Algebraic cobordism revisited

Details Algebraic cobordism revisited Algebraic cobordism revisited

2006-05-08

arxiv.org/abs/math/0605196v1

Mathematics

Algebraic Geometry

62 pages

Scientific paper

We define a cobordism theory in algebraic geometry based on normal crossing degenerations with double point singularities. The main result is the equivalence of double point cobordism to the theory of algebraic cobordism previously defined by Levine and Morel. Double point cobordism provides a simple, geometric presentation of algebraic cobordism theory. As a corollary, the Lazard ring given by products of projective spaces rationally generates all nonsingular projective varieties modulo double point degenerations. Double point degenerations arise naturally in relative Donaldson-Thomas theory. We use double point cobordism to prove all the degree 0 conjectures in Donaldson-Thomas theory: absolute, relative, and equivariant.

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