The decomposition theorem, perverse sheaves and the topology of algebraic maps

@article{Cataldo2009TheDT,
  title={The decomposition theorem, perverse sheaves and the topology of algebraic maps},
  author={Mark Andrea de Cataldo and Luca Migliorini},
  journal={Bulletin of the American Mathematical Society},
  year={2009},
  volume={46},
  pages={535-633}
}
We give a motivated introduction to the theory of perverse sheaves, culminating in the decomposition theorem of Beilinson, Bernstein, Deligne and Gabber. A goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of algebraic varieties. While most proofs are omitted, we discuss several approaches to the decomposition theorem, indicate some important applications and examples. 

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References

SHOWING 1-10 OF 280 REFERENCES
The Hodge theory of algebraic maps
Hodge-theoretic aspects of the Decomposition Theorem
Given a projective morphism of compact, complex, algebraic varieties and a relatively ample line bundle on the domain we prove that a suitable choice, dictated by the line bundle, of the
A geometric proof of the existence of Whitney stratifications
A stratification of a singular set, e.g. an algebraic or analytic variety, is, roughly, a partition of it into manifolds so that these manifolds fit together "regularly". A classical theorem of
The perverse filtration and the Lefschetz Hyperplane Theorem
The perverse filtration in cohomology and in cohomology with compact supports is interpreted in terms of kernels of restrictions maps to suitable subvarieties by using the Lefschetz hyperplane
Motivic decomposition and intersection Chow groups, I
For a quasiprojective variety S, we define a category CHM(S) of pure Chow motives over S. Assuming conjectures of Grothendieck and Murre, we show that the decomposition theorem holds in CHM(S). As a
Quivers, perverse sheaves, and quantized enveloping algebras
1. Preliminaries 2. A class of perverse sheaves on Ev 3. Multiplication 4. Restriction 5. Fourier-Deligne transform 6. Analysis of a sink 7. Multiplicative generators 8. Compatibility of
The Chow motive of semismall resolutions
We consider proper, algebraic semismall maps f from a complex algebraic manifold X. We show that the topological Decomposition Theorem implies a "motivic" decomposition theorem for the rational
Etale Cohomology and the Weil Conjecture
I. The Essentials of Etale Cohomology Theory.- II. Rationality of Weil ?-Functions.- III. The Monodromy Theory of Lefschetz Pencils.- IV. Deligne's Proof of the Weil Conjecture.- Appendices.- A I.
Hodge theory and complex algebraic geometry
Introduction. Part I. The Topology of Algebraic Varieties: 1. The Lefschetz theorem on hyperplane sections 2. Lefschetz pencils 3. Monodromy 4. The Leray spectral sequence Part II. Variations of
Hodge genera of algebraic varieties I
The aim of this paper is to study the behavior of Hodge‐theoretic (intersection homology) genera and their associated characteristic classes under proper morphisms of complex algebraic varieties. We
...
...