On the cohomology of Kähler and hyper-Kähler manifolds
Let \(M\) be a compact complex manifold of complex dimension \(n\). The Hodge number \(h^{p,q}\) of \(M\) is the dimension of the Dolbeault cohomology space \(H^{p,q}\) of \(M\). There are some well-known symmetries between the Hodge numbers, when \(M\) is a Kähler manifold. When \(M\) has complex dimension \(2m\) and has holonomy contained in \(\text{Sp} (m)\), then \(M\) is called a hyper-Kähler manifold. In this case the Hodge numbers of \(M\) are ``invariant by mirror symmetry'' which means that \(h^{p,q} = h^{2m - p,q}\). The Riemann-Roch theorem provides a number of relations between the Hodge numbers of \(M\) and the Chern numbers of \(M\). The author uses these facts to deduce a number of interesting results on the cohomology and on the Chern numbers of those manifolds. We quote only two examples. Let \(\chi(t) = \sum_p \chi_p t^p\) with \(\chi_p = \sum_q (-1)^q h^{p,q}\). Then \(\chi(-1)\) is known to be equal to the Euler-characteristic of \(M\) and equal to the top Chern class \(c_n\) evaluated on the fundamental cycle of \(M\). The author proves that for any \(k > 0\) the Taylor coefficient \(\chi^{(2k)} (-1)/(2k)!\) is expressible as a linear combination of Chern numbers each of which involves at least one \(c_i\) with \(i > n - 2k\). For hyper-Kähler manifolds of real dimension \(4m\) it is shown that the invariance by mirror symmetry of the Hodge numbers leads to a constraint on the Betti numbers of \(M\), and that 24 is a divisor of \(m_\chi\), where \(\chi\) denotes the Euler characteristic of \(M\).
scientific article; zbMATH DE number 863697