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Paradan’s wall crossing formula for partition functions and Khovanski-Pukhlikov differential operator

Arzu Boysal, Michèle Vergne (2009)

Annales de l’institut Fourier

Let P ( s ) be a family of rational polytopes parametrized by inequations. It is known that the volume of P ( s ) is a locally polynomial function of the parameters. Similarly, the number of integral points in P ( s ) is a locally quasi-polynomial function of the parameters. Paul-Émile Paradan proved a jump formula for this function, when crossing a wall. In this article, we give an algebraic proof of this formula. Furthermore, we give a residue formula for the jump, which enables us to compute it.

Positivity properties of toric vector bundles

Milena Hering, Mircea Mustaţă, Sam Payne (2010)

Annales de l’institut Fourier

We show that a torus-equivariant vector bundle on a complete toric variety is nef or ample if and only if its restriction to every invariant curve is nef or ample, respectively. Furthermore, we show that nef toric vector bundles have a nonvanishing global section at every point and deduce that the underlying vector bundle is trivial if and only if its restriction to every invariant curve is trivial. We apply our methods and results to study, in particular, the vector bundles L that arise as the...

Pulling back cohomology classes and dynamical degrees of monomial maps

Jan-Li Lin (2012)

Bulletin de la Société Mathématique de France

We study the pullback maps on cohomology groups for equivariant rational maps (i.e., monomial maps) on toric varieties. Our method is based on the intersection theory on toric varieties. We use the method to determine the dynamical degrees of monomial maps and compute the degrees of the Cremona involution.

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