Nef complexity of klt Fano varieties: the full spectrum and classification below two
Abstract
Let X be a complex klt Fano variety. We study the rational total index τX and the nef complexity cX=dim X+ρ(X)−τX. Previous joint work of the author with Enwright, Filipazzi, Gongyo, Moraga, Svaldi, and Wang established the corresponding nonnegativity and zero-case characterization in the broader framework of a special class of generalized log Calabi--Yau pairs. Here we give a direct proof in the Fano setting, prove that there is no value strictly between zero and one, and classify all klt Fano varieties with cX<2, without assuming ℚ-factoriality. At complexity one, after splitting off projective-space factors, the remaining factor is a normal quadric of Picard number one, a linear blow-up of projective space, or an integral (1,1) hypersurface in a product of projective spaces. If 1<cX<2, only two weighted-projective families occur; in particular, no smooth Fano variety has nef complexity strictly between one and two. We also determine the spectrum when the dimension is allowed to vary: {0}∪{2−2/q middle| q∈ℤ≥2} ∪(ℚ∩[2,∞)). Most proofs in this paper were developed from arguments generated by OpenAI's ChatGPT through discussions initiated and guided by the author; see the statement on the use of generative AI.
Related
- The inverse Monge--Ampère flow on klt Fano varieties
- The Miyaoka-Yau inequality and the delta invariant for Fano varieties I -klt Fano case-
- Fano Generalized Bott-Samelson Varieties
- Completed cohomology of Hilbert modular varieties below middle degree
- Non-commutative Fano varieties and three-dimensional minifolds
BibTeX
@misc{watanabe2026complexity,
title = {{Nef complexity of klt Fano varieties: the full spectrum and classification below two}},
author = {Kiwamu Watanabe},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27676},
eprint = {2610.27676},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27676},
}
RIS
TY - UNPB
TI - Nef complexity of klt Fano varieties: the full spectrum and classification below two
AU - Kiwamu Watanabe
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27676
UR - https://arxiv.org/abs/2610.27676
AB - Let X be a complex klt Fano variety. We study the rational total index τ_X and the nef complexity c_X=dim X+ρ(X)−τ_X. Previous joint work of the author with Enwright, Filipazzi, Gongyo, Moraga, Svaldi, and Wang established the corresponding nonnegativity and zero-case characterization in the broader framework of a special class of generalized log Calabi--Yau pairs. Here we give a direct proof in the Fano setting, prove that there is no value strictly between zero and one, and classify all klt Fano varieties with c_X<2, without assuming ℚ-factoriality. At complexity one, after splitting off projective-space factors, the remaining factor is a normal quadric of Picard number one, a linear blow-up of projective space, or an integral (1,1) hypersurface in a product of projective spaces. If 1<c_X<2, only two weighted-projective families occur; in particular, no smooth Fano variety has nef complexity strictly between one and two. We also determine the spectrum when the dimension is allowed to vary: {0}∪{2−2/q middle| q∈ℤ_≥2} ∪(ℚ∩[2,∞)). Most proofs in this paper were developed from arguments generated by OpenAI's ChatGPT through discussions initiated and guided by the author; see the statement on the use of generative AI.
ER -