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Let$M$be a smooth hypersurface of constant signature in$CP$^{$n$},$n$≥3. We prove the regularity foron$M$in bidegree (0,1). As a consequence, we show that there exists no smooth hypersurface in$CP$^{$n$},$n$≥3, whose Levi form has at least two zero-eigenvalues.

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We derive a formula for the virtual class of the moduli space of rubber maps to [P^1/G] pushed forward to the moduli space of stable maps to BG. As an application, we show that the Gromov-Witten theory of [P^1/G] relative to 0 and ∞ are determined by known calculations.

We consider surfaces M2 immersed in En c × R, where En c is a simply connected n-dimensional complete Riemannian manifold with constant sectional curvature c 6= 0, and assume that the mean curvature vector of the immersion is parallel in the normal bundle. We consider further a Hopf-type complex quadratic form Q on M2, where the complex structure of M2 is compatible with the induced metric. It is not hard to check that Q is holomorphic (see [3], p.289). We will use this fact to give a reasonable description of immersed surfaces in En c ×R that have parallel mean curvature vector.