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L-equivalences via Symplectic and $F_4$ Grassmannians

Ivan Noden

TL;DR

The paper constructs zero divisors in the Grothendieck ring of varieties by producing nontrivial L-equivalences between Calabi–Yau pairs cut out from homogeneous roofs, specifically in symplectic (type $C_{3r-1}$) and $F_4$ Grassmannians. Using Kanemitsu’s roof framework and detailed cohomological analysis, it shows that the bases $F_1$ and $F_2$ have equal classes while the Calabi–Yau pairs $(Z_1,Z_2)$ are nonisomorphic, yielding $\mathbb{L}^{r-1}([Z_1]-[Z_2])=0$ with the appropriate exponent ($\mathbb{L}^{2r-1}$ for $C_{3r-1}$ and $\mathbb{L}^2$ for $F_4$). Central to the argument are vanishing results from Borel–Weil–Bott computations and dimension comparisons of global sections, which distinguish $Z_1$ and $Z_2$ while keeping the bases indistinguishable in $K_0(\mathrm{Var}_{\mathbb{C}})$. This extends prior $G_2$-based constructions and enriches the connection between derived equivalence and L-equivalence in algebraic geometry.

Abstract

Using a construction of Kanemitsu from [9] and observations by Rampazzo in [19], we find examples of zero divisors in the Grothendieck ring of varieties by taking the zero loci of sections of vector bundles over symplectic and $F_4$ Grassmannians. These zero divisors yield instances of non-trivially L-equivalent Calabi-Yau varieties. This methodology is inspired by a similar process performed by Ito et al. on $G_2$ Grassmannians in [8].

L-equivalences via Symplectic and $F_4$ Grassmannians

TL;DR

The paper constructs zero divisors in the Grothendieck ring of varieties by producing nontrivial L-equivalences between Calabi–Yau pairs cut out from homogeneous roofs, specifically in symplectic (type ) and Grassmannians. Using Kanemitsu’s roof framework and detailed cohomological analysis, it shows that the bases and have equal classes while the Calabi–Yau pairs are nonisomorphic, yielding with the appropriate exponent ( for and for ). Central to the argument are vanishing results from Borel–Weil–Bott computations and dimension comparisons of global sections, which distinguish and while keeping the bases indistinguishable in . This extends prior -based constructions and enriches the connection between derived equivalence and L-equivalence in algebraic geometry.

Abstract

Using a construction of Kanemitsu from [9] and observations by Rampazzo in [19], we find examples of zero divisors in the Grothendieck ring of varieties by taking the zero loci of sections of vector bundles over symplectic and Grassmannians. These zero divisors yield instances of non-trivially L-equivalent Calabi-Yau varieties. This methodology is inspired by a similar process performed by Ito et al. on Grassmannians in [8].

Paper Structure

This paper contains 14 sections, 17 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weights, Roots and the Weyl Group
  4. Bruhat Decomposition
  5. Representations
  6. The Semisimple Case
  7. Representations of Parabolic Subgroups
  8. Homogeneous Vector Bundles
  9. Borel-Weil-Bott
  10. Example: Symplectic Group
  11. Homogeneous Roofs
  12. L-equivalences via Symplectic Grassmannians
  13. Cohomology Calculations
  14. L-equivalences via F4 Grassmannians

Key Result

Theorem 1.2

[theorem]th:type_c_zero Let $r\geq 2$ and $(Z_1, Z_2)$ a Calabi-Yau pair associated to the homogeneous roof of type $C_{3r-1}$. Then, $[Z_1] \neq [Z_2]$ and yet

Theorems & Definitions (37)

  • Definition 1.1: kuznetsov_shinder:2018
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: Borel-Weil-Bott
  • proof
  • Remark 2.2
  • Definition 3.1
  • Example 3.2: Roof of type $C_{3r-1}$
  • Proposition 3.3: rampazzo:2021
  • Remark 3.4
  • ...and 27 more