Frankel conjecture

Remarks

Frankel stated the conjecture in 1961 for compact Kähler manifolds of positive sectional curvature; the version above uses the weaker hypothesis of positive bisectional curvature. It was proved in 1979 by Mori, as a consequence of his proof of the Hartshorne conjecture that a smooth projective variety with ample tangent bundle is a projective space. Independently, Siu and Yau gave a differential-geometric proof in 1980, using energy-minimizing harmonic maps from $\mathbb{CP}^1$ to produce rational curves.

Example: the Fubini–Study metric on complex projective space has constant positive holomorphic sectional curvature and positive bisectional curvature, so $\mathbb{CP}^n$ satisfies the hypothesis.

Positive Ricci curvature is not enough. The product $\mathbb{CP}^1 \times \mathbb{CP}^1$ with the product of Fubini–Study metrics has positive Ricci curvature, but its bisectional curvature vanishes for $u$ tangent to the first factor and $v$ to the second. It is not biholomorphic to $\mathbb{CP}^2$, since its second Betti number is $2$, while that of $\mathbb{CP}^2$ is $1$.

Going deeper

For a student

Holomorphic bisectional curvature compares two complex directions at a point; it is weaker than sectional curvature and stronger than Ricci curvature. Mori's proof works in algebraic geometry: positivity makes the tangent bundle ample, and then one finds and bends rational curves to recognize $\mathbb{CP}^n$. Mori's method of bending rational curves became a basis of the minimal model program. The theorem was later extended by Mok to compact Kähler manifolds of nonnegative bisectional curvature, which are built from projective spaces, Hermitian symmetric spaces and flat tori.