We construct an intersection product on tropical cycles contained in the Bergman fan of a matroid. To do this we first establish a connection between the operations of deletion and restriction in matroid theory and tropical modifications as defined by Mikhalkin in [also provides an alternative procedure for intersecting cycles which is not based on intersecting with Cartier divisors. Also, we simplify the definition in the case of one-dimensional fan cycles in two-dimensional matroidal fans and give an application of the intersection product to realiability questions in tropical geometry.
We establish a canonical isomorphism between two bigraded cohomology theories for polyhedral spaces: Dolbeault cohomology of superforms and tropical cohomology. Furthermore, we prove Poincaré duality for cohomology of tropical manifolds, which are polyhedral spaces locally given by Bergman fans of matroids.Furthermore, the authors would like to thank the Graduierten Kolleg "GRK 1692" by the Deutsche Forschungsgemeinschaft for making possible the lecture series by the second author that inspired this collaboration.
SuperformsIf we consider the sections of this diagram over a basic open subset, then the first row is exact by Proposition 3.11. By Lemma 4.32 the second row is also exact. This shows that both rows are exact sequences of sheaves on X. Thus we have a commutative diagram of acyclic resolutions of L p , thus PD induces isomorphisms on the cohomology of the complexes of global sections. This precisely means that X has PD.When X is a compact tropical manifold, the above theorem immediately implies the following.Corollary 4.34. Let X be a compact tropical manifold of dimension n. Then
For a tropical manifold of dimension n we show that the tropical homology
classes of degree (n-1, n-1) which arise as fundamental classes of tropical
cycles are precisely those in the kernel of the eigenwave map. To prove this we
establish a tropical version of the Lefschetz (1, 1)-theorem for rational
polyhedral spaces that relates tropical line bundles to the kernel of the wave
homomorphism on cohomology. Our result for tropical manifolds then follows by
combining this with Poincar\'e duality for integral tropical homology.
Comment: 27 pages, 6 figures, published version
We determine the tropicalizations of very affine surfaces over a valued field that are obtained from del Pezzo surfaces of degree 5, 4 and 3 by removing their (−1)-curves. On these tropical surfaces, the boundary divisors are represented by trees at infinity. These trees are glued together according to the Petersen, Clebsch and Schläfli graphs, respectively. There are 27 trees on each tropical cubic surface, attached to a bounded complex with up to 73 polygons. The maximal cones in the 4-dimensional moduli fan reveal two generic types of such surfaces.
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