Milnor Genus of Hilbert Schemes

• The paper demonstrates that the Milnor genus—computed as integrals of Chern monomials—distinguishes products of Hilbert schemes and underpins the rational Spin-bordism basis.
• Hilbert schemes of K3 surfaces serve as explicit geometric generators, forming a free polynomial algebra in degrees 4n through their hyperkähler structure.
• The Oberdieck–Song–Voisin theorem and Pontryagin class pairings confirm the linear independence and completeness of these generators in complex and Spin-bordism.

The Milnor genus of Hilbert schemes figures centrally in the explicit construction of geometric bases for rational Spin-bordism using hyperkähler manifolds, as established via the interplay between the topology of Hilbert schemes of $K3$ surfaces and characteristic number pairings. The decisive theorem of Oberdieck, Song, and Voisin demonstrates that the integrals of Chern class monomials—Milnor genera—distinguish products of these Hilbert schemes, ensuring their independence in complex and Spin-bordism. This leads to a new, fully explicit basis for the rational Spin-bordism ring in terms of familiar geometric spaces.

1. Rational Spin-Bordism and Its Algebraic Structure

The spin-bordism group $\Omega_n^{Spin}$ classifies closed $n$-dimensional manifolds whose stable normal bundles admit Spin structures. Rational Spin-bordism comprises the group $\Omega_*^{Spin} \otimes \mathbb Q$, equivalently described via the Pontryagin–Thom isomorphism as

$$\Omega_*^{Spin} \otimes \mathbb Q \cong H_*(B\,Spin;\mathbb Q) \cong \mathbb Q[p_1, p_2, p_3, \ldots], \quad |p_i| = 4i,$$

a graded commutative $\mathbb Q$-algebra. This structure is that of a polynomial algebra with one generator in each degree $4,8,12,\ldots$ (Buchanan et al., 26 Jan 2026).

2. Hilbert Schemes of $K3$ Surfaces as Geometric Generators

The paper constructs, for every $n\geq 1$, the Hilbert scheme of $n$ points on a $K3$ surface, denoted $K3^{[n]}$. These are compact, hyperkähler manifolds of dimension $4n$. In each degree $4n$, explicit basis elements for rational Spin-bordism are given by products of these Hilbert schemes: $$B_{4n} = \left\{ \prod_{i=1}^a K3^{[n_i]} \;\Bigm|\; (n_1,\ldots,n_a)\in\mathcal P(n) \right\}$$ where $\mathcal P(n)$ denotes the set of integer partitions of $n$.

3. Ring Structure and Explicit Basis

As an algebra, rational Spin-bordism is isomorphic to the polynomial ring

$$\Omega_*^{Spin}\otimes\mathbb Q \cong \mathbb Q[\alpha_1,\alpha_2,\alpha_3,\ldots],\quad\deg(\alpha_n)=4n,$$

with $$\alpha_n = [K3^{[n]}] \in \Omega_{4n}^{Spin}\otimes\mathbb Q$$. In degree $4n$, all monomials

$$\alpha_{n_1} \alpha_{n_2} \cdots \alpha_{n_a}\quad \bigl((n_1,\ldots,n_a)\in\mathcal P(n)\bigr)$$

form a basis, and every element is a unique $\mathbb Q$-linear combination of such products. There are no polynomial ring relations beyond commutativity and graded compatibility (Buchanan et al., 26 Jan 2026).

4. Oberdieck–Song–Voisin Theorem and the Milnor Genus

The independence and spanning property of the Hilbert scheme basis hinge on a reduction to rational complex bordism and a fundamental result by Oberdieck, Song, and Voisin: In rational complex bordism, every class with vanishing odd Chern classes is a $\mathbb Q$-linear combination of products of the $K3^{[n]}$. The Milnor genus—defined as the integral of monomials in Chern classes—serves to distinguish all such products. Thus, the Milnor genera provide a complete invariant for this basis (Buchanan et al., 26 Jan 2026).

5. Pairing with Pontryagin Class Monomials and Triangular Structure

Basis elements $\prod_j K3^{[n_j]}$ are paired with dual basis elements in $H^{4n}(B\,Sp;\mathbb Q)$, namely monomials $p_{m_1}p_{m_2}\cdots p_{m_r}$. The OSV result ensures that the pairing matrix is block-lower-triangular by degree, with nonzero diagonal blocks. Explicitly,

$$\int_{K3^{[m]}} p_{m'} = \begin{cases} \neq 0, & m'=m, \ 0, & m'>m, \end{cases}$$

generalizing to products of Hilbert schemes by splitting off toral factors with trivial Pontryagin classes. The matrix determinant is nonzero in $\mathbb Q$, confirming the basis elements' linear independence and completeness via rank comparison.

6. Structural Consequences and Explicit Description

The rational Spin-bordism ring $\Omega_*^{Spin} \otimes \mathbb Q$ is a free commutative $\mathbb Q$-algebra generated by classes $[K3^{[n]}]$ in degrees $4n$. In the identification $$\Omega_*^{Spin}\otimes\mathbb Q\cong \mathbb Q[p_1,p_2,\ldots]$$, the class $[K3^{[n]}]$ matches the primitive polynomial generator $\alpha_n$. All ring relations follow those of the polynomial algebra, with characteristic number (Milnor genus) computations providing the necessary invariants to detect the basis. The explicit geometric basis in terms of hyperkähler manifolds is thus established for the first time (Buchanan et al., 26 Jan 2026).