Chern numbers and diffeomorphism types of projective varieties

Abstract

In 1954, Hirzebruch asked which linear combinations of Chern numbers are topological invariants of smooth complex projective varieties. We give a complete answer to this question in small dimensions, and also prove partial results without restrictions on the dimension.As the right-hand side takes on only finitely many values, the same is true for c 4 1 , and then for c 2 1 c 2 as well.Remark 2. Pasquotto [9] recently raised the question of the topological invariance of Chern numbers of symplectic manifolds, particularly in (real) dimensions 6 and 8. Our results for Kähler manifolds, of course, show that Chern numbers of symplectic manifolds are not topological invariants. In the Kähler case, we have used Hodge theory to argue that the variation of Chern numbers is quite restricted, often to finitely many possibilities. It would be interesting to know whether any finiteness results hold in the symplectic non-Kähler category.

Euler Characteristics of Crepant Resolutions of Weierstrass Models

“…With that restriction, the question was answered by Kotschick in the following theorem. The statement about surfaces is a consequence of Seiberg-Witten's theory [42,43].…”

Algebraic structures with unbounded Chern numbers

“…of positive square. Since the intersection pairing on S 3 has type (1,9), it follows that the orthogonal complement of ω is negative definite. Hence, G 2 q = 0 implies ω · G q = 0 for all q.…”

Chern numbers and the geometry of partial flag manifolds

“…This problem, originally raised in [11], was recently resolved completely in complex dimensions strictly smaller than 5, see [16], and we hope that the calculations performed in this paper will be useful in studying this problem in higher dimensions. The calculations in complex dimension 5, that is for F 2 , summarized in Table 1 might lead one to speculate about what happens for arbitrary n. In order to test such speculations we completed all the calculations for n D 3, that is in complex dimension 7.…”