2007/07/29 by Jean-Yves Welschinger, Welschinger, Jean-Yves
Computer Science · Mathematics · #14N35 #14P99 #53D45 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Polynomial and algebraic computation #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.0707.4317
openalex publication_date 2007/07/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper follows a previous one in which were introduced deformation invariants χdr, d ∈ H2 (X ; \Z), r ∈ \N, of closed real symplectic four-manifolds (X, ω, cX), invariants which produced lower bounds in real enumerative geometry. We prove here using methods of symplectic field theory that the lower bounds are sharp when r ≤ 1 and the real locus of the manifold contains a sphere, torus or real projective plane (under stronger assumptions in this last case). We also prove that a big power of two divides χdr as soon as r is not too big and when the real locus contains a sphere or real projective plane (under the same stronger assumptions in this last case). We finally present some explicit computations in the case of the projective plane or quadric ellipsoid surface as well as the general formulas used to get them, formulas which involve some relative invariants that we first define.