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.


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


Zbl 1082.14052
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