论文标题
在一类Simplectic $ 4 $ -Orbifolds上，带有消失的规范类别
On a class of symplectic $4$-orbifolds with vanishing canonical class
论文作者
论文摘要
对某些符合性$ 4 $ -Orbifolds的研究开始了一项消失的规范类别。我们表明，对于任何此类符合性$ 4 $ -orbifold $ x $，都有一个规范构造的符号符号$ 4 $ -orbifold $ y $，以及覆盖$ y \ y \ rightarrow x $的环状Orbifold，因此，$ y $在最多孤立的du Val Singularities singularities singularities singularities and trivial orbifold canonical Canonical Canonical Canonical Canonical Canonical Canonical canonical canonical canonelical canonelical canonelical bundle。 $ y $的最低分辨率是由$ \ tilde {y} $表示的，是一种符合性的calabi-yau $ 4 $ - manifold，具有自然的符号有限循环作用，扩展了Orbifold覆盖$ y Y friendarrow x $的甲板转换。此外，我们表明，当$ b_1（x）> 0 $，$ \ tilde {y} $是$ t^2 $ -Bundle af $ t^2 $带有符号纤维的$ t^2 $，而当$ b_1（x）= 0 $，$ \ tilde {y} $是一种积分的家具$ k3 $ k3 $ k3 $ k3 $ tillation $ k3 $ timation $在后一种情况下，$ x $的单数集已完全分类。为了进一步调查$ x $的拓扑，我们引入了一项一般连续的符号吹入程序，这可能具有独立的兴趣。在适当的假设下，该过程使我们能够将给定的符合性有理$ 4 $ -Manifold撤销到$ CP^2 $，在此过程中，我们可以从规范上将符号表面的给定配置转换为$ cp^2 $中的Pseudoholomormorphicerphic curves的“符号排列”。该过程是可逆的；通过一系列连续的爆破顺序，人们可以将符号表面的原始配置恢复到光滑的同位素。
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.
