Hilbert schemes of K3 surfaces, generalized Kummer, and cobordism classes of hyper-Kähler manifolds
The cobordism ring \(\mathrm{MU}^*(pt)\) is the ring which in degree \(i\) consists of the free abelian group generated by \(i\)-dimensional compact complex manifolds \(M\) with a stable complex structure \(\alpha\), i.e. a structure of complex vector bundle on \(T_M \oplus \mathbb R^k\), modulo the subgroup generated by boundaries, that is \(i+1\)-dimensional manifolds \(N\) with boundary and a stable complex structure on \(T_N|_{\partial N}\). It is known that the cobordism class of a pair \((M, \alpha)\) with \(M\) of dimension \(2i\) is determined by the Chern numbers \[ \int_M P_I(c_l(M, \alpha)) \] where \(c_l(M, \alpha)\) are the Chern classes of the complex vector bundle \(T_M \oplus \mathbb R^k\) and the polynomials \(P_I\) generate the space of degree \(2i\) weighted homogeneous polynomials. Suppose now that \(X\) is a Hyperkähler manifold of complex dimension \(2n\) (and \(M\) is the oriented real manifold defined by \(X\)). Since the odd Chern classes are trivial, the cobordism class is determined by the Chern numbers \(\int_M P_I(c_{2l}(X))\) where the polynomials \(P\) generate the space of degree \(4n\) weighted homogeneous polynomials. This space of polynomials is generated by monomials determined by partitions \(I=(n_1, \dots, n_k)\) of \(n\). Given a surface \(S\) one can associate to a partition \(I\) the manifold \[ S^[I] := S^{[n_1]} \times \dots \times S^{[n_k]} \] where \(S^{[d]}\) is the Hilbert scheme of \(d\) points on \(S\). The first main result of this paper states the cobordism class of any Hyperkähler manifold is a unique combination with rational coefficients of classes \(S^{[I]}\) where \(S\) is a \(K3\) surface. In this theorem the \(K3\) surface \(S\) can be replaced by any compact complex surface with \(c_2(S) \neq 0\) and \(c_1^2(S)=0\), since the cobordism class of \(S^{[I]}\) only depends on these two numbers. The authors also show that the same statement holds if we replace the product of Hilbert schemes by a product of generalized Kummer varieties. Using complex cobordism theory the main theorem can be reduced to showing that if \(S\) is a compact complex surface with the properties above, the Milnor genus \[ M(S^{[n]}) = \int_{S^{[n]}}\mathrm{ch}_{2n}(S^{[n]}) \] is non-zero. In the last section the authors raise a number of questions about the cobordism class and Chern numbers of Hyperkähler manifolds.
scientific article; zbMATH DE number 7608374