Tropical Hopf manifolds and contracting germs

Ruggiero, Matteo; Shaw, Kristin

doi:10.1007/s00229-016-0849-8

Cited by 4 publications

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“…This tropical complex is a tropical analogue of Hopf surfaces, which are non-algebraic complex surfaces. Tropical Hopf manifolds are explored at greater length, and in a different framework, in [RS16]. For the edges forming the top and bottom circles, the structure constants are indicated by the numbers adjacent to each endpoint.…”

Section: Linear Seriesmentioning

confidence: 99%

A specialization inequality for tropical complexes

Cartwright¹

2021

Compositio Math.

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Section: Linear Seriesmentioning

confidence: 99%

A specialization inequality for tropical complexes

Cartwright¹

2021

Compositio Math.

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“…We can find an analogy in non-standard analysis: the tropical line is the hyperreal line, the modification at a point is an approaching this point with an infinitesimal telescope, see Figure 10 and Section 4.2. In order to define tropical Hopf manifolds one should also use the modifications to study certain germs [49].…”

Section: Intuitions and Interpretationsmentioning

confidence: 99%

A guide to tropical modifications

Kalinin¹

2015

Preprint

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The last version is available on http://mathcenter.spb.ru/nikaan/modification.pdf This is an ongoing survey on tropical modifications and I would be glad to receive any comments (e.g. about papers using modifications that I should mention here) and suggestions (about expositions). If you have any questions about tropical modifications, do not hesitate to contact me by email nikaanspb on gmail.com.Substance is by nature prior to its modifications. ... nothing is granted in addition to the understanding, except substance and its modifications.Ethics. Benedictus de Spinoza.This paper surveys tropical modifications, which have already become a folklore in tropical geometry. Tropical modifications are used in tropical intersection theory and in study of singularities. They admit interpretations in various contexts such as hyperbolic geometry, Berkovich spaces, and non-standard analysis.We cite [10]: "Tropical modifications ... can be seen as a refinement of the tropicalization process, and allows one to recover some information ... sensitive to higher order terms."One must say that the name "modification" is used in two different senses: the modification as a well-defined operation (defined, e.g. in [37]); and a modification along N as a method that reveals a behavior of other varieties in an infinitesimal neighborhood of N . Namely, performing the modification of M along N ⊂ M , we will see how M changes, but the objects of codimension one in M may behave differently, depending on their behavior near N . We will clarify this distinction with examples.Our main goal is to mention different points of view, to give references, and to demonstrate the abilities of tropical modifications. We assume that the reader have already met "tropical modifications" somewhere and wants to understand them better.There are novelties here: a new obstruction (Theorem 2.29) for realizability of non-transversal intersections and a tropical version of Weil's reciprocity law (Theorem 2.10). A generalization of tropical momentum is given in Section 2.6.As a preliminary introduction to tropical geometry, see [8], [9] and [40], where tropical modifications are also discussed. We are glad to mention other texts, promoting modifications from different perspectives: [10] (examples, construction of curves with inflection points), [13] (repairing the j-invariant of elliptic curves), [14] (tropical genus two curves), [20] (tropical quartics curves), [50] (intersection theory on tropical surfaces), [56] (tropical K3 surfaces), [22] (tropical Dolbeault cohomology), [15] (presenting metric graphs as tropical planar curves), [23] (gonality of tropical curves), [12] (lifting divisors), [47] (constructions of real algebraic varieties), [5] (constructing tropical curves of big genus). The questions related to tropical singular points (cf. Chapter 1 in [25]) are treated here from the perspective of tropical modifications, see Section 3.4.This paper is organised as follows. We define tropical modifications via multivalued operations. Then we discuss several ex...

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“…This tropical complex is a tropical analogue of Hopf surface, which are non-algebraic complex surfaces. Tropical Hopf manifolds are explored at greater length, and in a different framework, in [RS16].…”

Section: Linear Seriesmentioning

confidence: 99%