A Simple Proof of the Formula for the Blowing up of Chern Classes

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics. Notation and Conventions. All varieties involved are quasi-projective varieties over an algebraically closed field of arbitrary characteristic. The equivalence relation used is the rational equivalence in Chow rings.Let E be the normal bundle (of rank r=codimx Y) N(Y,X) of Y in X. Then Y' is a copy of the projective bundle P(E) whose projection is PE and we Manuscript

A formula for the Chern classes of symplectic blow-ups

“…We regard N as a submanifold of M and interpret the embedding i : N → M as an inclusion map. The name of the following lemma (and several other results below) derives from the corresponding statement in intersection theory as proved (or quoted) in [7] or [9]. Proof.…”

“…The arguments in [9] can be transcribed to the topological setting without any particular difficulties, except perhaps for the part that appears in the present paper as Lemma 12. Those arguments, however, rely in an essential way on several lemmas from [7], giving relations in the Chow rings of various spaces under consideration.…”

Chern classes of blow-ups

“…In [9], Hansjörg Geiges and Federica Pasquotto extend the classic blow-up formula of Section 1·1 to the case of symplectic, complex, and real manifolds; their method follows closely the proof of Lascu and Scott in [12], whose algebro-geometric ingredients they transfer to the topological environment.…”

Symplectic geography in dimension 8.

“…Let M denote the blow-up of the symplectic manifold M along a symplectic submanifold N. In [9] the authors prove a formula for the Chern classes of the blow-up of an algebraic variety along a subvariety. This formula in fact holds for symplectic manifolds as well and gives an expression for the total Chern class of M in terms of those of M and N. From the corresponding expressions for the individual Chern classes, a straightforward computation shows that in dimension 8, the parameters (a, m, j, k, b) transform under blow-up as follows [13].…”

“…His method relies on symplectic constructions such as blow-up and connected symplectic sum. We follow his strategy of proof and make use of a general formula for computing the Chern classes of blow-up, which was known for algebraic varieties [9] and whose proof can easily be modified so that it applies to symplectic manifolds. Moreover, we apply Donaldson's theorem about existence of symplectic submanifolds to the total space of some symplectic fibrations.…”