By Rolf-Peter Holzapfel, Muhammed Uludag, M. Yoshida

This quantity contains lecture notes, survey and examine articles originating from the CIMPA summer season institution mathematics and Geometry round Hypergeometric capabilities held at Galatasaray collage, Istanbul, June 13-25, 2005. It covers quite a lot of themes with regards to hypergeometric capabilities, hence giving a large point of view of the state-of-the-art within the box.

**Read or Download Arithmetic and Geometry Around Hypergeometric Functions: Lecture Notes of a CIMPA Summer School held at Galatasaray University, Istanbul, 2005 PDF**

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**Extra info for Arithmetic and Geometry Around Hypergeometric Functions: Lecture Notes of a CIMPA Summer School held at Galatasaray University, Istanbul, 2005**

**Example text**

A structure of a complex manifold X on a smooth manifold gives a decomposition of the complexiﬁed tangent bundle TM ⊗ C into a holomorphic and anti∂ holomorphic part with local basis ∂z and ∂∂z¯i , respectively. We denote the i holomorphic part by TX . An additional structure of a hermitian complex manifold Moduli of K3 Surfaces 47 on X is given by a choice of a holomorphically varying structures of a hermitian vector space on tangent spaces TX,x deﬁned by a tensor hij (z)dzi ⊗ d¯ zj . 3) It allows one to deﬁne the adjoint operator δ of d , the Laplace operator Δ = δ d + d δ , and the notion of a harmonic form of type (p, q) (an element of the kernel of the Laplace operator).

In this case geodesic triangles are planar triangles in the euclidean geometry with ﬁnite area. The latter property is equivalent to positivity of all angles. Suppose that λ+μ+ν > 1. From spherical geometry it follows that a spherical triangle exists if and only if our condition is satisﬁed. We let W (Δ) be the group of isometries of S generated by the 3 reﬂections through the edges of a geodesic triangle Δ. First we look at subgroups generated by reﬂection in two intersecting geodesics. 25. Let ρ, σ be two geodesics intersecting in a point P with an angle πλ.

Keeping in mind the diverse background of the audience we include a general introduction to the theory of Hodge structures and period domains with more emphasis on abelian varieties and K3 surfaces. So, an expert may start reading the notes from Section 6. Research of the ﬁrst author is partially supported by NSF grant 0245203. Research of the second author is partially supported by Grant-in-Aid for Scientiﬁc Research A-14204001, Japan. 44 Igor V. Dolgachev and Shigeyuki Kond¯ o It has been known for more than a century that a complex structure on a Riemann surface of genus g is determined up to isomorphism by the period matrix Π = ( γj ωi ), where (γ1 , .