On a class of symplectic $4$-orbifolds with vanishing canonical class

Abstract

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 $Y\rightarrow X$, 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 $\tilde{Y}$, is a symplectic Calabi-Yau $4$-manifold endowed with a natural symplectic finite cyclic action, extending the deck transformations of the orbifold covering $Y\rightarrow X$. Furthermore, we show that when $b_1(X)>0$, $\tilde{Y}$ is a $T^2$-bundle over $T^2$ with symplectic fibers, and when $b_1(X)=0$, $\tilde{Y}$ is either an integral homology $K3$ surface or a rational homology $T^4$; 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 $CP^2$, during which process we can canonically transform a given configuration of symplectic surfaces to a "symplectic arrangement" of pseudoholomorphic curves in $CP^2$. 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.