论文标题
数值和kodaira尺寸
Numerical and kodaira dimensions of cotangent bundles
论文作者
论文摘要
我们推测数值和kodaira尺寸的平等$ν_1^*（x）$和$κ_1^*（x）$用于紧凑型kähler歧管$ x $的cotangent束，从而概括了规范束的经典案例。我们将其显示或将其显示为某些特殊流形的规范捆绑包的经典案例：其中，与KLT奇异性和琐碎的首个Chern阶级的合理连接或各种分辨率的分辨率，在那时我们显示$ν_1^**（x）=κ__1^*（x）=κ__1^*（x）= q'（x） $ x $的有限étale封面的不规则性。证明基于Beauville-Bogomolov的分解，并直接计算有限组的复杂托里的平滑模型$ a/g $。我们推测，当$ x $是“特殊”时，这些平等性成真，更普遍。 Fumio Sakai在[43]中已经引入和研究了不变的$κ_1^*$，当$κ_1^*（x）= - dim（x）$时，在[29]中介绍并研究了上述猜想的特定情况。
We conjecture the equality of the numerical and Kodaira dimensions $ν_1^*(X)$ and $κ_1^*(X)$ for the cotangent bundle of compact Kähler manifolds $X$, generalising the classical case of the canonical bundle. We show or reduce it to the classical case of the canonical bundle for some peculiar manifolds: among them, the rationally connected ones, or resolutions of varieties with klt singularities and trivial first Chern class, in which case we show that $ν_1^*(X)=κ_1^*(X)=q'(X)-dim(X)$, where $q'(X)$ is the maximal irregularity of a finite étale cover of $X$. The proof rests on the Beauville-Bogomolov decomposition, and a direct computation for smooth models of quotients $A/G$ of complex tori by finite groups. We conjecture that these equalities hold true, much more generally, when $X$ is `special'. The invariant $κ_1^*$ was already introduced and studied by Fumio Sakai in [43], the particular case of the preceding conjecture when $κ_1^*(X)=-dim(X)$ was introduced and studied in [29].
