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

By Kazuya Kato, Sampei Usui

In 1970, Phillip Griffiths anticipated that issues at infinity will be further to the classifying house D of polarized Hodge constructions. during this ebook, Kazuya Kato and Sampei Usui detect this dream through making a logarithmic Hodge idea. They use the logarithmic constructions all started through Fontaine-Illusie to restore nilpotent orbits as a logarithmic Hodge structure.

The publication specializes in primary themes. First, Kato and Usui build the fantastic moduli house of polarized logarithmic Hodge buildings with extra buildings. Even for a Hermitian symmetric area D, the current idea is a refinement of the toroidal compactifications by means of Mumford et al. For basic D, effective moduli areas could have slits brought on by Griffiths transversality on the boundary and be now not in the neighborhood compact. moment, Kato and Usui build 8 enlargements of D and describe their relatives through a basic diagram, the place 4 of those enlargements dwell within the Hodge theoretic zone and the opposite 4 dwell within the algebra-group theoretic quarter. those components are attached through a continual 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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If A is an additive category, then CpAq is additive. 39 Proof. Abelian group: Let f, g : K ‚ Ñ L‚ be two morphisms in CpAq. The following diagram commutes di´2 A‚ ... Ai´1 di´1 A‚ Ai ... B i´1 ai`2 f i `g i f i´1 `g i´1 i´2 dB ‚ diA‚ di´i B‚ Bi di`1 A‚ ... f i`1 `g i`1 diB ‚ B i`1 di`1 B‚ ... so f ` g :“ pf i ` g i qiPZ is a morphism of complexes. If α, β : L‚ Ñ M ‚ are two morphisms in CpAq we have ppf ` gq ˝ pα ` βqqi “ pf i ` g i q ˝ pαi ` β i q “ pf i ˝ αi q ` pf i ˝ β i q ` pg i ˝ αi q ` pg i ˝ β i q “ pf ˝ αqi ` pf ˝ βqi ` pg ˝ αqi ` pg ˝ βqi .

Let A be an abelian category and f “ me a morphism of A where m is a monomorphism and e is an epimorphism. Then ker f – ker e and coker f – coker m. Proof. By duality it suffices to show that ker f – ker e. Let k1 : ker f Ñ f and k2 : ker e Ñ e be the corresponding morphisms. Then f k1 “ 0 implies ek1 “ 0 and ek2 “ 0 implies f k2 “ 0. From the universal properties if follows that ker f – ker e. The rest of this section is devoted for a quick introduction to exact sequences in abelian categories.

Exactness at coker h: It suffices to show that g˜2 is an epimorphism. By commutativity g˜2 c2 “ c3 n where c2 , c3 , and n are epimorphisms. Hence g˜2 is an epimorphism. 9) is not in general the zero morphism, consider the following morphism of short exact sequences over Ab. 9) the last morphisms between kernels 0 – kerpZ Ñ Zq Ñ kerpZ Ñ Z{2q – 2Z – Z is not Id surjective and Z – cokerp0 Ñ Zq Ñ cokerpZ Ñ Zq – 0 is not injective. 9) is exact when δ˜ “ IdZ , which is not the zero morphism. If one embeddes this diagram to complexes over Ab in degree 0, one gets ˜ now between complexes, is not the zero morphism.

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