The Lawson-Yau Formula and its generalization

TL;DR

This paper provides a direct elementary computation and generalization of the Euler characteristic for Chow varieties over any algebraically closed field, including cases with group actions and quaternionic cycles.

Contribution

It introduces a direct elementary method to compute the Euler characteristic and extends the Lawson-Yau formula to the l-adic Euler-Poincare characteristic for broader contexts.

Findings

Computed the Euler characteristic of Chow varieties over any algebraically closed field.

Generalized the Lawson-Yau formula to the l-adic Euler-Poincare characteristic.

Calculated the Euler characteristic for Chow varieties with group actions, including quaternionic cycles.

Abstract

The Euler characteristic of Chow varieties of algebraic cycles of a given degree in complex projective spaces was computed by Blaine Lawson and Stephen Yau by using holomorphic symmetries of cycles spaces. In this paper we compute this in a direct and elementary way and generalize this formula to the l-adic Euler-Poincare characteristic for Chow varieties over any algebraically closed field. Moreover, the Euler characteristic for Chow varieties with certain group action is calculated. In particular, we calculate the Euler characteristic of the space of right quaternionic cycles of a given dimension and degree in complex projective spaces.