Eigenvector Mapping for a Free Rigid Body and Some Involutions of the Kummer Surface of a Self-Product Abelian Surface

多羅間大輔

12月16日(金)13：30

This talk deals with an algebro-geometric interpretation of the results in [1].
In [1], it is shown, as one of the main results, that the eigenvector mapping for a free rigid body factors as the composition of two double-coverings through a Kummer surface of a self-product Abelian surface. The eigenvector mapping is regarded as a rational mapping to the projective plane from the product of the integral curve of the Euler equation and the spectral curve associated with the Manakov equation, after a suitable complexification. This result can be proved by using two sequences of double-coverings of the Kummer surface onto $P_2(\mathbb{C})$.
In this talk, it will be shown that these sequences correspond to two sequences of Galois groups generated by some holomorphic involutions of the Kummer surface. Moreover, one can see that these involutions give rise to a Nikulin involution of the Kummer surface. As a result, one has another K3 surface by taking the quotient of the Kummer surface with respect to the Nikulin Involution. The geometric properties of these involutions will be presented as well as their relation to certain elliptic pencils of the surfaces appearing in the above sequences of double-coverings.

Reference
[1] I. Naruki and D. Tarama, Algebraic Geometry of the Eigenvector Mapping for a Free Rigid Body, Diff. Geom. Appl., 29, S1, 2011, S170-S182.