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Blowdowns and McKay correspondence on four dimensional quasitoric orbifolds

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Ganguli, S. and Poddar, M. Osaka J. Math. 50 (2013), 397–415 BLOWDOWNS AND MCKAY CORRESPONDENCE ON FOUR DIMENSIONAL QUASITORIC ORBIFOLDS SAIBAL GANGULI and MAINAK PODDAR (Received December 1, 2010, revised August 18, 2011) Abstract We prove the existence of torus invariant almost complex structure on any posi- tively omnioriented four dimensional primitive quasitoric orbifold. We construct pseudo-holomorphic blowdown maps for such orbifolds. We prove a version of McKay correspondence when the blowdowns are crepant. 1. Introduction Quasitoric orbifolds are generalizations or topological counterparts of simplicial pro- jective toric varieties. They admit an action of the real torus of half dimension such that the orbit space has the combinatorial type of a simple convex polytope. Davis and Januskiewicz [4], who introduced the notion of quasitoric space, showed that the for- mula for the cohomology ring of a quasitoric manifold, and hence of any nonsingular projective toric variety, may be deduced by purely algebraic topology methods. This was generalized to the orbifold case in [10]. In general quasitoric manifolds do not have integrable or almost complex structure. However, they always have stable almost complex structure. Moreover, positively omni- oriented quasitoric manifolds have been known to have an almost complex structure, see [2]. It was recently proved by Kustarev [6, 7] that any positively omnioriented quasi- toric manifold has an almost complex structure which is torus invariant. We extend his result to four dimensional primitive quasitoric orbifolds, see Theorem 3.1. Note that for higher dimensional positively omnioriented quasitoric orbifolds the existence of an al- most complex structure, torus invariant or otherwise, remains an open problem. We hope to address this in future. Inspired by birational geometry of toric varieties, we introduce the notion of blow- down into the realm of quasitoric orbifolds. Our blowdown maps contract an embedded orbifold sphere (exc

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