A simply connected surface of general type with p g =0 and K 2=2

Abstract

In this paper we construct a simply connected, minimal, complex surface of general type with pg = 0 and K 2 = 2 using a rational blow-down surgery and a Q-Gorenstein smoothing theory.In this section we develop a theory of Q-Gorenstein smoothing for projective surfaces with special quotient singularities, which is a key technical ingredient in our result.Definition. Let X be a normal projective surface with quotient singularities. Let X → ∆ (or X /∆) be a flat family of projective surfaces over a small disk ∆. The one-parameter family of surfaces X → ∆ is called a Q-Gorenstein smoothing of X if it satisfies the following three conditions; (i) the general fiber X t is a smooth projective surface,

Smooth structures and Einstein metrics on

“…For each such k, the natural smooth structure as the complex projective plane blown-up at k generic points, can be endowed [Ti90] by a Kähler-Einstein metric of positive scalar curvature. On the other hand, Lee and Park [LePa07] and more recently Park, Park and Shin [PPS07] constructed new exotic smooth structures on CP 2 #kCP 2 , for any 6 ≤ k ≤ 8. Moreover, these admit complex structures yielding interesting examples of minimal surfaces of general type.…”

“…A common feature of all of the examples appearing in [LePa07,PPS07] is that the pullback of the canonical divisor of the singular variety X to its minimal resolution is effective. For instance, in the example described above, we can write:…”

“…Moreover, the method proposed by J. Park [Pa05] was later refined [LePa07,PPS07] to produce interesting examples of minimal complex surfaces of general type. In this paper we show how we can use these constructions in regard to the existence or non-existence of Einstein metrics.…”

“…Based on ideas from [Pa05], Lee and Park [LePa07] and more recently Park, Park, and Shin [PPS07] constructed new examples of simply connected, minimal surfaces of general type with p g = 0, K 2 = 1, 2 or 3. We show that their examples satisfy the ampleness condition: Theorem 1.1.…”

Symplectic surgeries and normal surface singularities

“…Notice that we do not require the singular point to have a holomorphic model in X sing as in McCarthy and Wolfson [14] (where the analytic structure near the singular point is also assumed to be modeled by the holomorphic situation) -we just require the existence of a diffeomorphism. For "globalizing" local deformations in the holomorphic category in a similar context, see Lee and Park [12].…”