On Stringy E-functions and the Non-negativity Conjecture for Determinantal Varieties
Yifan Chen, Huaiqing Zuo
TL;DR
The paper computes stringy E-functions for determinantal varieties in the equal-rank case ($r=s$) and their projectivizations, expressing the results in closed form via Grassmannians and affine factors. The approach combines motivic integration with a detailed orbit decomposition of arc spaces and an explicit embedded resolution to compute log discrepancies, yielding $E_{st}(D^k)=E(\mathbb{L}^{kr}\cdot G(k,r))$ and $E_{st}(\hat{D}^k_{r,r})=\frac{(uv)^{kr}-1}{uv-1}\cdot E(G(k,r))$. This also allows verification of Batyrev’s non-negativity conjecture for the stringy Hodge numbers of the projective determinantal varieties. The results provide explicit, computable invariants linking determinantal geometry to Grassmannians and show non-negativity in a broad, well-structured class of singular varieties, with potential implications for mirror-symmetry-inspired invariants and stringy cohomology theories.
Abstract
We compute the stringy E-functions of determinantal varieties and establish that the stringy E-function of a determinantal variety coincides with the E-function of the product of a Grassmannian and an affine space. Furthermore, a similar result holds for the projectivization of determinantal varieties. As an application, we verify the non-negativity conjecture for the stringy Hodge numbers of these varieties.