## 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

Zbl 1082.14052
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