Volume-equality rigidity for tangent-bundle-semistable Fano manifolds

Determine whether every smooth Fano manifold X of dimension n whose tangent bundle T_X is slope-semistable with respect to the anticanonical polarization -K_X and whose anticanonical volume satisfies (-K_X)^n=(n+1)^n is isomorphic to projective space \mathbb{P}^n.

Background

The paper proves that if T_X is slope-semistable with respect to -K_X, then the finite-level anticanonical Hilbert-function bound h^0(X,-mK_X)\leq\binom{n+m(n+1)}{n} holds for every integer m\geq1, and equality at any one finite level implies X\simeq\mathbb P^n. However, equality of the volume estimate (-K_X)^n\leq(n+1)^n determines only the leading coefficient of the anticanonical Hilbert polynomial. It may therefore coexist with a strict deficit in the finite-level inequalities of order O(m^{n-1}) for every m, so the finite-level rigidity theorem does not resolve volume-level rigidity under tangent-bundle semistability alone.

Under the stronger hypothesis of K-semistability, Fujita's equality theorem gives an affirmative answer. The unresolved issue is whether slope semistability of the tangent bundle by itself suffices to characterize projective space in the equality case of the anticanonical volume comparison.