Surfaces of general type with $p_g=1$ and $q=0$

Park, Heesang; Park, Jong-Il; Shin, Dong Yun

doi:10.48550/arxiv.0906.5195

Cited by 4 publications

(12 citation statements)

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“…This paper is an addendum to the authors' work [5], in which we constructed a family of minimal complex surfaces of general type with p g = 1, q = 0, and 1 ≤ K 2 ≤ 2 and simply connected surfaces with p g = 1, q = 0, and 3 ≤ K 2 ≤ 6 using a Q-Gorenstein smoothing theory. We extend the results to the K 2 = 8 case in this paper.…”

Section: Introductionmentioning

confidence: 99%

A simply connected surface of general type withp_g= 0 andK²= 3

Park

Shin

2009

Geom. Topol.

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Recently, the second author [18] constructed a simply connected symplectic 4-manifold with b C 2 D 1 and K 2 D 2 using a rational blow-down surgery, and then Y Lee and the second author [9] constructed a family of simply connected, minimal, complex surfaces of general type with p g D 0 and 1 Ä K 2 Ä 2 by modifying Park's symplectic 4-manifold. After this construction, it has been a natural question whether one can find a new family of surfaces of general type with p g D 0 and K 2 3 using the same technique.The aim of this article is to give an affirmative answer to this question. Precisely, we are able to construct a simply connected, minimal, complex surface of general type with p g D 0 and K 2 D 3 using a rational blow-down surgery and a Q-Gorenstein smoothing theory developed in Lee and Park [9]. The key ingredient for the construction of K 2 D 3 case is to find a rational surface Z which contains several disjoint chains of curves representing the resolution graphs of special quotient singularities. Once we have the right candidate Z for K 2 D 3, the remaining argument is parallel to that of the K 2 D 2 case which appeared in Lee and Park [9]. That is, we contract these chains of curves from the rational surface Z to produce a projective surface X with special quotient singularities. We then prove that the singular surface X has a Q-Gorenstein smoothing and the general fiber X t of the Q-Gorenstein smoothing is a simply connected minimal surface of general type with p g D 0 and K 2 D 3. The main result of this article is the following. Theorem 1.2 There exists a simply connected symplectic 4-manifold with b C 2 D 1 and K 2 D 4 which is homeomorphic, but not diffeomorphic, to a rational surfaceIt is a very intriguing question whether the symplectic 4-manifold constructed in Theorem 1.2 above admits a complex structure. One way to approach this problem is to use Q-Gorenstein smoothing theory as above. But since the cohomology H 2 .T 0 X / is not zero in this case, it is hard to determine whether there exists a Q-Gorenstein smoothing. Therefore we need to develop more Q-Gorenstein smoothing theory in order to investigate the existence of a complex structure on the symplectic 4-manifold constructed in Theorem 1.2. We leave this question for future research. Q-Gorenstein smoothingIn this section we briefly review a theory of Q-Gorenstein smoothing for projective surfaces with special quotient singularities and we quote some basic facts developed in Lee and Park [9].Geometry & Topology, Volume 13 (2009)A simply connected surface of general type with p g D 0 and K 2 D 3 745

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Section: Introductionmentioning

confidence: 99%

A simply connected surface of general type withp_g= 0 andK²= 3

Park

Shin

2009

Geom. Topol.

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“…By using the double covering E(2) → E(1), together with the methods developed in [15], one can produce simply connected minimal surfaces of general type with p g = 1 and q = 0. For example, such surfaces with 1 ≤ K 2 ≤ 6 are constructed in [22].…”

Section: The Case Of Elliptic K3 Surfaces With a Sectionmentioning

confidence: 99%

Construction of surfaces of general type from elliptic surfaces via $${\mathbb{Q}}$$ -Gorenstein smoothing

2012

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“…Recent years have witnessed a drastic increase in our understanding of the topology and geometry of 4-manifolds and complex surfaces. The newest developments can be exemplified by the construction of simply connected surfaces of general type with small topology [22,23], by the unveiling of a myriad of exotic smooth structures on small 4-manifolds [1], and by how these manifolds have provided an adequate environment for the study of fundamental questions in Riemannian geometry that were previously out of reach.…”

Section: Introductionmentioning

confidence: 99%

“…

In this paper, first we consider the existence and non-existence of Einstein metrics on the topological 4-manifolds 3CP 2 #kCP 2 (for k ∈ {11, 13, 14, 15, 16, 17, 18}) by using the idea of [24] and the constructions in [23]. Then, we study the existence or non-existence of non-singular solutions of the normalized Ricci flow on the exotic smooth structures of these topological manifolds by employing the obstruction developed in [13].

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confidence: 99%

On Einstein metrics, normalized Ricci flow and smooth structures on $3\mathbb{CP}^2 # k \bar{\mathbb{CP}}^2$

Torres

2010

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Surfaces of general type with $p_g=1$ and $q=0$

Cited by 4 publications

References 7 publications

A simply connected surface of general type withp_g= 0 andK²= 3

A simply connected surface of general type withp_g= 0 andK²= 3

Construction of surfaces of general type from elliptic surfaces via $${\mathbb{Q}}$$ -Gorenstein smoothing

On Einstein metrics, normalized Ricci flow and smooth structures on $3\mathbb{CP}^2 # k \bar{\mathbb{CP}}^2$

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Surfaces of general type with $p_g=1$ and $q=0$

Cited by 4 publications

References 7 publications

A simply connected surface of general type withpg= 0 andK2= 3

A simply connected surface of general type withpg= 0 andK2= 3

Construction of surfaces of general type from elliptic surfaces via $${\mathbb{Q}}$$ -Gorenstein smoothing

On Einstein metrics, normalized Ricci flow and smooth structures on $3\mathbb{CP}^2 # k \bar{\mathbb{CP}}^2$

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A simply connected surface of general type withp_g= 0 andK²= 3

A simply connected surface of general type withp_g= 0 andK²= 3