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A study on the dimension of global sections of adjoint bundles for polarized manifolds. II. (English) Zbl 1225.14007
Let $$X$$ be a smooth complex projective variety and $$L$$ an ample vector bundle on $$X$$. It is of interest to study the adjoint linear series of the type $$|K_X+tL|$$, in particular their dimension. For this purpose in the paper under review invariants $$A_j:=A_j(X,L)$$ ($$j=0,\dots, n:=\dim(X)$$) are introduced: consider $$t$$ a positive integer, $$F_0(t):=h^0(K_X+tL)$$ and $$F_i(t):=F_{i-1}(t+1)-F_{i-1}(t)$$ for $$1 \leq i \leq n$$; $$A_j$$ is defined to be $$F_{n-j}(1)$$. They lead to the formula $$h^0(K_X+tL)=\Sigma_{j=0}^n {t-1 \choose n-j}A_j$$. In Section 3, nonnegativity of $$A_1$$ and $$A_2$$; and positivity of $$A_3$$ (when Kodaira dimension is positive) are proved. Some other lower bounds and classifications of the extremal cases are also provided (see Thm 3.1.1 for precise statements). In Section 4 these results on the $$A_j$$’s are used to study $$h^0(K_X+tL)$$ (essentially giving lower bounds) under certain assumptions on the pair $$(X,L)$$.
For part I, cf. [J. Algebra 320, No. 9, 3543–3558 (2008; Zbl 1160.14002)].

##### MSC:
 14C20 Divisors, linear systems, invertible sheaves 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 14J30 $$3$$-folds 14J35 $$4$$-folds 14J40 $$n$$-folds ($$n>4$$)
##### Keywords:
polarized manifolds; adjoint bundles; Hilbert coefficients
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