Odd-Degree Rigidity and a Numerical Characterization of Complete Flag Varieties
Abstract
Let F=G/B be the complete flag variety of a connected semisimple complex linear algebraic group. We prove that if a smooth Fano variety X has the same dimension and Picard number as F, then the existence of a continuous map X→ F of odd topological degree implies X is isomorphic to F. In particular, a smooth Fano variety homotopy equivalent to F is isomorphic to F. The algebraic input is a numerical characterization of complete flags: a positive-dimensional smooth Fano variety X is a complete flag variety if and only if its pseudoindex ιX is at least two and ker(Sym2N1(X)ℝ→ N2(X)ℝ) contains a positive-definite quadratic form. The pseudoindex condition can equivalently be replaced by the divisibility of −KX by two in Pic(X). The key observation is that a single definite quadratic relation excludes surfaces in every fiber of any elementary contraction; standard contraction theorems then yield smooth ℙ1-fibrations and the characterization of complete flags applies. For the odd-degree theorem, Poincaré duality and a mod-two Wu-class argument show that the anticanonical class is divisible by two; comparison of the integral coroot lattices then identifies the flag. Most proofs were developed through extensive discussions with OpenAI's ChatGPT (GPT-5.6 Sol and GPT-6), based on mathematical directions supplied by the author and arguments suggested by ChatGPT; see the AI Disclosure.
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BibTeX
@misc{watanabe2026odddegree,
title = {{Odd-Degree Rigidity and a Numerical Characterization of Complete Flag Varieties}},
author = {Kiwamu Watanabe},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41237},
eprint = {2610.41237},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41237},
}
RIS
TY - UNPB
TI - Odd-Degree Rigidity and a Numerical Characterization of Complete Flag Varieties
AU - Kiwamu Watanabe
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41237
UR - https://arxiv.org/abs/2610.41237
AB - Let F=G/B be the complete flag variety of a connected semisimple complex linear algebraic group. We prove that if a smooth Fano variety X has the same dimension and Picard number as F, then the existence of a continuous map X→ F of odd topological degree implies X is isomorphic to F. In particular, a smooth Fano variety homotopy equivalent to F is isomorphic to F. The algebraic input is a numerical characterization of complete flags: a positive-dimensional smooth Fano variety X is a complete flag variety if and only if its pseudoindex ι_X is at least two and ker(Sym^2N^1(X)_ℝ→ N^2(X)_ℝ) contains a positive-definite quadratic form. The pseudoindex condition can equivalently be replaced by the divisibility of −K_X by two in Pic(X). The key observation is that a single definite quadratic relation excludes surfaces in every fiber of any elementary contraction; standard contraction theorems then yield smooth ℙ^1-fibrations and the characterization of complete flags applies. For the odd-degree theorem, Poincaré duality and a mod-two Wu-class argument show that the anticanonical class is divisible by two; comparison of the integral coroot lattices then identifies the flag. Most proofs were developed through extensive discussions with OpenAI's ChatGPT (GPT-5.6 Sol and GPT-6), based on mathematical directions supplied by the author and arguments suggested by ChatGPT; see the AI Disclosure.
ER -