A generalized Frankel conjecture via the Yang-Mills flow
Jiangtao Li
TL;DR
This work introduces the $2$-positive bisectional curvature condition on compact Kähler manifolds and proves a Frankel-type classification: for $n=2$, $M^2$ must be biholomorphic to a del Pezzo surface, and for $n\ge3$, $M^n$ must be biholomorphic to either $\mathbb{C}P^n$ or the smooth complex hyperquadric $Q^n$. The authors develop a Yang-Mills flow approach on $\mathbb{C}P^1$ and establish a Hamilton-type tensor maximum principle for $\sqrt{-1}\Lambda_\omega F_A$, enabling a key estimate that forces the anti-canonical degree of rational curves to satisfy $-K_M\cdot C\ge n$, hence the pseudoindex $i(M)\ge n$. Combining this with the Miyazaki–Mori classification yields the rigidity result, linking curvature positivity to Fano manifold structure. The paper also discusses generalizations to $m$-positive bisectional curvature and situates the results in the broader context of curvature conditions and orthogonal Ricci positivity, suggesting avenues for extending the differential-geometric classification program.
Abstract
In this note, we introduce a new curvature condition called the $2-$positive bisectional curvature on compact Kähler manifolds. We then deduce a characterization theorem for manifolds with $2-$positive bisectional curvature, which can be regarded as a variant of the classical Frankel conjecture (cf.\cite{Fra61,SY80}) and its generalizations (cf.\cite{Siu80,Mok88}).