Quasi-projective moduli for polarized manifolds
This book discusses two subjects of quite different nature: Construction methods for quotients of quasi-projective schemes by group actions or by equivalence relations and properties of direct images of certain sheaves under smooth morphisms. Both methods together allow to prove the central result of the text, the existence of quasi-projective moduli schemes, whose points parametrize the set of manifolds with ample canonical divisors or the set of polarized manifolds with a semi-ample canonical divisor. Starting with A. Grothendieck's construction of Hibert schemes, including the basics of D. Mumford's geometric invariant theory and an introduction to M. Artin's theory of algebraic spaces, the reader finds the tools for the construction of moduli, usually not contained in textbooks on algebraic geometry.
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Moduli Problems and Hilbert Schemes
Weakly Positive Sheaves and Vanishing Theorems
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action of G algebraic moduli space algebraic space ample invertible sheaf ample sheaf arbitrary base change assume canonical singularities canonically polarized Cartier divisor characteristic zero choose closed subscheme coarse moduli scheme commutative compactification condition connected component consider Corollary defined definition denotes desingularization diagram direct factor direct image embedding equivalence relation etale exists finite covering finite morphism G x H G-action G-invariant G-linearization geometric quotient given global sections Gorenstein Hence Hilbert scheme holds true implies induced morphism isomorphism Kollar Lemma Let us write locally closed locally free sheaf moduli functor moduli problem moduli spaces neighborhood non-singular normal crossing divisor normal varieties notations numerically effective obtains open dense subscheme open subscheme polarized manifolds projective Proof proper properties pullback Q-Gorenstein schemes quasi-projective moduli scheme quasi-projective scheme rational singularities reduced scheme replace restriction satisfying the assumptions scheme H semi-ample semi-stable smooth surjective universal family Vanishing Theorem weakly positive