The finite group action and the equivariant determinant of elliptic operators

Abstract

Let $M$ be an almost complex manifold and $g$ a periodic automorphism of $M$ of order $p$. Then the rotation angles of $g$ around fixed points of $g$ are naturally defined by the almost complex structure of $M$. In this paper, under the assumption that the fixed points of $g^k$ $(1\leq k\leq p-1)$ are isolated, a calculation formula is provided for the homomorphism $I_D: {\Bbb Z}_p \to {\Bbb R}/{\Bbb Z}$ defined in [8]. The formula gives a new method to study the periodic automorphisms of almost complex manifolds. As examples of the application of the formula, we show the nonexistence of the ${\Bbb Z}_p$-action of specific isotropy orders and examine whether specific rotation angles exist or not.