On the Betti Numbers of Irreducible Compact Hyperkähler Manifolds of Complex Dimension Four

Abstract

The study of higher dimensional hyperkahler manifolds has attracted much attention: we have [Wk], [Bg1,2,3,4], [Fj1,2], [Bv1], [Vb1,2], [Sl1,2], [HS], [Huy], [Gu3,4,5] etc. It is evident that there are only a few known examples of these manifolds and the obvious question is: can we classify them as in the case of complex dimension 4? The Riemann-Roch formula plays an important role in the surface case, which yields K-3 surfaces as the only irreducible examples. However, for the higher dimensional case, the Riemann-Roch formula is not enough to give a picture of both the Hodge diamond and the existence of holomorphic sections of line bundles. In [Gu5], we combined the results of the Riemann-Roch formula in [Sl1,2] (see also [LW]) and the representations generated by the Kahler classes (see [Vb2], [LL], [Bg4]) to give a picture of the Hodge diamonds of irreducible compact hyperkahler manifolds of complex dimension 4. Theorem 1 (reproduced here) gives an upper bound b2 ≤ 23 for the second Betti number and was obtained independently by Beauville [Bv2] (unpublished). He kindly let me publish alone. The bound is obtained by applying Verbitsky’s work. In [HS] there is also an upper bound for the Euler characteristic but there seems as yet no lower bound, nor any bound for the Betti numbers. However, the method in [HS] actually gives us a way to calculate what we call generalized Chern numbers (which are only defined on hyperkahler manifolds) by Rozansky-Witten invariants, some of which in turn can be calculated as Chern numbers. Combining this approach with the method in [Bg4] we obtain an inequality in the opposite direction to the one in [HS] and apply it to our situation. Surprisingly, once we already have the bound on b2 this gives a more natural and much stronger inequality than the one we manipulated from the Riemann-Roch formula in [Gu5]. Therefore, we obtain our: Main Theorem. If M is an irreducible compact hyperkahler manifold of complex dimension 4, then 3 ≤ b2 ≤ 23. Moreover,