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.