Talk:Mixed Hodge structure

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.

• 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

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

• 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

• 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: $x^2w^\{n-1\}\; +\; y^2w^\{n-1\}\; +\; z^\{n+1\}$ is a smooth compactification, check on the chart $z\; \backslash neq\; 0$

• 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

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

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

• (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

• 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.

• 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)

• 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

• 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)‎