算术Riemann-Roch定理之应用

The Riemann-Roch theorem in classic algebraic geometry describes the relationship between the Euler-Poincare characteristic of a vector bundle on an algebraic variety and the Todd class of corresponding tangent bundle. After Grothendieck set up the scheme theory of algebraic geometry, the Riemann-Roch theorem has been extended to be a very general form, used to describe the non-commutativity of characteristic maps between algebraic K-theory and suitable cohomology theory with corresponding push-forwards. This project focuses on the arithmetic Riemann-Roch theorem in Arakelov geometry, this theorem introduces some topological and geometric invariants into the computation of the arithmetic Euler-Poincare characteristics of hermitian vector bundles on algebraic varieties over rings of algebraic integers, it has very fruitful applications in number theory and arithmetic algebraic geometry. The contents of this project are two applications of the arithmetic Riemann-Roch theorem, one is the construction and proof of the canonical key formula for projective abelian schemes in Arakelov geometry, another is the arithmetic interpretation and computation of the Yoshikawa invariant of K3 surfaces with an automorphism of order 2, these two applications are both useful for the construction of modular forms.

经典代数几何中的Riemann-Roch定理描述了代数簇上向量丛的Euler-Poincare特征与该代数簇切丛的Todd示性类之间的关系。在Grothendieck建立代数几何概型理论之后，Riemann-Roch定理已经被推广到了非常一般的形式，用于刻画代数K-理论与适当上同调理论之间的特征映射与相应推出映射之间的不交换性。本项目着眼于Arakelov几何中的算术Riemann-Roch定理，该定理将复流形上的一些拓扑和几何不变量引入到了代数整数环上代数簇赋范向量丛算术Euler-Poincare特征的计算当中，在数论和算术代数几何中有着十分丰富的应用。本项目的研究内容为算术Riemann-Roch定理的两个应用，一是Arakelov几何中射影阿贝尔概型典范钥公式的构造和证明，二是带有一个二阶自同构的K3曲面Yoshikawa不变量的算术几何解释与计算，这两个应用均有助于模形式的构造。

算术Riemann-Roch定理是Arakelov几何中的重要理论成果，与代数几何中的情形类似，它描述了K-理论和一些常见的上同调理论之间与陈类等示性类相关的特征映射的重要性质，这些特征映射被看成是特定范畴上函子间的自然变换。Gillet和Soule发展了高维几何的算术Grothendieck-Riemann-Roch定理，Roessler发展了算术Adams-Riemann-Roch定理，Koehler与Roessler发展了算术Lefschetz-Riemann-Roch定理，这些形式的Riemann-Roch定理将一些重要的拓扑和几何不变量联系在一起，在数论和算术几何中具有十分丰富的应用。在本项目中，项目负责人利用算术Adams-Riemann-Roch定理构造和证明了Arakelov几何中射影阿贝尔概型的Moret-Bailly典范钥公式，推广了Moret-Bailly，Chai和Faltings等人的工作，对Maillot和Roessler提出的阿贝尔概型钥公式是否具有典范构造和算术模拟的问题给出了肯定的回答。