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Hodge Theory (MN-49)$
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Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, and Lê Dung Tráng

Print publication date: 2014

Print ISBN-13: 9780691161341

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691161341.001.0001

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The Hodge Theory of Maps

The Hodge Theory of Maps

Chapter:
(p.257) Chapter Five The Hodge Theory of Maps
Source:
Hodge Theory (MN-49)
Author(s):

Mark Andrea de Cataldo

Luca Migliorini Lectures 1–3

Luca Migliorini

, Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, Lê Dũng Tráng
Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691161341.003.0005

This chapter summarizes the classical results of Hodge theory concerning algebraic maps. Hodge theory gives nontrivial restrictions on the topology of a nonsingular projective variety, or, more generally, of a compact Kähler manifold: the odd Betti numbers are even, the hard Lefschetz theorem, the formality theorem, stating that the real homotopy type of such a variety is, if simply connected, determined by the cohomology ring. Similarly, Hodge theory gives nontrivial topological constraints on algebraic maps. This chapter focuses on the latter, as it considers how the existence of an algebraic map f : XY of complex algebraic varieties is reflected in the topological invariants of X.

Keywords:   Hodge theory, algebraic maps, nontrivial topological constraints, topological invariants, smooth case, mixed Hodge structures, invariant cycle theorem

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