By Jean-Pierre Demailly

This quantity is a spread of lectures given by means of the writer on the Park urban arithmetic Institute (Utah) in 2008, and on different events. the aim of this quantity is to explain analytic thoughts worthwhile within the research of questions relating linear sequence, multiplier beliefs, and vanishing theorems for algebraic vector bundles. the writer goals to be concise in his exposition, assuming that the reader is already a bit of familiar with the elemental innovations of sheaf conception, homological algebra, and intricate differential geometry. within the ultimate chapters, a few very fresh questions and open difficulties are addressed--such as effects concerning the finiteness of the canonical ring and the abundance conjecture, and effects describing the geometric constitution of Kahler kinds and their confident cones.

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**Example text**

Mν = b1 − 1). Suppose that the metrics (hk,ν )ν 1 and hk have been constructed and let us proceed with the construction of (hk+1,ν )ν 1 . We do this again by induction on ν, assuming that hk+1,ν is already constructed and that dim V (Á(hk+1,ν )) > 0. We start in fact the induction with ν = 0, and agree in this case that Á(hk+1,0 ) = 0 (this would correspond Jean-Pierre Demailly, Analytic methods in algebraic geometry 54 to an infinite metric of weight identically equal to −∞). By Nadel’s vanishing theorem applied to Fm = aKX + mL = (aKX + bk L) + (m − bk )L with the metric hk ⊗ (hL )⊗m−bk , we get H q (X, Ç((a + 1)KX + mL) ⊗ Á(hk )) = 0 for q 1, m bk .

Since then, a number of other proofs have been given, one based on connections with logarithmic singularities [EV86], another on Hodge theory for twisted coefficient systems [Kol85], a third one on the Bochner technique [Dem89] (see also [EV92] for a general survey). Since the result is best expressed in terms of multiplier ideal sheaves (avoiding then any unnecessary desingularization in the statement), we feel that the direct approach via Nadel’s vanishing theorem is extremely natural. If D = αj Dj 0 is an effective Q-divisor, we define the multiplier ideal sheaf Á(D) to be equal to Á(ϕ) where ϕ = αj |gj | is the corresponding psh function defined by generators gj of Ç(−Dj ).

The first observation is that Á(ϕ) can be computed easily if ϕ has the form ϕ = αj log |gj | where Dj = gj−1 (0) are nonsingular irreducible divisors with normal crossings. Then Á(ϕ) is the sheaf of functions h on open sets U ⊂ X such that |h|2 |gj |−2αj dV < +∞. U Since locally the gj can be taken to be coordinate functions from a local coordinate m system (z1 , . . e. mj ⌊αj ⌋ (integer part). Hence Á(ϕ) = Ç(−⌊D⌋) = Ç(− ⌊αj ⌋Dj ) § 5. L2 estimates and existence theorems 37 where ⌊D⌋ denotes the integral part of the Q-divisor D = αj Dj .