$K$-Equivalence and Integral Cohomology

We introduce an integral version of the Hodge polynomial, which encodes the integral cohomology of smooth projective varieties. We prove it extends to a function which is well-defined on the Grothendieck ring of varieties and we obtain as a consequence that $K$-equivalent smooth projective varieties have isomorphic integral cohomology groups.

💡 Research Summary

The paper introduces an integral refinement of the Hodge polynomial, denoted (H_{\mathrm{vir},\mathbb Z}), which encodes the full integral cohomology—including torsion—of smooth projective complex varieties. The authors first define a “torsion Poincaré function” (T(X)) that records, for each prime (p) and each cohomological degree (i), the ranks of the successive (p)-power torsion subquotients of (H_i(X,\mathbb Z)). This data is packaged into a formal series in variables (s_p, r_j, t, x) belonging to a carefully constructed commutative ring (S).

Next they augment (T(X)) with the usual Hodge numbers (h_{p,q}(X)) via additional variables (u, v), obtaining the integral Hodge function \