通过联络与留数的曲面等变黎曼-罗赫定理
Equivariant Riemann-Roch for Surfaces via Connections and Residues
AI总结：
本文通过联络与留数方法，结合除子剥离和双有理技巧，为有限群作用下的光滑射影曲面上的等变欧拉示性数提供了统一计算程序，并给出ADE奇点修正类及实例验证。
AI中文摘要：
设$X$是$\mathbb{C}$上的光滑射影曲面，具有有限群$G$的有效作用，$L$是$G$-线性化线丛。我们发展了一种基于留数和双有理的方法来计算等变欧拉示性数$\chi_G(X,L)$。每个这样的$L$都有等变除子表示。除子剥离法逐层应用除子正合序列，将整个$G$-轨道分量一起处理。它将问题归结为结构层和低维等变贡献。在$X/G$的一个解消上，等变直像层的自然延拓保持欧拉示性数，但它们诱导的联络不必是对数的，一个显式的$D_4$计算表明了这一点。在取商之前爆破不动点，将局部计算归结为对角作用和循环Hirzebruch-Jung奇点。我们获得了ADE奇点的显式修正类函数，以及包括拟反射在内的一般对角作用的统一程序。在对角情形中，自然延拓是对数的，其留数由Hirzebruch-Jung解消上的单项式赋值计算。将这些留数计算与分支曲线项和除子剥离相结合，给出了计算每个$G$-线性化线丛的等变欧拉示性数的程序。我们将局部类与全纯Lefschetz公式以及Lim和Rota的Kleinian系数进行比较，并给出了$\mathbb{P}^2$上的例子，其中一个例子包含固定曲线。
英文摘要：
Let $X$ be a smooth projective surface over $\mathbb{C}$ with an effective action of a finite group $G$, and let $L$ be a $G$-linearized line bundle. We develop a residue-theoretic and birational approach to computing the equivariant Euler characteristic $χ_G(X,L)$. Every such $L$ admits an equivariant divisor presentation. Divisor peeling applies the divisor exact sequences one layer at a time, treating whole $G$-orbits of components together. It reduces the problem to the structure sheaf and lower-dimensional equivariant contributions. On a resolution of $X/G$, the natural extensions of the isotypic direct-image sheaves preserve Euler characteristics, but their induced connections need not be logarithmic, as an explicit $D_4$ calculation shows. Blowing up fixed points before taking the quotient reduces the local calculation to diagonal actions and cyclic Hirzebruch-Jung singularities. We obtain explicit correction class functions for the ADE singularities and a uniform procedure for general diagonal actions, including quasi-reflections. In the diagonal case, the natural extensions are logarithmic, and their residues are computed from monomial valuations on the Hirzebruch-Jung resolution. Combining these residue calculations with the branch-curve terms and divisor peeling gives a procedure for computing the equivariant Euler characteristic of every $G$-linearized line bundle. We compare the local classes with the holomorphic Lefschetz formula and the Kleinian coefficients of Lim and Rota, and give examples on $\mathbb{P}^2$, including one with fixed curves.
