By Jean-Pierre Demailly

This quantity is a selection 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 options valuable within the examine of questions referring to 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 little accustomed to the elemental ideas of sheaf thought, homological algebra, and complicated differential geometry. within the ultimate chapters, a few very fresh questions and open difficulties are addressed--such as effects on the topic of the finiteness of the canonical ring and the abundance conjecture, and effects describing the geometric constitution of Kahler types and their optimistic cones.

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First suppose that f actually coincides with g throughout an entire neighborhood V of N. Let h : sv - y -+ Rp be stereographic projection. Then the homotopy F(x, t) = f(x) F(x, t) = h - l [t · h(f(x)) + (I - t) · h(g(x))] for x t V for x £ lJ1 - N proves that f is smoothly homotopic to g. Thus is suffices to deform f so that it coincides with g in some small neighborhood of N, being careful not to map any new points into y during the deformation. Choose a product representation N X Rv-. V C M for a neighborhood V of N, where Vis small enough so that f(V) and *For example, 0 for t < 1 and ;\(t) = 0 fort 2: 1.

Thus is suffices to deform f so that it coincides with g in some small neighborhood of N, being careful not to map any new points into y during the deformation. Choose a product representation N X Rv-. V C M for a neighborhood V of N, where Vis small enough so that f(V) and *For example, 0 for t < 1 and ;\(t) = 0 fort 2: 1. 49 The Pontryagin construction g(V) do not contain the antipode y of y.

Two framed submanifolds (N, b) and (N', ltl) are framed cobordant if there exists a cobordism X C M X [0, 1] between N and N' and a framing u of X, so that u;(x, t) = (v;(x), 0) u;(x, t) = (w'(x), 0) for (x, t) t: N X [0, e) for (x, t) t: N' X (1 - e, 1]. Again this is an equivalence relation. Now consider a smooth map f : M-+ sv and a regular value y t: sv. The map f induces a framing of the manifold 1 (y) as follows: Choose a positively oriented basis b = (v\ · · · , vv) for the tangent space T(Sv) •.

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