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2015 | OriginalPaper | Chapter

# An Introduction to Hodge Structures

Authors: Sara Angela Filippini, Helge Ruddat, Alan Thompson

Publisher: Springer New York

## Abstract

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 Kähler 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.
Footnotes
1
A third characterization of Hodge structures is given in terms of certain representations of $$\mathrm{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{C}^{{\ast}}$$ (see, for example, [31]). More precisely, a rational Hodge structure of weight n on a $$\mathbb{Q}$$-vector space H can be identified with an algebraic representation $$\rho: \mathbb{C}^{{\ast}}\rightarrow GL(H_{\mathbb{R}})$$, where $$H_{\mathbb{R}}:= H \otimes _{\mathbb{Q}}\mathbb{R}$$, such that the restriction of ρ to $$\mathbb{R}^{{\ast}}$$ is given by $$\rho (\lambda ) =\lambda ^{n}$$. From this point of view, it is clearly completely natural to use constructions from multi-linear algebra to produce new Hodge structures.

2
Note that here we use the original notation by Steenbrink [30]; the two indices p, q are swapped in [25].

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Title
An Introduction to Hodge Structures
Authors
Sara Angela Filippini
Helge Ruddat
Alan Thompson