Talk:Mixed Hodge structure

Resources and suggestions for improving article

If you're interested in helping improve this article, I've outlined resources and data points which should be added. These additions will make a more "feature complete" article making mixed Hodge structures more accessible to a (mature enough) general mathematics audience.

Examples section

http://geometry.ma.ic.ac.uk/acorti/wp-content/uploads/2016/07/mixedLSGNT.pdf
https://arxiv.org/abs/1412.8499
https://www.math.bgu.ac.il/~kernerdm/eTexts/Steenbrink.Mixed.Hodge.Structure.Singularities.Survey.pdf
Mixed Hodge Structures book for intermediate Jacobians

abstract

Tate structure $H^{1}(\mathbb {G} _{m};\mathbb {Z} )=\mathbb {Z} (-1)$ and its dual
extensions, use for curves
Mixed hodge structure for Cohomology of a smooth projective variety

complements

Elliptic curves minus points (or algebraic curves minus points) -> need Hypercohomology, check out https://mathoverflow.net/questions/21483/question-about-hypercohomology-spectral-sequence-of-a-complex-of-almost-acycl

resolutions of singularities

Mention proper base change
Add example of normalization of curves
Add example of resolution of singularities for an A_n singular surface
https://arxiv.org/abs/alg-geom/9602006 Pg 82: $x^{2}w^{n-1}+y^{2}w^{n-1}+z^{n+1}$ is a smooth compactification, check on the chart $z\neq 0$

Theorems

Deligne's global invariant cycle theorem
Monodromy Weight filtration/theorem
Bounding the weights — Preceding unsigned comment added by Wundzer (talk • contribs) 03:33, 12 August 2020 (UTC)[reply]
Mixed hodge module's theorem for intersection cohomology

Monodromy Weight filtration

Hodge Theory of maps Part I Milgiorini contains the relevant material.

pg 265 (pg 281 in pdf) contains example of decomposition theorem
continuing from there the rest contains everything required for monodromy weight filtration
https://www.math.purdue.edu/~arapura/preprints/limitmhs.pdf has more useful information
good examples can be found using stable reduction, or use a Lefschetz fibration

Nilpotent orbit theorems

pg 294 (pdf 310) of Hodge theory of Cattani, Zein, Griffiths, Trang

Singularities

(MULTIPLIER IDEALS, MILNOR FIBERS, AND OTHER SINGULARITY INVARIANTS) - https://pdfs.semanticscholar.org/d683/1f275409bb35b5704619bebe7b20a784ef16.pdf
https://arxiv.org/abs/1012.3150 (pg 5-6 helps give intuition for why rational indices are used while constructing/defining mixed Hodge modules)
http://www.numdam.org/article/CM_1995__97_1-2_285_0.pdf - monodromy and weight filtration + signature of intersection forms for isolated singularities
https://arxiv.org/abs/1703.07146 - COMPUTING MILNOR FIBER MONODROMY FOR SOME PROJECTIVE HYPERSURFACES (Dimca)
Chapter 3 of Singularities and Topology of Hypersurfaces
https://math.unice.fr/~dimca/sing.pdf
http://www.numdam.org/item/AIF_2007__57_3_775_0/ - Movasati - MHS of isolated singularities
MHM intro - http://www.numdam.org/article/AST_1989__179-180__145_0.pdf - contains references to all relevant papers

Monodromy of Milnor fibers

Computing the eigenvalues for the monodromy of Milnor fibers can be done by looking at b-functions. The Multiplier ideals, milnor fibers, and other singularity invariants contains an intro with some of the related theorems. Maybe this material should be contained in a spin-off article about Milnor fibers.

In Mirror Symmetry

https://projecteuclid.org/euclid.atmp/1312998216
http://swc.math.arizona.edu/aws/2004/notes.html — Preceding unsigned comment added by Wundzer (talk • contribs) 03:30, 12 August 2020 (UTC)[reply]

Other

Grothendieck Symbol gives the cycle class of a variety: checkout https://hal.archives-ouvertes.fr/hal-01271554/document
That document contains examples for relative and local cohomology, also good examples with curves: e.g. compactifying a curve with points gives an extension of mixed Hodge structures

Applications

Langlands

http://www-users.math.umn.edu/~kwlan/articles/iccm-2016.pdf

— Preceding unsigned comment added by Wundzer (talk • contribs) 19:43, 3 August 2020 (UTC)‎[reply]