The Hodge theory of algebraic maps

By M. A. Cataldo, L. Migliorini. 2003

185 citations recorded by the source corpus.

Abstract We give a geometric proof of the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber for the direct image of the intersection cohomology complex under a proper map of complex algebraic varieties. The method rests on new Hodge-theoretic results on the cohomology of projective varieties which extend naturally the classical theory and provide new applications.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1769e5b247f0da8294777ad190728abd62eb751e

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

The decomposition theorem, perverse sheaves and the topology of algebraic maps (2007, methodology): “In the 2000’s, we re-proved these results in [48, 51].”
An overview of Morihiko Saito's theory of mixed Hodge modules (2014, background): “sheavesunderlying polarizableHodge modules. This result, which contains [BBD82, Th´eor`eme 6.2.5] as a special case, is one of the major accomplishments of Saito’s theory.5 5De Cataldo and Migliorini [dCM05] later found a more elementary Hodge-theoretic proof for the decomposition theorem in the spe”
Topology of Hitchin systems and Hodge theory of character varieties (2010, methodology): “, by (1.2.9), MˇDol and Mˆ Dol are homeomorphic to MˇB and MˆB respectively, which are aﬃne complex varieties of dim 6g− 6. Let us recall that the reﬁned intersection forms on the ﬁbres of a map (see [15], §3.4 for a general discussion) is the composite of the two maps H12g−12−r(ˇχ −1(0)) −→ Hr(Mˇ D”
Hodge genera of algebraic varieties I (2006, methodology): “([5], [12]) (1) Decomposition: Let f : X → Y m be a proper map of algebraic varieties of indicated dimensions.”
The perverse filtration and the Lefschetz Hyperplane Theorem (2008, methodology): “Y;Z/ of a quasi projective variety with a mixed Hodge structure and to extend all the results of [8] to the case of quasi projective varieties.”
Global topology of the Hitchin system (2011, methodology): “ght 4 and type (2;2) class, instead of the usual weight 2 and type (1;1) class of the Hard Lefschetz theorem. However there is a situation where a similar Hard Lefschetz theorem was observed. Namely, [CM1] introduce the perverse ltration: P 0 ˆˆ P iˆ:::P k(X) ˘=Hk(X) for f: X!Y proper Xand Y smooth,”
Wild Harmonic Bundles and Wild Pure Twistor D-modules (2008, methodology): “And, de Cataldo-Migliorini [14] gave another proof of the original result of Beilinson-Bernstein-Deligne-Gabber by using their own Hodge theoretic method but without Saito’s method.”
Light ladders and clasp conjectures (2015, unknown): “25h) κ(b,c,d),(0001) = [4]− κ(b,c+1,d−2),(0010) κ(b,c,d−1),(1101) − κ(b+1,c−1,d−1),(0100) κ(b,c,d−1),(1001) − 1 κ(b,c,d−1),(0111) .”
Hitchin fibrations, abelian surfaces, and the P=W conjecture (2019, methodology): “∗(X,Q) = M k,d GkH d(X,Q) splits the perverse ﬁltration, if PkH d(X,Q) = M i≤k GiH d(X,Q), ∀k,d. 1.3. The decomposition theorem. Perverse ﬁltrations can be described through the decomposition theorem [7, 17]. By applying the decomposition theorem to the morphism π : X → Y, we obtain an isomorphism (”