifolds: such an integer may be computed, and $n(D)\to\infty$ as $D\to\infty$. We consider the problem for another class of varieties, namely, toric varieties. For smooth toric varieties and certain singular ones, $n(D)$ may be computed, and $n(D)\to\infty$ as $D\to\infty$. For other singular toric varieties, however, it turns out that $n(D)$ cannot always be made arbitrarily large by a suitable choice of $D$.

math.AG↗

arXiv1994

The topology of the space of rational curves on a toric variety

Let $X$ be a compact toric variety. Let $Hol$ denote the space of based holomorphic maps from $CP^1$ to $X$ which lie in a fixed homotopy class. Let $Map$ denote the corresponding space of continuous maps. We show that $Hol$ has the same homotopy groups as $Map$ up to some (computable) dimension. The proof uses a description of $Hol$ as a space of configurations of labelled points, where the labels lie in a partial monoid determined by the fan of $X$.

alg-geom↗