2020/05/10 by Chen Weimin, Chen, Weimin
Mathematics · #57S17 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Primary 57R18 #Secondary 57R17 #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2005.04688
openalex publication_date 2020/05/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A study of certain symplectic 4-orbifolds with vanishing canonical class is initiated. We show that for any such symplectic 4-orbifold X, there is a canonically constructed symplectic 4-orbifold Y, together with a cyclic orbifold covering Y→ X, such that Y has at most isolated Du Val singularities and a trivial orbifold canonical line bundle. The minimal resolution of Y, to be denoted by Y, is a symplectic Calabi-Yau 4-manifold endowed with a natural symplectic finite cyclic action, extending the deck transformations of the orbifold covering Y→ X. Furthermore, we show that when b1(X)>0, Y is a T2-bundle over T2 with symplectic fibers, and when b1(X)=0, Y is either an integral homology K3 surface or a rational homology T4; in the latter case, the singular set of X is completely classified. To further investigate the topology of X, we introduce a general successive symplectic blowing-down procedure, which may be of independent interest. Under suitable assumptions, the procedure allows us to successively blow down a given symplectic rational 4-manifold to CP2, during which process we can canonically transform a given configuration of symplectic surfaces to a "symplectic arrangement" of pseudoholomorphic curves in CP2. The procedure is reversible; by a sequence of successive blowing-ups in the reversing order, one can recover the original configuration of symplectic surfaces up to a smooth isotopy.