Symplectic structures of algebraic surfaces and deformation

Abstract

Friedman and Morgan made the "speculation" that deformation equivalence and diffeomorphism should coincide for algebraic surfaces. Counterexamples, for the hitherto open case of surfaces of general type, have been given in the last years by Manetti, by Kharlamov-Kulikov and in my cited article. For the latter much simpler examples, it was shown that there are surfaces $S$ which are not deformation equivalent to their complex conjugate. However, by Seiberg-Witten theory, any (oriented) diffeomorphism of minimal surfaces carries the canonical class K to + K or to - K, and deformation equivalence implies the existence of a diffeomorphism carrying K to +K. In fact, as observed by a referee, the bulk of the proof was to show that our surfaces have no selfhomeomorphism carrying K to - K (the same for the K-K surfaces). In this note we show that Manetti's surfaces provide indeed a counterexample to the reinforced conjecture, since they are symplectomorphic. Our result is that a surface of general type has a canonical symplectic structure (up to symplectomorphism) which is invariant for deformation and for certain degenerations to normal surfaces. Since moreover no simply connected counterexamples to the conjecture are known, we provide explicit families of 1-connected surfaces, which are obtained by glueing together two fixed manifolds with boundary, are not deformation equivalent, but are homeomorphic under a homeomorphism carrying K to +K. We also give as application the existence of symplectically equivalent, but not deformation equivalent cuspidal plane curves.