Towards relative invariants of real symplectic 4-manifolds

Abstract

Let $(X,\omega ,c_X)$ be a real symplectic 4-manifold with real part $RX$. Let $L\subset RX$ be a smooth curve such that $[L]=0\in H_1(RX;Z/2Z)$. 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.

21 pages, 8 figures

References in corpus (1)

• Invariants of real symplectic 4-manifolds out of reducible and cuspidal curves

Cited by in corpus (1)

• Invariants of real symplectic 4-manifolds out of reducible and cuspidal curves