Série de Courts Exposés

Combinatorial Lefschetz Section Theorems

by Prof. Karim Alexander ADIPRASITO (Free University Berlin & IHÉS)

Amphitéâtre Léon Motchane (IHES)

Amphitéâtre Léon Motchane


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