\(L^ 2\) vanishing theorems for positive line bundles and adjunction theory. (English) Zbl 0883.14005

Demailly, Jean-Pierre

Catanese, F. (ed.) et al., Transcendental methods in algebraic geometry. Lectures given at the 3rd session of the Centro Internazionale Matematico Estivo (CIME), Cetraro, Italy, July 4–12, 1994. Cetraro: Springer. Lect. Notes Math. 1646, 1-97 (1996).

Main goal of the paper is to describe a few analytic tools which are useful to study questions such as linear series and vanishing theorems for algebraic vector bundles. Also, algebraic and analytic proofs of some results are compared. One of the first applications of the analytic method in algebraic geometry is Kodaira’s use of the Bochner technique (1950-60) to relate cohomology and curvature via harmonic forms. Well known is the Akizuki-Kodaira-Nakano theorem (1954): If \(X\) is a nonsingular projective algebraic variety and \(L\) is a holomorphic line bundle on \(X\) with positive curvature, then \(H^q(X,\Omega^p_X\otimes L)=0\) for \(p+q>\) dim\(X\). Hörmander (1965) used a refinement of this technique to obtain a fundamental \(L^2\) estimate, concerning solutions of the Cauchy-Riemann operator. Except vanishing theorems, more precise quantative information about solutions of \(\bar{\partial}\)-equations was obtained. Main tools to relate analytic and algebraic geometry are the multiplier ideal sheaf \(I(\phi)\) and positive currents. \(I(\phi)\) is defined as a sheaf of germs of holomorphic functions \(f\) such that \(|f|^2e^{-2\phi}\) is locally summable, where \(\phi\) is a (locally defined) plurisubharmonic function. Since \(I(\phi)\) is a coherent algebraic sheaf over \(X\), we have a direct correspondence between analytic and algebraic objects which takes into account singularities efficiently. Currents, introduced by Lelong (1957), play the role of algebraic cycles, and many classical results of intersection theory apply to currents. Also an analytic interpretation of the Seshadri constant of a line bundle is given and it represents a measure of local positivity. One of the motivations for this work was the conjecture of Fujita: If \(L\) is an ample (i.e. positive) line bundle on a projective \(n\)-dimensional algebraic variety \(X\) then \(K_X+(n+2)L\) is very ample. Reider (1988) gave a proof of the Fujita conjecture in the case of surfaces.
Using an analytic approach, in the paper under review it is shown that \(2K_X+L\) is very ample under suitable numerical conditions for \(L\). The first part of the proof is to choose an appropriate metric using a complex Monge-Ampère equation and the Aubin-Calabi-Yau theorem. Solution \(\phi\) of the equation assumes logarithmic poles and they are controlled using the intersection theory of currents. Detailed relations to the existing algebraic proofs of similar results are given (Ein-Lazarsfeld, Fujita, Siu). In the last section, a proof of the effective Matsusaka big theorem obtained by Y.-T. Siu [Ann. Inst. Fourier 43, No. 5, 1387-1405; Zbl 0803.32017)] is presented. Siu’s proof is based on the very ampleness of \(2K_X+mL\) together with the theory of holomorphic Morse inequalities [J.-P. Demailly, Ann. Inst. Fourier 35, No. 4, 189-229 (1985; Zbl 0565.58017)]. Long and detailed preliminary sections dedicated to the basic facts of complex differential geometry are included which make the main ideas of the paper easier to understand.
For the entire collection see [Zbl 0855.00017].

Reviewer: N.Blažić (Beograd)

MSC:

14F17	Vanishing theorems in algebraic geometry
32L05	Holomorphic bundles and generalizations
14F43	Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
14F05	Sheaves, derived categories of sheaves, etc. (MSC2010)
32L20	Vanishing theorems
32C30	Integration on analytic sets and spaces, currents
32W20	Complex Monge-Ampère operators

Keywords:

positive line bundle; linear series; vanishing theorems; Lelong number; intersection theory; Bochner technique; \(L^ 2\) estimates; Seshadri constant; numerically effective line bundle; Fujita conjecture; Monge-Ampere equation; very ample line bundle; algebraic vector bundles; effective Matsusaka big theorem

Citations:

Zbl 0565.58017; Zbl 0803.32017; Zbl 0575.58013