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

By Jean-Yves Welschinger. 2003

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We first build the moduli spaces of real rational pseudo-holomorphic curves in a given real symplectic 4-manifold. Then, following the approach of Gromov and Witten [3, 19, 11], we define invariants under deformation of real symplectic 4-manifolds. These invariants provide lower bounds for the number of real rational J-holomorphic curves which realize a given homology class and pass through a given real configuration of points.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ca0f736c825658dad02c2c4dc0602af4163d4db5

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Citations in the corpus

Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions (2006, background): “In [20, 21, 22], for strongly semi-positive real symplectic manifolds, Welschinger defined invariants counting real rational J-holomorphic curves intersecting a generic 0-invariant collection of marked points.”
A Caporaso-Harris type formula for Welschinger invariants of real toric Del Pezzo surfaces (2006, background): “) Due to the Welschinger theorem [10, 11] (and the genericity of the complex structure on Σ), the number”
Labeled floor diagrams for plane curves (2009, background): “Theorem 8.1 was recently used in [2] to obtain a Caporaso–Harris-type recurrence for Welschinger invariants.”
Welschinger invariants of real Del Pezzo surfaces of degree ≥ 3 (2011, background): “They were introduced in [24], [25] and count, with appropriate signs, the real rational pseudo-holomorphic curves which pass through given real collections of points in a given real rational symplectic four-fold.”
On Block-Goettsche multiplicities for planar tropical curves (2012, methodology): “It was shown in [19] that the number of rational real curves passing through a generic configuration of 3d− 1 points and enhanced with these signs does not depend on the choice of configuration.”
Floor decompositions of tropical curves : the planar case (2008, background): “However, Welschinger proved in [Wel05] that when g = 0 and Tor(∆) is Fano, one can define an invariant.”
Welschinger invariant and enumeration of real rational curves (2003, background): “formation invariance of Welschinger numbers (see [ 16 , 17]), Theorem 1.1 extends to the”
Logarithmic equivalence of Welschinger and Gromov-Witten invariants (2004, background): “Welschinger [ 10 , 11] of a way to attribute weights of ±1 to real solutions so that the number of solutions counted with weights is independent of the configuration of points; more precisely, a real collection w may even contain complex conjugated pairs of points, and then the result depends only o”
Quantum Structures for Lagrangian Submanifolds (2007, result): “, x5? It would of course be also interesting to understand the relation of the above to the works of Welschinger [68, 69, 70] and of Solomon [65].”
A tropical calculation of the Welschinger invariants of real toric Del Pezzo surfaces (2004, background): “One of the important questions of real enumerative geometry is: for a real algebraic surface Σ, how many real rational curves in an ample linear system |D| pass through a conjugation-invariant configuration of −KΣD − 1 distinct generic points in Σ? Similarly to the complex case, where the answer is ”
Refined curve counting on complex surfaces (2012, background): “The analogues of the Gromov–Witten invariants in real algebraic geometry are the Welschinger invariants [72].”
Brief introduction to tropical geometry (2015, background): “He proved that the result is invariant on the choice of points in general position in the case g = 0 (see [Wel05]).”
Disk enumeration on the quintic 3-fold (2006, background): “t of standard Gromov-Witten invariants in many respects. In good situations, the invariants obtained from Mf Σ(X/L,d) are actually enumerative. For example, Welschinger’s signed counts of real curves [18, 19] arise as specializations of the theory [15]. 1.3. Conventions. 1.3.1. Coordinates. Let z0,.”
Enumeration of curves via floor diagrams (2007, unknown): “Nevertheless, Welschinger proved in [7], [8] that in the case when n = 2, 3, g = 0 and lj = 0 for any j > 0, there is a consistent choice of signs ±1 such that the number of the corresponding real curves counted with the sign depends only on d (note that the assumption lj = 0 for j > 0 allows us to ”