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

This EMS quantity comprises elements. the 1st half is dedicated to the exposition of the cohomology idea of algebraic types. the second one half bargains with algebraic surfaces. The authors have taken pains to provide the fabric carefully and coherently. The ebook includes a number of examples and insights on a number of topics.This booklet could be immensely necessary to mathematicians and graduate scholars operating in algebraic geometry, mathematics algebraic geometry, complicated research and comparable fields.The authors are recognized specialists within the box and I.R. Shafarevich is additionally recognized for being the writer of quantity eleven of the Encyclopaedia.

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**Extra info for Algebraic geometry 02 Cohomology of algebraic varieties, Algebraic surfaces**

**Example text**

Pick a holomorphic coordinate neighborhood U and compact neighborhoods A ⊂ W of A and B ⊂ B ⊂ W ∩ U of B with A ⊂ int A ⊂ A ⊂ W , such that φ is smooth on A . By the preceding discussion, there exists a smooth J-convex function φδ : B → R with |φδ (x) − φ(x)| < ε/2 for all x ∈ B . Pick smooth cutoff functions g, h : V → [0, 1] such that g = 1 on A, g = 0 outside A , h = 1 on B , and h = 0 outside B . Define a continuous function φ : V → R, φ := φ + (1 − g)h(φδ − φ). The function φ is smooth on A ∪ B , |φ(x) − φ(x)| < ε/2 for all x ∈ V , φ = φδ on B \ A , and φ = φ on A and outside B .

If the function f is (strictly) convex, then T2 f (z, 0) + T2 f (iz, 0) = 2 aij zi z¯j ij is positive for all z = 0, so the Levi form is positive definite. This shows that geometric convexity of Σ implies i-convexity. 7 below. It is true, however, locally after a biholomorphic change of coordinates. 12 (Narasimhan). A hypersurface Σ ⊂ Cn is i-convex if and only if it can be made geometrically convex in a neighborhood of each of its points by a biholomorphic change of coordinates. Proof. The ‘if’ follows from the discussion above and the invariance of iconvexity under biholomorphic maps.

21) i=1 ¯ | ≤ |dz ψu | |Z| |W |. Ψzi w¯ Zi W Combining all these relations we estimate n−1 4 Ψz z¯ Zi Z¯j + 2 Re LΣ (X) = |∇Ψ| i,j=1 i j n−1 i=1 ¯ + Ψww¯ |W |2 Ψzi w¯ Zi W 1 Hψmin |Z|2 − 2|dz ψu | |Z| |W | − |ψuu | |W |2 |∇Ψ| |Z|2 ≥ Hψmin |Ψw |2 − 2|dz ψu | |Ψz | |Ψw | − |ψuu | |Ψz |2 |∇Ψ| |Ψw |2 1 ≥ Hψmin (1 + ψu2 ) − 2|dz ψu | |dz ψ| 1 + ψu2 − |ψuu | |dz ψ|2 . 21). Since |∇Ψ|2 = 1 + ψu2 + |dz ψ|2 , this concludes the proof. 3 Smoothing In this chapter we develop some techniques for constructing J-convex functions.