Pantheon SEMPARIS Le serveur des séminaires parisiens Paris

Status Confirmed
Seminar Series MATH-IHES
Subjects math
Date Friday 22 March 2019
Time 11:00
Institute IHES
Seminar Room Amphithéâtre Léon Motchane
Speaker's Last Name Adiprasito
Speaker's First Name Karim
Speaker's Email Address
Speaker's Institution Hebrew University of Jerusalem & IHES
Title Combinatorial Lefschetz Theorems Beyond Positivity
Abstract Consider a simplicial complex that allows for an embedding into R^d. How many faces of dimension d/2or higher can it have? How dense can they be? This basic question goes back to Descartes' "Lost Theorem" and Euler's work on polyhedra. Using it and other fundamental combinatorial problems, we introduce a version of the Kähler package beyond positivity, allowing us to prove the hard Lefschetz theorem for toric varieties (and beyond) even when the ample cone is empty. A particular focus lies on replacing the Hodge-Riemann relations by a non-degeneracy relation at torus-invariant subspaces, allowing us to state and prove a generalization of theorems of Hall and Laman in the setting of toric varieties and, more generally, the face rings of Hochster, Reisner and Stanley. This has several applications including full characterization of the possible face numbers of simplicial rational homology spheres, a generalization of the crossing lemma.)
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