Invariant count of holomorphic disks in the cotangent bundles of the two-sphere and real projective plane
The present note is a further step in the author's theory of enumerative invariants of real symplectic and algebraic varieties [Invent. Math. 162, No. 1, 195--234 (2005; Zbl 1082.14052)]. This time the author considers the \(1\)-neighborhood \(U\) of the two-sphere and of the real projective plane in their cotangent bundles, equipped with the natural symplectic structure and an antisymplectic involution, which keeps the zero-section invariant. The main result is that the count with appropriate weights \(\pm 1\) of pseudo-holomorphic disks with punctures, immersed into \(U\) so that the boundary goes to the zero section and the limits near punctures turn into (multiple) geodesics on \(\partial U\), and matching suitable number of constraints (as passing through fixed points in \(U\) or merging to given geodesics in \(\partial U\)), gives a number independent of the choice of the configuration of the constraints, but determined by only combinatorial data. This result is closely related to the invariant enumeration of real rational curves in the projective plane and the ellipsoid quadric (so-called Welschinger invariants). As a consequence, the author derives some congruences for the Welschinger invariants of the plane and the quadric, as well as proves the negativity of certain Welschinger invariants of these surfaces, whereas all previously computed Welschinger invariants were nonnegative.
scientific article; zbMATH DE number 5142254
scientific article