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Summary: NOTES ON D-MODULES AND CONNECTIONS WITH HODGE
THEORY
DONU ARAPURA
These notes are almost entirely expository and result from my attempt to learn
this material. The first part summarizes D-module theory up to Riemann-Hilbert.
The second part discusses vanishing cycles in its various forms. This is needed in
the next part which summarizes the basics of Morihiko Saito's theory of Hodge
modules. My main motivation for going through all this was to convince myself
that Saito's methods and the more naive construction in [A] yield the same mixed
Hodge structure on the cohomology of a geometric variation of Hodge structure.
The proof of this is given in part 4. Readers interested in just this part, may find
the note [A2] on the"ArXiv" more convenient.
I gave some informal talks on this material at KIAS in Seoul in 2005 and TIFR
Mumbai in 2008. I would them for giving me the opportunity to do so.
Contents
1. D-modules 2
1.1. Weyl Algebra 2
1.6. Dn-modules 3
1.22. Inverse and direct image 5
1.29. Differential operators on affine varieties 7
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