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An introduction to Hodge structures
journal contributionposted on 08.10.2018, 12:40 authored by Sara Angela Filippini, Helge Ruddat, Alan ThompsonAlan Thompson
We begin by introducing the concept of a Hodge structure and give some of its basic properties, including the Hodge and Lefschetz decompositions. We then define the period map, which relates families of Kahler manifolds to the families of Hodge structures defined on their cohomology, and discuss its properties. This will lead us to the more general definition of a variation of Hodge structure and the Gauss-Manin connection. We then review the basics about mixed Hodge structures with a view towards degenerations of Hodge structures; including the canonical extension of a vector bundle with connection, Schmid's limiting mixed Hodge structure and Steenbrink's work in the geometric setting. Finally, we give an outlook about Hodge theory in the Gross-Siebert program.
A. Thompson was supported by a Fields-Ontario-PIMS postdoctoral fellowship with funding provided by NSERC, the Ontario Ministry of Training, Colleges and Universities, and an Alberta Advanced Education and Technology Grant.
- Mathematical Sciences