Invariant count of holomorphic disks in the cotangent bundles of the two-sphere and real projective plane
type: Item
instance of: scholarly article
title: Invariant count of holomorphic disks in the cotangent bundles of the two-sphere and real projective plane
zbMATH Open document ID: 1118.53057
author: Jean-Yves Welschinger
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2007-04-13T00:00:00Z
review text: The present note is a further step in the author's theory of enumerative invariants of real symplectic and algebraic varieties [Invent. Math. 162, No. 1, 195--234 (2005; Zbl 1082.14052)]. This time the author considers the \(1\)-neighborhood \(U\) of the two-sphere and of the real projective plane in their cotangent bundles, equipped with the natural symplectic structure and an antisymplectic involution, which keeps the zero-section invariant. The main result is that the count with appropriate weights \(\pm 1\) of pseudo-holomorphic disks with punctures, immersed into \(U\) so that the boundary goes to the zero section and the limits near punctures turn into (multiple) geodesics on \(\partial U\), and matching suitable number of constraints (as passing through fixed points in \(U\) or merging to given geodesics in \(\partial U\)), gives a number independent of the choice of the configuration of the constraints, but determined by only combinatorial data. This result is closely related to the invariant enumeration of real rational curves in the projective plane and the ellipsoid quadric (so-called Welschinger invariants). As a consequence, the author derives some congruences for the Welschinger invariants of the plane and the quadric, as well as proves the negativity of certain Welschinger invariants of these surfaces, whereas all previously computed Welschinger invariants were nonnegative.
Mathematics Subject Classification ID: 53D45
Mathematics Subject Classification ID: 14N10
Mathematics Subject Classification ID: 14N35
Mathematics Subject Classification ID: 14P25
zbMATH DE Number: 5142254
zbMATH Keywords: cotangent bundle
zbMATH Keywords: symplectic manifold
zbMATH Keywords: pseudo-holomorphic disks
zbMATH Keywords: enumerative geometry
reviewed by: Eugenii I. Shustin
MaRDI profile type: Publication
full work available at URL: j.crma.2007.01.005
OpenAlex ID: W2040957643
cites work: Q2739348
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DOI: 10.1016/J.CRMA.2007.01.005
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description: scientific article; zbMATH DE number 5142254