Cohomological Characterizations of Projective Spaces and Hyperquadrics

Projective spaces and hyperquadrics are the simplest projective algebraic varieties, and they can be characterized in many ways. The aim of this paper is to provide a new characterization of them in terms of positivity properties of the tangent bundle (Theorem 1.1). The first result in this direction was Mori’s proof of the Hartshorne conjecture in [Mor79] (see also Siu and Yau [SY80]), that characterizes projective spaces as the only manifolds having ample tangent bundle. Then, in [Wah83], Wahl characterized projective spaces as the only manifolds whose tangent bundles contain ample invertible subsheaves. Interpolating Mori’s and Wahl’s results, Andreatta and Wiśniewski gave the following characterization: Theorem [AW01]. Let X be a smooth complex projective n-dimensional variety. Assume that the tangent bundle TX contains an ample locally free subsheaf E of rank r. Then X ≃ Pn and either E ≃ OPn(1) or r = n and E = TPn . We note that earlier, in [CP98], Campana and Peternell obtained the same result for r ≥ n− 2. Let E be an ample locally free subsheaf of TPn of rank p < n. By taking its determinant, we obtain a nonzero section in H(P,∧TPn ⊗OPn(−p)). On the other hand, most sections in H(P,∧TPn ⊗OPn(−p)) do not come from ample locally free subsheaves of TPn . This motivates the following characterization of projective spaces and hyperquadrics, which was conjectured by Beauville in [Bea00]. Here Qp denotes a smooth quadric hypersurface in Pp+1, and OQp(1) denotes the restriction of OPp+1(1) to Qp. When p = 1, (Q1,OQ1(1)) is just (P ,OP1(2)). Theorem 1.1. Let X be a smooth complex projective n-dimensional variety and L an ample line bundle on X. If H(X,∧TX ⊗ L −p) 6= 0 for some positive integer p, then either (X,L ) ≃ (P,OPn(1)), or p = n and (X,L ) ≃ (Qp,OQp(1)). The statement of this theorem can be interpreted in the following way. Let X be a smooth complex projective n-dimensional variety and L an ample line bundle on X. Consider the sheaf TL := TX ⊗L −1. Then Wahl’s theorem [Wah83] says that if H(X,TL ) 6= 0 then X ≃ Pn. Theorem 1.1 generalizes this statement to the case when one only assumes that H(X,∧TL ) 6= 0 for some 0 < p ≤ n. In order to prove Theorem 1.1, first notice that X is uniruled by [Miy87, Corollary 8.6]. Next observe that if the Picard number of X is 1, then it is necessarily a Fano variety. If the Picard number is larger than 1, then we fix a minimal covering family H of rational curves on X, and follow the strategy in [AW01] of looking at the H-rationally connected quotient π : X◦ → Y ◦ of X (see Section 2 for definitions). We show that any non-zero section s ∈ H(X,∧TX⊗L −p) restricts to a non-zero section s◦ ∈ H(X,∧TX◦/Y ◦⊗L −p), except in the very special case when p = 2 and X ≃ Q2. This is achieved in Section 5. Afterwards we need to deal with two cases: the case when X is a Fano manifold with Picard number 1, and the case in which the H-rationally connected quotient π : X◦ → Y ◦ is either a projective space bundle or a quadric bundle, and H(X,∧TX◦/Y ◦ ⊗ L −p) 6= 0. When X is a Fano manifold with Picard number ρ(X) = 1, the result follows from the following.