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Pré-Publication, Document De Travail Année : 2005

Towards relative invariants of real symplectic 4-manifolds

Résumé

Let $(X, \omega, c_X)$ be a real symplectic 4-manifold with real part $R X$. Let $L \subset R X$ be a smooth curve such that $[L] = 0 \in H_1 (R X ; Z / 2Z)$. 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 math.AG/0303145 and the count of reducible real rational curves done in math.SG/0502355. Finally, we show how these techniques also allow 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.

Dates et versions

hal-00012421 , version 1 (22-10-2005)

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Jean-Yves Welschinger. Towards relative invariants of real symplectic 4-manifolds. 2005. ⟨hal-00012421⟩

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