Lefschetz Properties in Algebra, Geometry and Combinatorics

2021 ◽  
Vol 17 (4) ◽  
pp. 1539-1600
Author(s):  
Martina Juhnke-Kubitzke ◽  
Juan C. Migliore ◽  
Rosa Maria Miró-Roig ◽  
Justyna Szpond
Keyword(s):  
Lefschetz Properties

AUTOMORPHIC LEFSCHETZ PROPERTIES FOR NONCOMPACT ARITHMETIC MANIFOLDS

2021 ◽  
pp. 1-48
Author(s):  
Arvind N. Nair ◽  
Ankit Rai
Keyword(s):  
Symmetric Spaces ◽  
Automorphic Forms ◽  
Hodge Theory ◽  
Hyperbolic Manifolds ◽  
Shimura Varieties ◽  
Restriction Maps ◽  
Locally Symmetric Spaces ◽  
Compact Congruence ◽  
Lefschetz Properties ◽  
Invent Math

Abstract We prove the injectivity of Oda-type restriction maps for the cohomology of noncompact congruence quotients of symmetric spaces. This includes results for restriction between (1) congruence real hyperbolic manifolds, (2) congruence complex hyperbolic manifolds, and (3) orthogonal Shimura varieties. These results generalize results for compact congruence quotients by Bergeron and Clozel [Quelques conséquences des travaux d’Arthur pour le spectre et la topologie des variétés hyperboliques, Invent. Math.192 (2013), 505–532] and Venkataramana [Cohomology of compact locally symmetric spaces, Compos. Math.125 (2001), 221–253]. The proofs combine techniques of mixed Hodge theory and methods involving automorphic forms.

scholarly journals The Weak and Strong Lefschetz properties for Artinian K-algebras

2003 ◽  
Vol 262 (1) ◽  
pp. 99-126 ◽  
Author(s):  
Tadahito Harima ◽  
Juan C. Migliore ◽  
Uwe Nagel ◽  
Junzo Watanabe
Keyword(s):  
Lefschetz Properties

scholarly journals Laplace equations, Lefschetz properties and line arrangements

2018 ◽  
Vol 222 (9) ◽  
pp. 2657-2666 ◽  
Author(s):  
Roberta Di Gennaro ◽  
Giovanna Ilardi
Keyword(s):  
Line Arrangements ◽  
Laplace Equations ◽  
Lefschetz Properties

scholarly journals PRINCIPAL RADICAL SYSTEMS, LEFSCHETZ PROPERTIES, AND PERFECTION OF SPECHT IDEALS OF TWO-ROWED PARTITIONS

2021 ◽  
pp. 1-41
Author(s):  
CHRIS MCDANIEL ◽  
JUNZO WATANABE
Keyword(s):  
Lower Bound ◽  
Complex Numbers ◽  
Arbitrary Field ◽  
Cherednik Algebras ◽  
Weak Lefschetz Property ◽  
Lefschetz Property ◽  
Lefschetz Properties ◽  
Bounded Below ◽  
Rational Cherednik Algebras

Abstract We show that the Specht ideal of a two-rowed partition is perfect over an arbitrary field, provided that the characteristic is either zero or bounded below by the size of the second row of the partition, and we show this lower bound is tight. We also establish perfection and other properties of certain variants of Specht ideals, and find a surprising connection to the weak Lefschetz property. Our results, in particular, give a self-contained proof of Cohen–Macaulayness of certain h-equals sets, a result previously obtained by Etingof–Gorsky–Losev over the complex numbers using rational Cherednik algebras.

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