×

Towards relative invariants of real symplectic four-manifolds. (English) Zbl 1107.53059

Summary: Let \((X,\omega,c_X)\) be a real symplectic four-manifold with real part \(\mathbb{R}X\). Let \(L \subset \mathbb{R}X\) be a smooth curve such that \([L] = 0 \in H_1 (\mathbb{R}X;\mathbb{Z}/2\mathbb{Z})\). We construct invariants under deformation of the quadruple \((X,\omega,c_X,L)\) by counting the number of real rational \(J\)-holomorphic curves which realize a given homology class \(d\), pass through an appropriate number of points and are tangent to L. As an application, we prove a relation between the count of real rational \(J\)-holomorphic curves done in [J.-Y. Welschinger, Invent. Math. 162, No. 1, 195–234 (2005; Zbl 1082.14052)] and the count of reducible real rational curves done in [J.-Y. Welschinger, Bull. Soc. Math. Fr. 134, No. 2, 287–325 (2006)]. Finally, we show how these techniques also allow us to extract an integer valued invariant from a classical problem of real enumerative geometry, namely about counting the number of real plane conics tangent to five given generic real conics.

MSC:

53D35 Global theory of symplectic and contact manifolds
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14P99 Real algebraic and real-analytic geometry
32Q65 Pseudoholomorphic curves

Citations:

Zbl 1082.14052
PDF BibTeX XML Cite
Full Text: DOI arXiv