Spinor states of real rational curves in real algebraic convex 3-manifolds and enumerative invariants
type: Item
instance of: scholarly article
title: Spinor states of real rational curves in real algebraic convex 3-manifolds and enumerative invariants
zbMATH Open document ID: 1084.14056
DOI: 10.1215/S0012-7094-04-12713-7
author: Jean-Yves Welschinger
published in: Duke Mathematical Journal
publication date: 2005-04-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0311466
review text: This paper makes a next step in the creation of the theory of enumerative invariants of real algebraic varieties, initiated by the author. In his pioneering work [Invent. Math. 162, No. 1, 195--234 (2005; Zbl 1082.14052)] the author has defined enumerative invariants of real rational symplectic four-folds, which count real rational pseudo-holomorphic curves, belonging to a given homology class and passing through a real configuration of points, with weights \(\pm 1\), and which depend only on the distribution of real fixed points among the connected components of the real part of the given symplectic variety. In the present paper, the author defines invariants of real convex smooth algebraic 3-folds equipped with a Pin\(^-\) structure. Namely, given such a variety \(X\) and a homology class \(d\in H_2(X)\) such that \(c_1(X)d\) is positive even, the set \(R(\omega)\) of real rational curves passing through a generic real configuration \(\omega\) of \(c_1(X)d/2\) points in \(X\) is finite. Each curve in \(R(\omega)\) gives a loop in the principal \(\text{ O}(3)\)-bundle on \({\mathbb R}X\), and the curve obtains weight \(+1\) or \(-1\) depending on whether the loop lifts to a loop in the respective Pin\(^-\)-bundle on \({\mathbb R}X\) or not. The main theorem states that the total sum of the weights of the curves counted does not depend on the chosen configuration \(\omega\), but only on the distribution of the fixed points among the connected components of \({\mathbb R}X\). As an immediate consequence, the absolute value of the invariant provides a lower bound for the number of real rational curves in the class \(d\) passing through any generic real configuration of \(c_1(X)d/2\) points with a given distribution among the components of \({\mathbb R}X\) (notice that the corresponding Gromov-Witten invariant serves as an upper bound). Another important result says that, in case \(X={\mathbb P}^3\), when a pair or real points in a configuration turns into a pair of ima…
reviewed by: Eugenii I. Shustin
Mathematics Subject Classification ID: 14N35
Mathematics Subject Classification ID: 14P25
Mathematics Subject Classification ID: 53D45
zbMATH DE Number: 2160595
zbMATH Keywords: real algebraic 3-manifolds
zbMATH Keywords: real rational curves
zbMATH Keywords: Pin\(^-\) structure
zbMATH Keywords: real enumerative invariants
zbMATH Keywords: Gromov-Witten invariants
MaRDI profile type: Publication
arXiv ID: math/0311466
cites work: Clifford modules
cites work: Notes on stable maps and quantum cohomology
cites work: Gromov-Witten invariants of blow-ups
cites work: Pseudo holomorphic curves in symplectic manifolds
cites work: Welschinger invariant and enumeration of real plane rational curves
cites work: Q3973298
cites work: Gromov-Witten classes, quantum cohomology, and enumerative geometry
cites work: Q5440951
cites work: Characteristic Classes. (AM-76)
cites work: Vector bundles on complex projective spaces
cites work: A mathematical theory of quantum cohomology
cites work: Q4428722
cites work: Invariants of real rational symplectic 4-manifolds and lower bounds in real enumerative geometry.
cites work: Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry
cites work: Q4012873
OpenAlex ID: W2949409233
description: scientific article