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# Combinatorial Lefschetz Section Theorems

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Amphitéâtre Léon Motchane (IHES)
### Amphitéâtre Léon Motchane

#### IHES

Le Bois Marie
35, route de Chartres
91440 Bures-sur-Yvette

Description

Intuitively, the classical variants of the Lefschetz Section Theorem relate a complex algebraic variety X to the intersection of X with a hyperplane H transversal to X (or, alternatively, to an ample divisor D of X). They are tremendously useful to compute invariants of the variety.
However, Lefschetz Section Theorems also hold for spaces that are constructed combinatorially rather than algebraically. Among other things, I will introduce Lefschetz theorems for -- certain real subspace arrangements and their complements, -- toric arrangements and their complements and, -- matroids and smooth tropical varieties (joint work with Anders Björner). These theorems translate results of Lefschetz, Hamm-Le, Andreotti--Frankel, Bott--Thom, Akizuki--Kodaira--Nakano and Kodaira--Spencer to a combinatorial setting.

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