Nef complexity of klt Fano varieties: the full spectrum and classification below two

Let $X$ be a complex klt Fano variety. We study the rational total index ${\tau }_X$ and the nef complexity $c_X=\dim X+\rho (X)-{\tau }_X$. 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 $\mathbb{Q}$-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: $$\{0\}\cup \{2-\frac{2}{q}|q\in {\mathbb{Z}}_{\ge 2}\}\cup (\mathbb{Q}\cap [2,\mathrm{\infty }))\mathrm{.}$$ 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.