Dolgachev surface

Abstract: A fake projective plane is a compact complex manifold of dimension 2 which has the same Betti numbers as the complex projective plane, but not isomorphic to the complex projective plane. As was shown by D. Mumford, there exists at least one such surface. In this paper we prove the existence of a fake projective plane which is birational to a cyclic cover of degree 7 of a Dolgachev surface.

Abstract: Starting with the Dolgachev surface E(1)2,3 we con- struct an infinite family of distinct exotic copies of the rational surface E(1), each of which admits a handlebody decomposition without 1- and 3- handles, and we draw these handlebodies.

Abstract: Given a 4-manifold X and an imbedding T2 × B2 ⊂ X, we describe an algorithm X ↦ Xp,q for drawing the handlebody of the 4-manifold obtained from X by (p, q)-logarithmic transforms along the parallel tori. By using this algorithm for relatively prime (p, q), we obtain a simple handle picture of the Dolgachev surface E(1)p,q, from that we deduce that the exotic copy of differs from the original one by a codimension zero simply connected Stein submanifold Mp,q. This gives examples of infinitely many small Stein manifolds which are exotic copies of each other (rel boundaries). Also by using the description of S2 × S2 as a union of two cusps glued along their boundaries, and by using this algorithm, we show that a pair of log transforms along the tori in these cusps gives back S2 × S2 or .

Abstract: Given a 4-manifold X and an imbedding of T^{2} x B^2 into X, we describe an algorithm X --> X_{p,q} for drawing the handlebody of the 4-manifold obtained from X by (p,q)-logarithmic transforms along the parallel tori. By using this algorithm, we obtain a simple handle picture of the Dolgachev surface E(1)_{p,q}, from that we deduce that the exotic copy E(1)_{p,q} # 5(-CP^2) of E(1) # 5(-CP^2) differs from the original one by a codimension zero simply connected Stein submanifold M_{p,q}, which are therefore examples of infinitely many Stein manifolds that are exotic copies of each other (rel boundaries). Furthermore, by a similar method we produce infinitely many simply connected Stein submanifolds Z_{p} of E(1)_{p,2} # 2(-CP^2)$ with the same boundary and the second Betti number 2, which are (absolutely) exotic copies of each other; this provides an alternative proof of a recent theorem of the author and Yasui [AY4]. Also, by using the description of S^2 x S^2 as a union of two cusps glued along their boundaries, and by using this algorithm, we show that multiple log transforms along the tori in these cusps do not change smooth structure of S^2 x S^2.