Talk:Mixed Hodge structure

Resources and suggestions for improving article

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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

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abstract

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  • Tate structure   and its dual
  • extensions, use for curves
  • Mixed hodge structure for Cohomology of a smooth projective variety

complements

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resolutions of singularities

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  • 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:   is a smooth compactification, check on the chart  

Theorems

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Monodromy Weight filtration

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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

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

Singularities

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Monodromy of Milnor fibers

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  • 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

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Other

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  • 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

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Langlands

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— Preceding unsigned comment added by Wundzer (talkcontribs) 19:43, 3 August 2020 (UTC)Reply