Abstract
Let M be an irreducible compact hyperkähler manifold of complex dimension six. We prove that the second Betti number of M is at most 23.
Dimension four
Let us recall how to bound the second Betti number in dimension four. Salamon [10] proved that the Betti numbers of a compact hyperkähler manifold of dimension 2n satisfy the relation 2 2n j=1 (−1) j (3j 2 − n)b 2n−j = nb 2n .
Theorem 1 (Beauville [2] , Guan [5] ) Let M be an irreducible compact hyperkähler manifold of complex dimension four. Then the second Betti number b 2 of M is at most 23.
Proof Irreducible hyperkähler manifolds are simply-connected, so b 1 = 0. Therefore Salamon's relation for n = 2 gives −2b 3 + 20b 2 + 92 = 2b 4 .
Verbitsky [11] proved that Sym k H 2 (M, R) injects into H 2k (M, R) for k ≤ n. In particular, we can write
and
where b The left-hand side is negative if b 2 > 23, whereas the right-hand side is clearly non-negative. Therefore the second Betti number b 2 can be at most 23.
Example Up to deformation, there are two known examples of irreducible compact hyperkähler manifolds of dimension four: the Hilbert scheme Hilb 2 S of two points on a K3 surface S (see Fujiki [3] ) and the generalized Kummer variety K 2 (A) of an abelian surface A (see Beauville [1] ). Their Hodge diamonds are Theorem 2 (Verbitsky [12] , Looijenga and Lunts [7] ) Let M be an irreducible compact hyperkähler manifold of dimension 2n with second Betti number b 2 . Then there is an action of so(4, b 2 − 2) on the real cohomology The irreducible representation with highest weight vector 1 ∈ H 0 (M, C) is precisely the subring of the cohomology generated by H 2 (M, C). In dimension six, the remainder of the cohomology comes from irreducible representations V 1 , V 2 , V 3 , V 4 , and V 5 whose highest weight vectors lie in the Hodge groups that are circled in Figure 1 and described in Table 1 . The dimensions of the spin representations are not needed for our arguments. Decompose the complex cohomology of M into irreducible representations of so(b 2 + 2, C), as above. Suppose that V 2 occurs with multiplicity c, V 3 occurs with multiplicity d, and V 5 occurs with multiplicity e. The contributions of V 2 and V 3 to H 4 (M, C) are of dimensions b 2 and 1, respectively. Including Sym 2 H 2 (M, C) and multiplicities, we deduce that
Similarly, the contributions of V 2 , V 3 , and V 5 to H 6 (M, C) are of dimensions (b 2 2 − b 2 + 2)/2, b 2 , and 1, respectively. Including Sym 3 H 2 (M, C) and multiplicities, we deduce that
Substituting the formulae for b 4 and b 6 into Salamon's relation gives 36
and after simplifying and rearranging we obtain 
Higher dimensions
When n = 4 Salamon's relation gives 2b 7 + 16b 6 − 46b 5 + 88b 4 − 142b 3 + 208b 2 + 376 = 4b 8 .
Thus in dimension eight, b 7 appears with a coefficient of the 'wrong' sign, and we cannot simply imitate the proof of Theorem 3. We can prove the following weaker result. Proof The proof is essentially the same as that of Theorem 3 so we omit most of the details. After decomposing the complex cohomology of M into irreducible representations of so(b 2 + 2, C), we obtain formulae for b 4 , b 6 , and b 8 in terms of b 2 and certain non-negative multiplicities c, d, e, f , . . .. Substituting these into Salamon's relation gives
The left-hand side is negative if b 2 ≥ 25 > 21+ √ 510.825.7109, whereas for b 2 ≥ 25 the right-hand side will be a non-negative linear combination of the non-negative multiplicities c, d, e, f , . . .. Therefore the second Betti number b 2 can be at most 24.
Remark We could have assumed that only b 7 vanishes, but because of the so(b 2 + 2, C)-action on the cohomology this would already imply that all odd Betti numbers vanish.
Remark The pattern appears to be that in dimension 2n, the polynomial in b 2 on the left-hand side has largest root
, so that an irreducible compact hyperkähler manifold whose odd Betti numbers all vanish must have second Betti number b 2 ≤ 21+ √ 433+96n 2
. The author has not rigorously verified this.

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