Complex cobordisms and singular manifolds arising from Chern classes

TL;DR

This paper explores the existence and properties of canonical complex cobordism classes associated with singular submanifolds, providing solutions for specific cases and examining their relations to Chern classes and resolution theories.

Contribution

It introduces new solutions for the canonical complex cobordism classes of certain singular submanifolds and analyzes their properties and relations to existing theories.

Findings

Defined complex cobordism classes $Q_r(\xi)$ and $P_r(\xi)$ for singular manifolds.

Showed these classes have properties like deformed sum formula.

Established relations between these classes and $IH$-small resolutions.

Abstract

This paper deals with the question of J.Morava on existence of canonical complex cobordism class of singular submanifold. We present several solutions of this question for $X_r(\xi)$ -- the set of points where $\dim\xi-r+1$ generic sections of a complex vector bundle $\xi$ are linearly dependent. The corresponding complex cobordism classes $Q_r(\xi)$ and $P_r(\xi)$ tend to have many nice properties, such as deformed sum formula, but they don't coincide with Chern classes $c_r^U(\xi)$. They also have relation to the theory of $IH$-small resolutions.