By J. Amoros

This publication is an exposition of what's presently recognized concerning the primary teams of compact Kahler manifolds. This classification of teams includes all finite teams and is exactly smaller than the category of all finitely presentable teams. For the 1st time ever, this ebook collects jointly the entire effects acquired within the previous few years which objective to characterise these countless teams which may come up as primary teams of compact Kahler manifolds. almost all these effects are unfavourable ones, asserting which teams don't come up. they're proved utilizing Hodge conception and its combos with rational homotopy conception, with $L^2$ -cohomology, with the idea of harmonic maps, and with gauge thought. there are various optimistic effects besides, showing fascinating teams as primary teams of Kahler manifolds, in truth, of gentle complicated projective types. The tools and methods used shape an enticing mixture of topology, differential and algebraic geometry, and intricate research. The ebook will be important to researchers and graduate scholars attracted to any of those parts, and it may be used as a textbook for a complicated graduate direction. considered one of its awesome positive factors is a huge variety of concrete examples. The ebook incorporates a variety of new effects and examples that have no longer seemed in other places, in addition to discussions of a few vital open questions within the box

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E iÂ1 ; : : : ; e iÂn / D e i m1 Â1 C Ci mn Ân . Then n 1 n cn >0 m 2 T . Let K be a compact neighborhood of 0 in R . m/ m : s f D f m2Zn \ K Example. Œ 1;1/ sN f is the Fourier sum of f . The following theorem (see [111], p. Rn /. Theorem 5. Let K be a compact convex neighborhood of 0 in Rn and 1 < p < 1. The following statements are equivalent. T n /, b D Œ1K . T / and for every 1 < p < 1. Consequently for every interval I of R and every b D Œ1I . Rn / with b T D Œ1C . But for D the unit ball in Rn (n > 1) and for b D Œ1D .

Dixmier, Chap. I, Sect. 3, no. 4, Corollaire 1, p. 42. The next result is Kaplansky’s density theorem. Theorem 2. H/ with B C. H/. S˛ / of B such that: 1: lim˛ S˛ D T strongly, 2: kS˛ k Ä kT k for every ˛. Proof. See Dixmier, [36], Chap. I, Sect. 3, no. 5, Th´eor`eme 3, p. 43–44. Let G be a locally compact group. In this paragraph, we denote by A the set of all /, where is a complex measure with finite support. G// with unit: 2G . / D 2G . G/ . The C following statement is straightforward. 2 G.

G/: Remark. We will extend this result to p 6D 2 for certain classes of locally compact groups. f˛ /. Chapter 3 The Figa–Talamanca Herz Algebra Let G be a locally compact group. G/, is a Banach algebra for the b pointwise product on G. G/. G / 0 Let G be a locally compact group and 1 < p < 1. G/. ıx /Œ p l; Œ p k : Definition 1. Let G be a locally compact group and 1 < p < 1. ln / < 1. G/. G/. nD1 A. 1007/978-3-642-20656-6 3, © Springer-Verlag Berlin Heidelberg 2011 33 34 3 The Figa–Talamanca Herz Algebra Definition 2.