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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeDec 19th 2012

created Hodge structure. Currently with nothing but a pointer to this nice book:

• Chris Peters, Jozef Steenbrink, Mixed Hodge Structures, Ergebisse der Mathematik (2007) (pdf)

Eventually I’d think we should move over Hodge-structure articles from Hodge theory to here. But not tonight.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeDec 19th 2012
• (edited Dec 19th 2012)

Earlier today I had given Hodge structure a basic Idea-section.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeDec 19th 2012
• (edited Dec 19th 2012)

To make this more interesting, even in the absence of any time to do this justice, let me say what I am driven by here:

There are the following two facts:

• a mixed Hodge structure is essentially (put a tad more generally than usually done)

1. equipped with the structure of a filtration on its Chern character, namely on its image in real/de Rham cohomology

this is precisely the structure needed to pass to the corresponding differential cohomology (Peters-Steenbrinck make that connection explicit, for ordinary cohomology, in section 7.2)

• motives are supposed to “embed fully faithfully into Hodge structures”.

At the beginning this starts out sounding familiar and boring. But if you read these items aloud consecutively, they seem to be hinting at some story to be made more explicit. I’d like to find more pieces of that story. Eventually.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeJun 5th 2014
• (edited Jun 5th 2014)

added the definition of Hodge filtration for a complex analytic space, together with some basic remarks.

Also created holomorphic de Rham complex

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJun 5th 2014
• (edited Jun 5th 2014)

expanded the technical section a tad more, and then I added the following paragraph to the Idea section at Hodge structure

By a central theorem of Hodge theory (recalled as theorem \ref{HodgeFiltrationForComplexSpaceReproducesKaehlerHodgeStructure} below) the traditional (and original) filtration on the complex cohomology of a Kähler manifold induced by the harmonic differential forms generalizes to a filtration of the complex-valued ordinary cohomology of any complex analytic space which is simply given by the canonical degree-filtration of the holomorphic de Rham complex.

This means that ordinary differential cohomology in the guise of Deligne cohomology is nothing but the homotopy pullback of a stage of the Hodge filtration along the “Chern character” map from integral to complex cohomology. (A point of view highlighted for instance in Peters-Steenbrink 08, section 7.2). Viewed this way Hodge structures are filtrations of stages of differential form cycle refinements of Chern characters that appear in the general definition/characterization of differential cohomology, as discussed at differential cohomology hexagon starting around the section de Rham coefficients

This modern point of view is also crucial for instance in the characterization of an intermediate Jacobian (see there) as the subgroup of Deligne cohomology that is in the kernel of the map to Hodge-filtering stage of ordinary cohomology. See at intermediate Jacobian – characterization as Hodge-trivial Deligne cohomology.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeJun 5th 2014

added the basic definition of the Hodge filtration on Kähler manifolds.

In the course of this I wanted to be able to link to Hodge isomorphism, so I have split off a brief entry for that.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeJun 16th 2014

I should say: the other day I had added some more standard stuff to Hodge structure.

• a remark on what the equivalence given by the degeneration of the Frölicher spectral sequence means in components

• some minimum on the actual abstract definition of Hodge structures on abelian groups

• CommentRowNumber8.
• CommentAuthorstilson
• CommentTimeNov 23rd 2015
Thanks for creating this entry Urs. Two questions:
1) Can you elaborate on Remark 2? It is unclear how the sequence is affected by the filtration as nothing is stated about what is to the right or left of the parts you have not ommitted. Further, should it be $\Omega_X^{p+1}$ as opposed to $\Omega^{p+1}$?

2) In definition 4, I think the end of the first sentence should be $H^{p,q}=\overbar{H^{q,p}}$ and not homotopic/isomorphic. I was/am confused by this. However, I think there can not be a symmetric monoidal unit in the category of pure Hodge structures unless that is an equality of sets and not an isomorphism of vector spaces. Also, this is how it is presented in the book you reference. I would be happy for you to explain why I am misundertanding this.

Thanks for your work on this site.
• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeNov 23rd 2015

I’ll try to find time to get back to this. But if there is anything that you recognize as wrong, please just go and fix it right away.

• CommentRowNumber10.
• CommentAuthorstilson
• CommentTimeNov 25th 2015
Urs, I will edit point 2. I hesitated because this is a point that confused me and I was unsure if I was correct until now.

I of course understand this will take time and I don't think this should be your highest priority. Thanks again for all of this stuff you have put up here. :)
• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeNov 25th 2015
• (edited Nov 25th 2015)

Okay, thanks. I’ll try to come back to it.