Projective toric polynomial generators in the unitary cobordism ring

Abstract

By the classical result of Milnor and Novikov, the unitary cobordism ring is isomorphic to a graded polynomial ring with countably many generators:. In this paper we solve the well-known problem of constructing geometric representatives for a i among smooth projective toric varieties, a n = [X n ], dim C X n = n. Our proof uses a family of equivariant modifications (birational isomorphisms) B k (X) → X of an arbitrary complex manifold X of (complex) dimension n (n 2, k = 0, . . . , n − 2). The key fact is that the change of the Milnor number under these modifications depends only on the dimension n and the number k and does not depend on the manifold X itself.

Quasitoric Totally Normally Split Representatives in the Unitary Cobordism Ring

“…For n odd, let a 0,n := s n ([M 1,n ] + 2[BF n ]), a 1,n−1 = s n ([M 1,n ] + [BF n ]). It follows from (14) and Proposition 3.20 that a 0,n = n + 1. For 0 < i j let a i,j := s i+j ([M i,j ]).…”

“…In this Section the proof of Theorem 1.3 is given. We use standard facts from unitary cobordism theory (for example, see [14]). An auxiliary statement from Number Theory is needed (see [6] for the proof).…”

“…Then for any prime p one has: Proof. If s = 1, then the desired manifold is N 0,n with Milnor number equal to n + 1 = p. Otherwise, for p = 2 we take BF n with Milnor number equal to 2 (see (14)). Now suppose s 2, p > 2.…”

“…manifolds (non-singular varieties, resp.). (See, for example, [14].) The natural complex structure on the bounded flag bundle BF (ξ n+1 , .…”

“…I use Segre class (i.e. the multiplicatively inverse to Chern class, see [14]) to compute the summands in the above expression:…”