Topologically invariant Chern numbers of projective varieties

Abstract

We prove that a rational linear combination of Chern numbers is an oriented diffeomorphism invariant of smooth complex projective varieties if and only if it is a linear combination of the Euler and Pontryagin numbers. In dimension at least three we prove that only multiples of the top Chern number, which is the Euler characteristic, are invariant under diffeomorphisms that are not necessarily orientation-preserving. These results solve a long-standing problem of Hirzebruch's. We also determine the linear combinations of Chern numbers that can be bounded in terms of Betti numbers.

11 pages; minor edits in final version, to appear in Adv. Math

References in corpus (3)

• Characteristic numbers of algebraic varieties
• Chern numbers and diffeomorphism types of projective varieties
• Chern numbers and the geometry of partial flag manifolds

Cited by in corpus (10)

• The Hodge ring of Kaehler manifolds
• Characteristic numbers of algebraic varieties
• Characteristic numbers of crepant resolutions of Weierstrass models
• Kaehler structures on spin 6-manifolds
• Algebraic structures with unbounded Chern numbers
• Examples of diffeomorphic complete intersections with different Hodge numbers
• Updates on Hirzebruch's 1954 Problem List
• The Riemann-Roch Theorem on higher dimensional complex noncommutative tori
• A characterization of the $\hat{A}$-genus as a linear combination of Pontrjagin numbers
• Homological congruence formulae for characteristic classes of singular varieties