Second Chern class and Fujiki constants of hyperkähler manifolds

Abstract

We study characteristic classes on hyperkähler manifolds with a view towards the Verbitsky component. The case of the second Chern class leads to a conditional upper bound on the second Betti number in terms of the Riemann-Roch polynomial, which is also valid for singular examples. We discuss the general structure of characteristic classes and the Riemann-Roch polynomial on hyperkähler manifolds using among other things Rozansky-Witten theory.

Computing Riemann-Roch polynomials and classifying hyper-Kähler fourfolds

“…Under a condition on some generalized Fujiki constants of X, Beckmann and Song prove in [BS] (see also [S2]) that the only possible Betti numbers b 2 (X), b 3 (X), and b 4 (X) for X are as in case (a) of Theorem 9.3. Moreover, by [BS,Proposition 5.6], [BS,Conjecture 1.2] would imply that the Fujiki constant c X is either 3 or 9. In particular, the second case in (b) should not occur.…”

Atomic Objects on hyper-Kähler manifolds

“…In the known examples, we proceed analogously making use of the fact that we know the generalized Fujiki constants C(c 2 (X) 2 ) and C(c 4 (X)) through knowing the Riemann-Roch polynomial [6,Cor. 2.7].…”

“…Combining these two equations, we obtain an equation involving C(c 2 (X) 2 ) and C(c 4 (X)) which is violated in all the known examples, see [6,Sec. 4].…”