By I.R. Shafarevich (editor), R. Treger, V.I. Danilov, V.A. Iskovskikh

This EMS quantity includes elements. the 1st half is dedicated to the exposition of the cohomology concept of algebraic kinds. the second one half bargains with algebraic surfaces. The authors have taken pains to offer the fabric carefully and coherently. The e-book comprises a number of examples and insights on a variety of topics.This ebook could be immensely important to mathematicians and graduate scholars operating in algebraic geometry, mathematics algebraic geometry, complicated research and similar fields.The authors are recognized specialists within the box and I.R. Shafarevich can also be recognized for being the writer of quantity eleven of the Encyclopaedia.

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4 (The classification of S13 -catenoids of parabolic type). 2]) fP = Fe3 F ∗ , where i F= √ 2 2 √ z 0 √ 0 1/ z 3 − log(z) −1 + log(z) . 1 + log(z) −3 − log(z) (30) (see Fig. ) Moreover, fP is complete with embedded ends. The end z = 0 lies in the upper ideal boundary ∂+ S13 and the end z = ∞ lies in the lower ideal boundary ∂− S13 ( see [5] for the definition of ∂± S13 ). Proof. We set gP := log z + 1 = X logz, log z − 1 i X := √ 2 1 1 , 1 −1 (31) which has PSU(1, 1)-monodromy. The above CMC-1 face fP has hyperbolic Gauss map G(= z), secondary Gauss map gP , and Hopf differential Q = dz2 /(4z2 ).

Then the secondary Gauss map g(z) satisfies |g(0)| = 1. If |g(0)| > 1, then we replace g by 1/g, and may assume that |g(0)| < 1. Since SU(1, 1) acts on the unit disk as an isometry of the Poincar´e disk transitively, there exists an a ∈ SU(1, 1) such that a g(0) = 0. By replacing g by a g, we may assume that g(0) = 0, which implies that b12 = 0 and b22 = 0. Then, by replacing g with 1/g, we may set g= b21 b22 −m−1 + z . b11 b11 By replacing the coordinate z with 1/z, we have that g= b21 b22 m+1 + z .

4. Non-co-orientable extended hyperbolic metrics on the 2-sphere with two regular singularities are obtained from the pullback of the spherical Poincar´e metric by the following meromorphic functions, defined on the universal covering of C \ {0}, g= ζ (2m−1+iτ )/2 − i ζ (2m−1+iτ )/2 + i (m ∈ Z, τ ≥ 0), (45) Hyperbolic Metrics and Space-Like CMC-1 Surfaces 43 where ζ is the canonical coordinate of C. If τ = 0 (resp. τ = 0), then the double covering of the metric has elliptic (resp. hyperbolic) monodromy.