# Analytic Methods in Algebraic Geometry (vol. 1 in the by Jean-Pierre Demailly

By Jean-Pierre Demailly

This quantity is a ramification of lectures given by way 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 precious within the examine of questions touching on 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 accustomed to the fundamental recommendations of sheaf concept, homological algebra, and complicated differential geometry. within the ultimate chapters, a few very fresh questions and open difficulties are addressed--such as effects with regards to the finiteness of the canonical ring and the abundance conjecture, and effects describing the geometric constitution of Kahler types and their confident cones.

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Complex Surfaces of General Type: Some Recent Progress 39 Fig. 3. A geometric base of π1 (C − {1, . . n}) Now, it is obvious that Bn acts on the free group π1 (C\{1, . . n}), which has a geometric basis (we take as base point the complex number p := −2ni) γ1 , . . γn as explained in ﬁgure 3. This action is called the Hurwitz action of the braid group and has the following algebraic description • σi (γi ) = γi+1 −1 • σi (γi γi+1 ) = γi γi+1 , whence σi (γi+1 ) = γi+1 γi γi+1 • σi (γj ) = γj for j = i, i + 1.

Remark 13. Obviously there is a lifting of α to Cn , the space of n-tuples of roots of polynomials of degree n and there are (continuous) functions wi (t) n such that wi (0) = i and αt (z) = i=1 (z − wi (t)). Then to each braid is associated a naturally deﬁned permutation τ ∈ Sn given by τ (i) := wi (1). 38 Ingrid C. Bauer, Fabrizio Catanese, and Roberto Pignatelli Fig. 2. Relation aba = bab on braids A very powerful generalization of Artin’s braid group was given by M. Dehn (cf. [Deh38], we refer also to the book [Bir74]).

Bm }) is chosen, determines a factorization of the identity in the mapping class group M apg τ1 ◦ τ2 ◦ · · · ◦ τm = Id as a product of Dehn twists. We are now ready to state the theorem of Kas (cf. [Kas80]). Theorem 31. , there are two diﬀeomorphisms u : M → M , v : P1 → P1 such that f ◦ u = v ◦ f ) if and only if the two corresponding factorizations of the identity in the mapping class group are equivalent (under the equivalence relation generated by Hurwitz equivalence and by simultaneous conjugation).