Tropical homology
I Itenberg, L Katzarkov, G Mikhalkin, I Zharkov - Mathematische Annalen, 2019 - Springer
Given a tropical variety X and two non-negative integers p and q we define a homology
group H_ p, q (X) H p, q (X) which is a finite-dimensional vector space over Q Q. We show …
group H_ p, q (X) H p, q (X) which is a finite-dimensional vector space over Q Q. We show …
Cohomologically tropical varieties
E Aksnes, O Amini, M Piquerez, K Shaw - arXiv preprint arXiv:2307.02945, 2023 - arxiv.org
Given the tropicalization of a complex subvariety of the torus, we define a morphism
between the tropical cohomology and the rational cohomology of their respective tropical …
between the tropical cohomology and the rational cohomology of their respective tropical …
Lefschetz section theorems for tropical hypersurfaces
C Arnal, A Renaudineau, K Shaw - Annales Henri Lebesgue, 2021 - numdam.org
We establish variants of the Lefschetz section theorem for the integral tropical homology
groups of tropical hypersurfaces of tropical toric varieties. It follows from these theorems that …
groups of tropical hypersurfaces of tropical toric varieties. It follows from these theorems that …
[HTML][HTML] Local h-polynomials, invariants of subdivisions, and mixed Ehrhart theory
E Katz, A Stapledon - Advances in Mathematics, 2016 - Elsevier
There are natural polynomial invariants of polytopes and lattice polytopes coming from
enumerative combinatorics and Ehrhart theory, namely the h-and h⁎-polynomials …
enumerative combinatorics and Ehrhart theory, namely the h-and h⁎-polynomials …
Tropical refined curve counting via motivic integration
We propose a geometric interpretation of Block and Göttsche's refined tropical curve
counting invariants in terms of virtual χ− y specializations of motivic measures of …
counting invariants in terms of virtual χ− y specializations of motivic measures of …
Real phase structures on tropical manifolds and patchworks in higher codimension
J Rau, A Renaudineau, K Shaw - arXiv preprint arXiv:2310.08313, 2023 - arxiv.org
This paper generalises the homeomorphism theorem behind Viro's combinatorial
patchworking of hypersurfaces in toric varieties to arbitrary codimension using tropical …
patchworking of hypersurfaces in toric varieties to arbitrary codimension using tropical …
Geometric monodromies, mixed Hodge numbers of motivic Milnor fibers and Newton polyhedra
K Takeuchi - arXiv preprint arXiv:2308.09418, 2023 - arxiv.org
We introduce the theory of geometric monodromies in various situations, focusing on its
relations with toric geometry and motivic Milnor fibers, and moreover in the modern …
relations with toric geometry and motivic Milnor fibers, and moreover in the modern …
Euler characteristic and signature of real semi-stable degenerations
E Brugallé - Journal of the Institute of Mathematics of Jussieu, 2023 - cambridge.org
EULER CHARACTERISTIC AND SIGNATURE OF REAL SEMI-STABLE DEGENERATIONS
Page 1 J. Inst. Math. Jussieu (2023), 22(5), 2447–2454 doi:10.1017/S1474748022000056 © …
Page 1 J. Inst. Math. Jussieu (2023), 22(5), 2447–2454 doi:10.1017/S1474748022000056 © …
Formulas for monodromy
A Stapledon - Research in the Mathematical Sciences, 2017 - Springer
Given a family X of complex varieties degenerating over a punctured disk, one is interested
in computing related invariants called the motivic nearby fiber and the refined limit mixed …
in computing related invariants called the motivic nearby fiber and the refined limit mixed …
Cellular Sheaf Cohomology in Polymake
L Kastner, K Shaw, AL Winz - … Geometry: Selected Papers From the 2016 …, 2017 - Springer
This is a guide to the polymake extension cellularSheaves. We first define cellular sheaves
on polyhedral complexes in Euclidean space, as well as cosheaves, and their (co) …
on polyhedral complexes in Euclidean space, as well as cosheaves, and their (co) …