Related papers: Spherical Tropicalization
We prove a complex polynomial plank covering theorem for not necessarily homogeneous polynomials. As the consequence of this result, we extend the complex plank theorem of Ball to the case of planks that are not necessarily centrally…
We establish a correspondence between simplicial fans, not necessarily rational, and certain foliated compact complex manifolds called LVMB-manifolds. In the rational case, Meersseman and Verjovsky have shown that the leaf space is the…
We construct a family of compactifications of the affine cone of the Grassmannian variety of 2-planes. We show that both the tropical variety of the Pl\"ucker ideal and familiar valuations associated to the construction of Newton-Okounkov…
Following DeMeyer, Ford & Miranda [DFM93], we define a topology on a fan by declaring open sets to be its subfans. Then, like Kato [Kat94], we make our fans into monoided spaces by associating a sheaf of monoids to each fan. (Our sheaf of…
Polytropes are both ordinary and tropical polytopes. We show that tropical types of polytropes in $\mathbb{TP}^{n-1}$ are in bijection with cones of a certain Gr\"{o}bner fan $\mathcal{GF}_n$ in $\mathbb{R}^{n^2 - n}$ restricted to a small…
We construct fans which give Kato-Usui partial compactifications of period domains of weight -1. Similarly to the fans of toroidal compactifications, these fans are given by polyhedral decompositions. We also show a fibration structure for…
We establish a canonical isomorphism between two bigraded cohomology theories for polyhedral spaces: Dolbeault cohomology of superforms and tropical cohomology. Furthermore, we prove Poincar\'e duality for cohomology of tropical manifolds,…
We study the tropicalization of the variety of symmetric rank two matrices. Analogously to the result of Markwig and Yu for general tropical rank two matrices, we show that it has a simplicial complex structure as the space of symmetric…
We prove the generalised Mukai conjecture for $\mathbb{Q}$-factorial spherical Fano varieties. In this case, a stronger inequality holds featuring an extra term - the minimum absolute complexity of a log Calabi-Yau pair - which measures how…
Given a spherical homogeneous space G/H of minimal rank, we provide a simple procedure to describe its embeddings as varieties with torus action in terms of divisorial fans. The torus in question is obtained as the identity component of the…
Let X be a plane in a torus over an algebraically closed field K, with tropicalization the matroidal fan Sigma. In this paper we present an algorithm which completely solves the question whether a given one-dimensional balanced polyhedral…
We define the tropical moduli space of covers of a tropical line in the plane as weighted abstract polyhedral complex, and the tropical branch map recording the images of the simple ramifications. Our main result is the invariance of the…
Kato and Usui developed a theory of partial compactifications for quotients of period domains D by arithmetic groups {\Gamma}, in an attempt to generalize the toroidal compactifications of Ash-Mumford-Rapoport-Tai to non-classical cases.…
We generalize classical results about the topology of toric varieties to the case of projective Q-factorial T-varieties of complexity one using the language of divisorial fans. We describe the Hodge-Deligne polynomial in the smooth case,…
We study the tropicalization of the moduli space of algebraic spin curves, exhibit its combinatorial stratification and prove that the strata are irreducible. We construct the moduli space of tropical spin curves, prove that it is…
We introduce the notion of tropical Lagrangian multi-sections over a fan and study its relation with toric vector bundles. We also introduce a "SYZ-type" construction for toric vector bundles which gives a reinterpretation of Kaneyama's…
The map which takes a square matrix to its tropical eigenvalue-eigenvector pair is piecewise linear. We determine the cones of linearity of this map. They are simplicial but they do not form a fan. Motivated by statistical ranking, we also…
We prove that if X, X' are closed subschemes of a torus T over a non-Archimedean field K, of complementary codimension and with finite intersection, then the stable tropical intersection along a (possibly positive-dimensional, possibly…
We translate the equivariant decomposition theorem (in the case of a proper morphism of toric varieties) in to the language of combinatorially defined ``shifted minimal complexes''.
It was shown in [S. Kaliman, M. Zaidenberg, Gromov ellipticity of cones over projective manifolds, Math. Res. Lett. (to appear), arXiv:2303.02036 (2023)] that the affine cones over flag manifolds and rational smooth projective surfaces are…