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 $X\to F$ 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 ${\rm ker}({\rm Sym}^2N^1(X)_{\mathbb R}\to N^2(X)_{\mathbb R})$ contains a positive-definite quadratic form. The pseudoindex condition can equivalently be replaced by the divisibility of $-K_X$ by two in ${\rm Pic}(X)$. 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 ${\mathbb P}^1$-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.