Doctor of Philosophy, The Ohio State University, 2018, Mathematics

The first steps in defining a notion of spherical tropicalization were recently taken by Tassos Vogiannou in his thesis and by Kiumars Kaveh and Christopher Manon in a related paper. Broadly speaking, the classical notion of tropicalization concerns itself with valuations on the function field of a toric variety that are invariant under the action of the torus. Spherical tropicalization is similar, but considers instead spherical G-varieties and G-invariant valuations. The core idea of my dissertation is the construction of the extended tropicalization of a spherical embedding. Vogiannou, Kaveh, and Manon only concern themselves with subvarieties of a spherical homogeneous space G/H. My thesis describes how to tropicalize a spherical embedding by tropicalizing the additional G-orbits of X and adding them to the tropicalization of G/H as limit points. This generalizes work done by Kajiwara and Payne for toric varieties and affords a means for understanding how to tropicalize the compactification of a subvariety of G/H in X. The extended tropicalization construction can be described from three different perspectives. The first uses the polyhedral geometry of the colored fan and the second extends the Grobner theory definition given by Kaveh and Manon. The third method works by embedding the spherical variety in a specially-constructed toric variety, tropicalizing there with the standard theory, and then applying a particular piecewise-projection map. This final perspective introduces a novel means for tropicalizing a homogeneous space that allows us to prove several statements about the structure of a spherical tropicalization by transferring results from the toric world where more is known. We also suggest a definition for the tropicalization of subvarieties of a homogeneous space whose defining equations have coefficients with non-trivial valuation. All the previous theory has been done in the constant coefficient case, i.e. when th (open full item for complete abstract)

Committee: Gary Kennedy (Advisor); David Anderson (Committee Member); Maria Angelica Cueto (Committee Member) Subjects: Mathematics