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A simple characterization is given of open subsets of a complex surface that smoothly perturb to Stein open subsets. As applications, ℂ2 contains domains of holomorphy (Stein open subsets) that are exotic smoothings of ℝ4, and others homotopy equivalent to S2 but cut out by smooth, compact 3-manifolds. Pseudoconvex embeddings of Brieskorn spheres and other 3-manifolds into complex surfaces are constructed, as are pseudoconcave holomorphic fillings (with disagreeing contact and boundary orientations). Pseudoconcave complex structures on Milnor fibers are found. A byproduct of this construction is a simple polynomial expression for the signature of the (p, q, npq−1) Milnor fiber. Akbulut corks in complex surfaces can always be chosen to be pseudoconvex or pseudoconcave submanifolds. The main theorem is expressed via Stein handlebodies (possibly infinite), which are defined holomorphically in all dimensions by extending Stein theory to manifolds with non-compact boundary. [ABSTRACT FROM AUTHOR]

We compute local Gromov–Witten invariants of cubic surfaces at all genera. We use a deformation a of cubic surface to a nef toric surface and the deformation invariance of Gromov–Witten invariants. [ABSTRACT FROM AUTHOR]