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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Morse theory and toric vector bundles


Author: David Treumann
Journal: Trans. Amer. Math. Soc. 369 (2017), 1-29
MSC (2010): Primary 14M25
DOI: https://doi.org/10.1090/tran/6511
Published electronically: August 22, 2016
MathSciNet review: 3557766
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Abstract: Morelli’s computation of the $K$-theory of a toric variety $X$ associates a polyhedrally constructible function on a real vector space to every equivariant vector bundle $\mathcal {E}$ on $X$. The coherent-constructible correspondence lifts Morelli’s constructible function to a complex of constructible sheaves $\kappa (\mathcal {E})$. We show that certain filtrations of the cohomology of $\kappa (\mathcal {E})$ coming from Morse theory coincide with the Klyachko filtrations of the generic stalk of $\mathcal {E}$. We give Morse-theoretic (i.e. microlocal) conditions for a complex of constructible sheaves to correspond to a vector bundle and to a nef vector bundle.


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David Treumann
Affiliation: Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02467-3806

Received by editor(s): August 21, 2012
Received by editor(s) in revised form: December 27, 2013, June 25, 2014, and July 12, 2014
Published electronically: August 22, 2016
Article copyright: © Copyright 2016 American Mathematical Society