Almost Complex Structures on Complex Projective Spaces

Abstract

In this paper we classify the almost complex structures on a complex projective space as roots of a certain polynomial equation.Introduction.A real oriented vector bundle E is said to admit an almost complex structure if there is a complex vector bundle F such that £-F as oriented vector bundles.An almost complex structure on F is a (complex) isomorphism class of such bundles F. When X is a 27?-dimensional oriented manifold and TX is its tangent bundle, an almost complex structure on TX is called an almost complex structure on X, and X together with an almost complex structure is called an almost complex manifold.A special case of this is when X is a complex manifold with complex tangent bundle Jx.Since (jX)R = TX, X always admits an almost complex structure.In this paper we study the generic complex manifolds, the complex projective spaces, and classify their almost complex structures as roots of a certain polynomial equation.In this context we give an answer to a question of Hirzebruch [7] posed in 1954 concerning which elements of cohomology arise as Chern classes of almost complex structures.The techniques used to solve this problem apply equally well in other cases (for example to Uin), S ", G2(C") = Grassmannian of 2-planes in C", S2m + l x Lnip)-wheie Lnip) is the lens space and p is sufficiently large, various products of these, etc.) and the details and calculations in these cases will appear later.For example, one can easily show that S " x S m* admits precisely one almost complex structure.(In fact this is the complex tangent bundle of the complex structure defined by Calabi and Eckmann [9].)1. Definitions and statement of main theorems.Let Zm be the product of 77? copies of the integers as an abelian group.We consider Zm C Zm+1 by the inclusions («x, •. •, am) H* (aj,.. •, am, 0), and put Z°°= IJ , Z™.For each 77?, let p ' Z -» Z be projection onto the 777th coordinate.