Surgery of real symplectic fourfolds and Welschinger invariants

Journal of Singularities
volume 17 (2018), 267-294

Received: 13 March 2018. Received in revised form: 17 July 2018.

DOI: 10.5427/jsing.2018.17l

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Abstract:

A surgery of a real symplectic manifold X_R along a real Lagrangian sphere S is a modification of the symplectic and real structure on X_R in a neighborhood of S. Genus 0 Welschinger invariants of two real symplectic 4-manifolds differing by such a surgery have been related. In the present paper, we explore some particular situations where general formulas greatly simplify. As an application, we reduce the computation of genus 0 Welschinger invariants of all del Pezzo surfaces to the cases covered in earlier works, and of all R-minimal real conic bundles to the cases covered by others. As a by-product, we establish the existence of some new relative Welschinger invariants. We also generalize earlier results to the enumeration of curves of higher genus, and give relations between hypothetical invariants defined in the same vein as previous works.

Real enumerative geometry, Welschinger invariants, symplectic sum

Mathematical Subject Classification (2010):

Primary 14P05, 14N10; Secondary 14N35, 14P25

Author(s) information:

Erwan Brugallé

CMLS, École polytechnique
CNRS, Université Paris-Saclay
91128 Palaiseau Cedex, France

Université de Nantes
Laboratoire de Mathématiques Jean Leray
2 rue de la Houssinière
F-44322 Nantes Cedex 3, France

email: erwan.brugalle@math.cnrs.fr