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

We give a new realization of crystal bases for finite-dimensional irreducible modules over special linear Lie algebras using the monomials introduced by H. Nakajima. We also discuss the connection between this monomial realization and the tableau realization given by Kashiwara and Nakashima.

Abstract. In this paper, we introduce the notion of abstract crystals for quantum generalized Kac-Moody algebras and study their fundamental properties. We then prove the crystal embedding theorem and give a characterization of the crystals B(∞) and B(λ).

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The Ba/Si(111) surface, previously known as a 3 x 1 phase, is found to have a 3 x 2 periodicity and a semiconducting band gap. The substrate reconstructs into the honeycomb chain-channel (HCC) structure with Ba atoms in the channel, as in the alkali-metal-induced Si(111)-(3 x 1). However, the metal coverage is determined to be 1/6 monolayers, half the alkali-metal coverage. We propose that the structure and the metal coverage determined for the Ba adsorbate is universal for other alkaline-earth-metal adsorbates. With the alkali-metal-induced 3 x 1 case, our results lead to a rule that one donated electron per 3 x 1 surface unit is necessary to stabilize the HCC reconstruction of Si.