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**Invariants of real symplectic four-manifolds out of reducible and cuspidal curves.**
*(English)*
Zbl 1118.53058

The Gromov-Witten invariants count holomorphic curves in complex algebraic varieties satisfiying various intersection constraints. A ‘real’ break-through occured at MSRI in 2004 in the theory of enumerative real algebraic geometry. This paper is part of that break-through. A real symplectic manifold is a symplectic manifold together with an anti-symplectic involution. The fixed point set of the involution is called the real part of the manifold. Assuming that the real part has \(N\) connected components, one can construct an invariant that counts the real rational curves that meet the \(k\)-th component in \(r_k\) points for \(k=1\dots N\) and represent the homology class \(d\).

In this paper, a similar invariant is constructed from real rational curves that either have: a unique real ordinary cusp, or have exactly two irreducible components, or are tangent to the correct collection of lines. These curves are counted with a sign given by the number of pairs of complex conjugate points in the inverse image of the real part. Gromov-Witten invariants can either be defined using algebraic geometry or elliptic analysis. These cuspidal invariants are defined using analysis. They are independent of the location of the points on the components of the real part and independent of the deformation class of the symplectic structure. For the complex projective plane these invariants agree with the curve counting invariants mentioned at the start of this review for the numbers of marked points in the correct range.

In this paper, a similar invariant is constructed from real rational curves that either have: a unique real ordinary cusp, or have exactly two irreducible components, or are tangent to the correct collection of lines. These curves are counted with a sign given by the number of pairs of complex conjugate points in the inverse image of the real part. Gromov-Witten invariants can either be defined using algebraic geometry or elliptic analysis. These cuspidal invariants are defined using analysis. They are independent of the location of the points on the components of the real part and independent of the deformation class of the symplectic structure. For the complex projective plane these invariants agree with the curve counting invariants mentioned at the start of this review for the numbers of marked points in the correct range.

Reviewer: David Auckly (Manhattan)