Cartan Determinants in algebraic geometry
Abstract
We formulate a dg-categorical analogue of the Cartan determinant conjecture using numerical Grothendieck groups. Let D be a smooth proper dg category over an algebraically closed field k. The Euler pairing on K0(D) descends to a nondegenerate bilinear form on the numerical Grothendieck group Knum(D), and hence defines a numerical Cartan determinant det Mχ(D). While the classical Cartan determinant conjecture predicts that det Mχ(A)=1 for finite-dimensional smooth algebras A, this equality is no longer expected for general smooth proper dg categories. However, we propose the following positivity question: det Mχ(D)>0. Our main result proves this positivity when D=Perf(X) for a large class of smooth projective varieties X. More precisely, we show that det Mχ(X)>0 if either dim X is odd, or X has even dimension and satisfies the numerical Hodge standard conjecture.
Related
- Einstein-Cartan and Palatini-Cartan Gravity from Graded Poisson Geometry
- A Cartan geometry from a correspondence space in Riemannian geometry
- Holomorphic Conformal Cartan Geometry: Conformal Gravity, Twistors, and Ambitwistors
- Explicit and asymptotically good constructions of Algebraic Geometry codes in the sum-rank metric
- Algebraic Magnetism: X-products of attractors via 𝔽1-geometry
BibTeX
@misc{liu2026cartan,
title = {{Cartan Determinants in algebraic geometry}},
author = {Yeqin Liu and Yu Shen and Ziyu Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29598},
eprint = {2609.29598},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29598},
}
RIS
TY - UNPB
TI - Cartan Determinants in algebraic geometry
AU - Yeqin Liu
AU - Yu Shen
AU - Ziyu Zhang
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.29598
UR - https://arxiv.org/abs/2609.29598
AB - We formulate a dg-categorical analogue of the Cartan determinant conjecture using numerical Grothendieck groups. Let D be a smooth proper dg category over an algebraically closed field k. The Euler pairing on K_0(D) descends to a nondegenerate bilinear form on the numerical Grothendieck group K_num(D), and hence defines a numerical Cartan determinant det M_χ(D). While the classical Cartan determinant conjecture predicts that det M_χ(A)=1 for finite-dimensional smooth algebras A, this equality is no longer expected for general smooth proper dg categories. However, we propose the following positivity question: det M_χ(D)>0. Our main result proves this positivity when D=Perf(X) for a large class of smooth projective varieties X. More precisely, we show that det M_χ(X)>0 if either dim X is odd, or X has even dimension and satisfies the numerical Hodge standard conjecture.
ER -