On Semipositivity of Sheaves of Differential Operators and the Degree of a Unipolar Q-fano Variety

Recall that a Q-Fano variety is by definition a normal projective variety X such that the anticanonical divisor class −K = −K X is Q-Cartier and ample. For such X we define the (Weil) index i = i(X) to be the largest integer such that K X /i exists as a Weil divisor (see [R] for a discussion of Weil divisors and reflexive sheaves, and also Lemma 2 below; NB i differs from the (industry-standard) Cartier index) , the divisor class group N (X) to be the group of Weil divisors modulo rational equivalence, and the Picard number ρ = ρ(X) to be the rank of N (X). When ρ = 1, X is said to be unipolar; note that in this case the singularities of X are automatically Q-factorial. The purpose of this paper is to prove the following theorem over C: Theorem 0. Let X be an n-dimensional unipolar Q-Fano variety of index i. Then (i) if X has log-terminal singularities, we have (0.1) (−K X) n ≤ (max(in, n + 1)) n ; (ii) if the tangent sheaf T X = (Ω X) * is semistable (with respect to −K X) (sin-gularities only assumed normal), we have