$K$-Equivalence and Integral Cohomology

Technical Summary: K-Equivalence and Integral Cohomology

Problem Statement
In 1995, Kontsevich established that K-equivalent smooth projective varieties share the same Hodge numbers. While this result addressed the rational or complex cohomological invariants (specifically the Hodge decomposition), the question remained whether K-equivalence implies an isomorphism of the integral cohomology groups, $H^i(X, \mathbb{Z})$. Unlike Hodge numbers, integral cohomology contains torsion information, which is not captured by the standard Hodge polynomial. This paper addresses the gap by extending Kontsevich's result to the integral level, proving that K-equivalent smooth projective varieties have isomorphic integral cohomology groups.

Methodology
The authors introduce a new invariant, the integral virtual Hodge function, denoted as $H_{\text{vir}, \mathbb{Z}}$. The construction proceeds in several technical stages:

1. Definition of the Integral Hodge Function ($H_{\mathbb{Z}}$):
   For a smooth projective variety $X$, the authors define a function $H_{\mathbb{Z}}(X)$ that encodes both the free and torsion parts of the cohomology.

   • Free Part: Captured by the usual Hodge numbers $h^{p,q}(X)$, weighted by variables $u$ and $v$.
   • Torsion Part: Captured by a "torsion Poincaré function" $T(X)$. This involves invariants $a_{p,i,j}(X)$, which represent the rank of the $\mathbb{Z}/p\mathbb{Z}$-vector space formed by the quotient $p^j H^i(X, \mathbb{Z})_{\text{tors}} / p^{j+1} H^i(X, \mathbb{Z})_{\text{tors}}$.
   • Target Ring: The function takes values in a specific ring $R$ generated by variables $s_p, r_j, u, v, t, x$ subject to relations that ensure the multiplicative behavior of the function. Crucially, the ring includes relations like $tx^2 = tx - x$ and $s_p^2 = s_p$ to handle the combinatorial structure of torsion and the interaction between free and torsion parts under the Künneth formula.

2. Extension to the Grothendieck Ring:
   The authors prove that $H_{\mathbb{Z}}$ extends to a well-defined function $H_{\text{vir}, \mathbb{Z}}$ on the Grothendieck ring of varieties, $K_0(\text{Var}_{\mathbb{C}})$. This requires demonstrating two key properties:

   • Multiplicativity: $H_{\mathbb{Z}}(X \times Y) = H_{\mathbb{Z}}(X)H_{\mathbb{Z}}(Y)$. The proof involves a detailed analysis of the Künneth formula for integral cohomology, showing that the torsion terms in the product match the product of the torsion terms in the individual factors, accounting for the interaction between free and torsion components.
   • Blow-up Relation: The function satisfies the relation $H_{\mathbb{Z}}(X) - H_{\mathbb{Z}}(Z) = H_{\mathbb{Z}}(\text{Bl}_Z(X)) - H_{\mathbb{Z}}(E)$ for a blow-up of $X$ along a smooth center $Z$ with exceptional divisor $E$. This allows the function to descend to the Grothendieck ring, which is generated by smooth projective varieties modulo these relations.

3. Degree Filtration and Completion:
   To handle K-equivalence, the authors utilize the completed Grothendieck ring $\widehat{K_0(\text{Var}_{\mathbb{C}})}[L^{-1}]$. They define a degree on the target ring $R$ and show that $H_{\text{vir}, \mathbb{Z}}$ factors through the completion. This allows the application of Kontsevich's result that K-equivalent varieties have equal classes in the completed ring.

Key Contributions and Results

• Theorem 1.1: The primary result states that if $X$ and $X'$ are K-equivalent smooth projective varieties over $\mathbb{C}$, then $H^i(X, \mathbb{Z}) \simeq H^i(X', \mathbb{Z})$ for all $i$. This is achieved by showing that $H_{\text{vir}, \mathbb{Z}}(X) = H_{\text{vir}, \mathbb{Z}}(X')$, and since the integral cohomology groups can be recovered from the coefficients of $H_{\text{vir}, \mathbb{Z}}$ (via the map $\phi_i$), the groups must be isomorphic.
• Recovery of Torsion: The paper provides a mechanism to reconstruct the integral cohomology groups, including their torsion subgroups, directly from the coefficients of the integral Hodge function.
• Torsion-Free Cohomology for Cell Decompositions: As a corollary, the authors provide a new proof that if a smooth projective variety admits an algebraic cell decomposition (e.g., varieties with a $\mathbb{C}^*$-action with finitely many fixed points), its integral cohomology is torsion-free. This follows because the function reduces to the standard Hodge polynomial in such cases.
• Independence of Crepant Resolutions: The paper establishes that for a projective variety $X$ with at worst Gorenstein canonical singularities, the integral cohomology of any crepant resolution $Y \to X$ is independent of the choice of resolution.

Significance and Claims
The paper claims to extend Kontsevich's 1995 result from Hodge numbers to the full integral cohomology groups. The significance lies in the construction of a multiplicative invariant on the Grothendieck ring that is sensitive to torsion, a feature previously lacking in virtual Hodge theory.

The authors note that while the result (Theorem 1.1) was previously shown in works by Gusein-Zade and Saito [GS96] and by Yasuda and Xie [XY18] (which they became aware of after posting the initial version), their contribution is the introduction of the integral virtual Hodge function $H_{\text{vir}, \mathbb{Z}}$. This function serves as a tool that not only proves the theorem but also encodes the integral cohomology structure in a way that is well-defined on the Grothendieck ring. The authors suggest this framework admits further generalizations, such as to stringy invariants using the Gorenstein measure or to the Grothendieck group of stacks.