On the Hodge and Betti Numbers of Hyper-Kahler Manifolds

In this survey article, we review past results (obtained by Hirzebruch, Libgober-Wood, Salamon, Gritsenko, and Guan) on Hodge and Betti numbers of Kahler manifolds, and more specifically of hyper-Kahler manifolds, culminating in the bounds obtained by Guan in 2001 on the Betti numbers of hyper-Kahler fourfolds. Let X be a compact Kahler manifold of dimension m. One consequence of the Hirzebruch-Riemann-Roch theorem is that the coefficients of the x(y)-genus polynomial pX(y) := Sigma(m)(p,q=0) (-1)(q)h(p,q)(X)y(p) is an element of Z[y] are (explicit) universal polynomials in the Chern numbers of X. In 1990, LibgoberWood determined the first three terms of the Taylor expansion of this polynomial about y = -1 and deduced that the Chern number integral (X) c(1)(X)c(m-1)(X) can be expressed in terms of the coefficients of the polynomial pX(y) (Proposition 2.1). When X is a hyper-Kahler manifold of dimension m = 2n, this Chern number vanishes. The Hodge diamond of X also has extra symmetries which allowed Salamon to translate the resulting identity into a linear relation between the Betti numbers of X (Corollary 2.5). When X has dimension 4, Salamon's identity gives a relation between b(2)(X), b(3)(X), and b(4)(X). Using a result of Verbitsky's on the injectivity of the cup-product map that produces an inequality between b(2)(X) and b(4)(X), it is easy to conclude b(2)(X) <= 23. Guan established in 2001 more restrictions on the Betti numbers (Theorem 3.6).