Involutions and Chern numbers of varieties

Abstract

Consider an involution of a smooth projective variety over a field of characteristic not two. We look at the relations between the variety and the fixed locus of the involution from the point of view of cobordism. We show in particular that the fixed locus has dimension larger than its codimension when certain Chern numbers of the variety are not divisible by two, or four. Some of those results, but not all, are analogues of theorems in algebraic topology obtained by Conner-Floyd and Boardman in the sixties. We include versions of our results concerning the vanishing loci of idempotent global derivations in characteristic two. Our approach to cobordism, following Merkurjev's [Mer02], is elementary, in the sense that it does not involve resolution of singularities or homotopical methods.

On the algebraic cobordism ring of involutions

“…The conclusion of the lemma means that P H (E) has vanishing b α -coefficient when α has length > r. It follows from (1.2.3.i) that whenever 0 → E 1 → E 2 → E 3 → 0 is an exact sequence of constant rank vector bundles, the lemma holds for E = E 2 if it holds for both E = E 1 and E = E 3 . By the splitting principle (see [Hau,(2.1.16)]), we may thus assume that r = 1, in which case the statement follows from (1.2.3.ii).…”

“…Let now t : X → E be a regular section whose zerolocus is Z. By homotopy invariance we have t * = s * , and the second statement follows from the first and the transversality axiom [Hau,(2.1.3.vii)].…”

“…Let X ∈ Sm k . For any n ∈ N, it follows from the transversality axiom [Hau,(2.1.3.v)] and [Hau,(2.1.9)] that c 1 (O(1)) ∈ H(P n × X) is the class of a linearly embedded P n−1 × X (we write P −1 = ∅). In particular c 1 (O(1)) n+1 = 0, and by the projective bundle axiom [Hau,(2.1.3.vi)], mapping x to c 1 (O(1)) induces a ring isomorphism…”

“…Proof. The first statement is true by definition when r = 1 (see [Hau, (2.1.9)]), and follows in general from the splitting principle [Hau,(2.1.16)] and the transversality axiom [Hau,(2.1.3.v)] in general. Let now t : X → E be a regular section whose zerolocus is Z.…”

“…A similar notation will be used with other sets of variables (such as a i , X i , or x i ). [Hau,(3.1.2), (3.1.4)].) For each vector bundle E over X ∈ Sm k , there is a class P H (E) ∈ H(X)[b 1 , .…”