The author shows that there is only a small number of possibilities for Betti numbers of an irreducible \(4\)-dimensional hyper-Kähler manifold \(M\). All Betti numbers of such a manifold are determined by \(b_2\) and \(b_3\). Examples are known only for \((b_2,b_3)=(23,0)\) or \((7,8)\). The author proves, in particular, that if \(b_2\geq 9\) then \(b_2=23\) and \(b_3=0\).
First he proves that \(b_2\leq 23\). The proof is an easy consequence of S. M. Salamon’s Riemann-Roch formula [see Topology 35, No. 1, 137-155 (1996; Zbl 0854.58004)] and M. Verbitsky’s result saying that \(S^2H^2(M)\hookrightarrow H^4(M)\) [see Geom. Funct. Anal. 6, No. 4, 601-611 (1996; Zbl 0861.53069)].
Then the author uses some results of N. Hitchin and J. Sawon in [Duke Math. J. 106, 599-615 (2001; Zbl 1024.53032)] and F. A. Bogomolov in Geom. Funct. Anal. 6, No. 4, 612-618 (1996; Zbl 0862.53050) to bound \(b_3\).