Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry

Abstract

:We first present the construction of the moduli space of real pseudo-holomorphic curves in a given real symplectic manifold. Then, following the approach of Gromov and Witten [3, 15, 10], we construct invariants under deformation of real rational symplectic 4-manifolds. These invariants provide lower bounds for the number of real rational J-holomorphic curves in a given homology class passing through a given real configuration of points.

Spinor states of real rational curves in real algebraic convex 3-manifolds and enumerative invariants

“…How to compute the invariant χ d,p r ? See [13] for a discussion of related problems in real enumerative geometry and [5] for an estimation of the similar invariants constructed in [14], [15].…”

Recursive Formulas for Welschinger Invariants of the Projective Plane

“…However the number obtained is no longer an invariant (see [Wel05a] or [IKS09]). See section 7.2 for this discussion in the tropical setting.…”

“…Welschinger invariants of symplectic 4-manifolds were introduced in [Wel03] (see also [Wel05a]), and since then they are attracting a lot of attention. One of the interests of these invariants is that they provide lower bounds in real enumerative geometry.…”

A Caporaso–Harris type formula for Welschinger invariants of real toric Del Pezzo surfaces

“…(In fact, the listed properties of R(D, ω) can be taken here as a definition of the term 'generic'.) Due to the Welschinger theorem [10,11] (and the genericity of the complex structure on Σ), the number…”

“…Welschinger invariants of real rational symplectic four-manifolds [10,11] represent one of the most interesting and intriguing objects in real enumerative geometry. In the case of a real unnodal (i.e., not containing any rational (−n)-curve, n ≥ 2) Del Pezzo surface Σ the Welschinger invariants count, with appropriate weights ±1, the real rational curves which belong to an ample linear system |D| and pass through a given generic conjugation-invariant set of c 1 (Σ) · D − 1 points in Σ.…”