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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)].

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 (\(n>4\))
Full Text: DOI Euclid