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V). All of these statements hold if the “classical” bundles involved are replaced by Q-twists (provided in (ii ) that the sum or extension is defined ). Proof. Statement (i) follows immediately from the corresponding facts for nef line bundles. For the first assertion in (ii), suppose that E1 and E2 are nef, and fix any ample Q-divisor class h ∈ N 1 (X)Q . 8 (v). Extensions are treated in the same manner. Turning to (iii), note that S k E <δ> = S k E< k1 δ> , and so the amplitude of S k E < h > for all ample Q-divisor classes h is equivalent to the amplitude of S k E for all ample classes h .

G. [252, Chapter 6, §1 (iv)] for a quick overview. 1 on the cohomology of low-codimensional subvarieties of projective space. ] extended some of these results to subvarieties of other homogeneous varieties. 4] also proves a convexity estimate for M − X when X ⊆ M is any smooth subvariety whose normal bundle is ample and globally generated. B Ample Cotangent Bundles and Hyperbolicity We now consider smooth projective varieties with ample cotangent bundles. Such varieties are hyperbolic, and the theme is that they exhibit strong forms of properties known or expected for hyperbolic varieties.

16 Chapter 6. Ample and Nef Vector Bundles (i). E is nef if and only if the following condition is satisfied: Given any finite map ν : C −→ X from a smooth irreducible projective curve C to X, and given any quotient line bundle L of ν ∗ E, one has deg L ≥ 0. (ii). Fix an ample divisor class h ∈ N 1 (X) on X. Then E is ample if and only if there exists a positive rational number δ = δh > 0 such that deg L ≥ δ C · ν ∗ h for any ν : C −→ X and ν ∗ E (*) L as above. Note that in (ii) the constant δ is required to be independent of C, ν and L.