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We begin by genera1izing the concept of positivity. 1) Definition. A herrnitian ho1omorphic 1ine bund1e E on an n-dimensiona1 comp1ex manifo1d X is k-positive [k-negative] at x E: X * -* i f the curvature 0 x E: Tx 0 Tx is semi-positive [semi-negative] and has at least n-k positive [negative] eigenva1ues. bund1e on A ho1omorphic line X is k-positive [k-negative] if it carries a herrnitian metric that is k-positive [k-negative] at all points of Note that negative. X. 2) Theorem. Let E be a k-negative ho1omorphic 1ine bund1e on a cornpact Kähler manifold X.

An (x) < ° (and thus E is semi-Eositive. > llq < A. q+ n q ° for [a jk ] ° [ll q- 0] ° [ll q- 0] E is positive If > for all q; if < for all q. 22). 27) Theorem. for some point Proof: Let X o e: ° ~ q ~ X, then We must show that n - 1. llq If ~ ° on X = 0. Hq(X, E) HO,q(X,E) 0. Let a e: HO,q(X,E). 58) ° =Da = olV'a + V'o'a - vCT i\(8" a). 28) vCT (81\ a, W " a) > 0. a, W 1\ a), By the -35- We now compute the pointwise inner product Let x € X and let {nl, •.. ,nn} <0 1\ CI, W " CI}. be an orthonormal basis for Tx* such that n L o (x) j=l AJ" (x) nJ" 1\ nj .

56) K~h1er The map is injective for k > n. The map Ak+2 Tc* L is injective for Proof: k < n. 42), L We show the first: s [A,L s ] > 1. Suppose Therefore if k > o and thus u = *A*w, it suffices to show either statement. 57) for manifo1ds. U E = sen - k - s + 1)Ls - 1 A k T* such that c Au o. (1 - k)(2 - k) ••• (n - k)u, O. D. We now consider the Lap1acians 6 6' + Cl -45- with respect to the trivial line bundle on metric. Then V =d and V' = 3. 64) (where ~ denotes the trivial 1ine bund1e). 69) Theorem.

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