Stein structures not determined by the contact boundary

Abstract

We construct two Stein structures on the same compact smooth four-manifold whose boundary contact forms are strictly identified and whose first Chern classes agree, but whose exact symplectic forms are not symplectomorphic. Moreover, no diffeomorphism makes the associated Weinstein structures homotopic. The examples are obtained by removing a tubular neighborhood of a smooth bicanonical curve from a fake projective plane and comparing the induced Stein structure with its conjugate. Their canonical Spinc structures differ by a nonzero class of order two. After the boundary forms are normalized using a common Boothby-Wang connection form, a hypothetical symplectomorphism preserves the oriented circle fiber and extends over the divisor cap, contradicting canonical-class rigidity. For the Weinstein statement, the required fiber-preservation is obtained from a monopole Floer grading asymmetry, using the Nelson-Weiler computation and Taubes's ECH-Seiberg-Witten correspondence. This gives an affirmative solution to a normalized version of the relative filling problem following Problem 4.96 in the K3 problem list.