By Olivier Debarre

The category concept of algebraic kinds is the point of interest of this e-book. This very lively sector of study continues to be constructing, yet an grand volume of information has accrued over the last 20 years. The authors aim is to supply an simply available advent to the topic. The publication begins with preparatory and conventional definitions and effects, then strikes directly to speak about quite a few facets of the geometry of gentle projective types with many rational curves, and finishes in taking the 1st steps in the direction of Moris minimum version application of class of algebraic forms through proving the cone and contraction theorems. The ebook is well-organized and the writer has saved the variety of techniques which are used yet now not proved to a minimal to supply a typically self-contained creation.

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**Example text**

Rig A principal G x -bundle Prig over an overconvergent space X† (this can either be taken literally as in [GK00] or requires an appropriate interpretation using the work of Berthelot), with a Frobenius. A specialization map Sp from Prig to the restriction of PdR to X† . rig All of this compatible with the corresponding structure on GdR x and G x . Now, for an aﬃne open V ∈ X, we can find a section δ : VK → F0 PdR , which we may reinterpret −1 as a trivialization v : PdR → GdR x × V with v(γ) = (δ(π(γ)) γ, π(γ)).

6 and Rem. 7] is the following. Suppose X = A2 and Di is defined by xi = 0 where xi , i = 1, 2, are the coordinates. One can start with a connection with log singularities along D1 ∪ D2 , take the residue along D1 , which can be interpreted again as a connection on A2 with logarithmic singularities along x1 = 0, x2 = 0, restrict to x1 = 1, take the residue at the point x2 = 0 and restrict to x2 = 1. Then this is exactly the same as taking the fiber at (1, 1) after taking the residue to N00 . Consequently, it is also the same as doing the above procedure {1,2} with the roles of 1 and 2 reversed.

Since p-adic multiple zeta values were defined using the multiple polylogarithm, it is perhaps not surprising that Furusho was only able to prove the analogues of the integral shuﬄe product formulae. The series product formulae were established in [BF06] and further work concerning generalized multiple zeta values, covering the case km = 1 as well, was later done in [FJ07]. The strategy for proving the series shuﬄe relation is rather simple, but certain intricacies have to be overcome by using the tangential base points and their generalizations.