Classifying spaces of degenerating polarized Hodge by Kazuya Kato, Sampei Usui

By Kazuya Kato, Sampei Usui

In 1970, Phillip Griffiths predicted that issues at infinity should be additional to the classifying house D of polarized Hodge constructions. during this publication, Kazuya Kato and Sampei Usui become aware of this dream by means of making a logarithmic Hodge thought. They use the logarithmic buildings all started via Fontaine-Illusie to restore nilpotent orbits as a logarithmic Hodge structure.

The publication makes a speciality of central subject matters. First, Kato and Usui build the high-quality moduli area of polarized logarithmic Hodge constructions with extra buildings. Even for a Hermitian symmetric area D, the current conception is a refinement of the toroidal compactifications by way of Mumford et al. For basic D, positive moduli areas could have slits as a result of Griffiths transversality on the boundary and be not in the neighborhood compact. moment, Kato and Usui build 8 enlargements of D and describe their family by way of a primary diagram, the place 4 of those enlargements reside within the Hodge theoretic region and the opposite 4 reside within the algebra-group theoretic quarter. those components are hooked up by way of a continuing map given via the SL(2)-orbit theorem of Cattani-Kaplan-Schmid. This diagram is used for the development within the first topic.

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Q the analytic set Zj = {Pj = 0} is irreducible, (iii) the set Sing (Zj ) is analytic set of dimension < n − 1 and Zj \ Sing (Zj ) is a connected analytic manifold of dimension n − 1 which is dense in Zj . Proof. Assume, first, that k = 1. We call ∆ = {S1 (P ) = 0, z ∈ B} the discriminant set of P. The complement B\∆ is connected. Note that the set Z\P −1 (∆) is an analytic manifold of dimension n − 1. Fix a point b ∈ B\∆ and consider the fundamental group π1 (B\∆) of loops through b. 2). , αm (z ) along this loop.

Mk (B) = mk+1 (B) . By Nakayama’s lemma we conclude that mk (B) = 0 which implies the inclusion mk (A) ⊂ I. This completes the proof. Problem 4. Show that always mk (A) ⊂ I holds for k = dimC A/I. , zn ] and algebraic varieties Z ⊂ Cn . Theorem 11 If the set Z of common roots of elements of an ideal I is empty, then I contains the unit element. Proof. , ps of I. They have no common root. , gs such that g1 p1 + 6 ... + gs ps = 1. We use induction in n. For n = 0 the statement is trivial. For n > 0 we choose a coordinate system in such a way that all the highest power of zn in pj has constant coefficient for all j.

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