Lectures on Vanishing Theorems by Helene Esnault, Eckart Viehweg

By Helene Esnault, Eckart Viehweg

This publication, a longer number of lectures introduced at "Schloss Reisenburg" in the course of the DMV-Seminar "Algebraic Geometry, 1991", goals at offering Kodaira's vanishing theorem and a number of other generalizations in a manner that's as algebraic as attainable. We boost the idea of logarithmic de Rham complexes, using the corresponding spectral series, lifting homes for manifolds and their Frobenius morphisms in attribute p zero and the facts of the degenration of the Hodge to de Rham spectral series with algebraic tools, because of P. Deligne and L. Illusie. We follow these how you can receive vanishing theorems. a number of common functions and the favourite theorems of M. eco-friendly and R. Lazarsfeld whole the image. The exposition is self-contained and obtainable to someone with heritage in smooth algebraic geometry. the required formalisms from cohomology thought are recalled in an appendix. The workshop equipped via the Deutsche Mathematiker-Vereinigung (German arithmetic Society) are meant to aid, specifically, scholars and more youthful mathematicians, to acquire an advent to fields of present learn. in the course of the technique of those well-organized seminars, scientists shape different fields is additionally brought to new mathematical rules. The e-book of those workshops within the sequence DMV SEMINAR will make the cloth to be had to a good higher viewers.

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Viehweg: Lectures on Vanishing Theorems By definition 0 ≡ c1 (N )ν+1 = c1 (N )ν · (c1 (φ∗ L) + D) for ν = ν(N ). Since c1 (φ∗ L) is also represented by an effective divisor, this is only possible if 0 ≡ c1 (N )ν · c1 (φ∗ L) = c1 (N )ν−1 · c1 (φ∗ L) · (c1 (φ∗ L) + D). The same argument shows that c1 (φ∗ L)2 · c1 (N )ν−1 ≡ 0 and after ν steps we get c1 (φ∗ L)ν · c1 (φ∗ L) + c1 (φ∗ L)ν ·D ≡ 0 and hence c1 (φ∗ L)ν · D = F · D = 0. ✷ §6 Differential forms and higher direct images The title of this lecture is a little bit misleading.

Hence 2 · (dim g −1 (Sa ) − dim Sa ) ≥ a and H b (W, Ra g∗ V ) = 0 for a + b > n + r(g) ≥ 2 · dim g −1 (Sa ) − dim Sa ≥ a + dim Sa . 13). ✷ 42 H. Esnault, E. 14. Remark. 6) one has cd(X, D) = dim U − 1, whereas r(g) = dim U − 2. 2). 2) directly, whenever it is possible. 2) is not yet complete. The necessary arguments needed to show the E1 -degeneration will only be presented in Lecture 10. Very quickly we will have to restrict ourselves to characteristic zero. ) are too much to ask for in characteristic p = 0.

11. Properties. a) r(g) = Max{ dim Γ − dim g(Γ) − codim Γ; Γ irreducible closed subvariety of U } dim (generic fibre of g |Γ ) − codim Γ; Γ irreducible closed subvariety of U } b) If b denotes the maximal fibre dimension for g, then r(g) ≤ Max{dim U − dim W ; b − 1}. c) If U ⊆ U is open and dense, then r(g |U ) ≤ r(g). d) If ∆ ⊆ U is closed then r(g |∆ ) ≤ r(g) + codimU (∆). §4 Vanishing theorems, the formal set-up. 41 Proof: a) and c) are obvious and b) follows from a). For d) one remarks that for Γ ⊂ ∆ one has codim∆ (Γ) = codimU (Γ) − codimU (∆).

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