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)
• 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)

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)‎