Lefschetz (1,1)-theorem in tropical geometry

Jell, Philipp; Rau, Johannes; Shaw, Kristin

doi:10.46298/epiga.2018.volume2.4126

Cited by 29 publications

(54 citation statements)

References 15 publications

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“…The tropical (co)homology groups with integral coefficients of a non-singular tropical hypersurface satisfy a variant of Poincaré duality [JRS18]. Using this we deduce in Section 4 that the tropical homology groups of a non-singular tropical hypersurface in a non-singular tropical toric variety which satisfy the assumptions below are torsion free, as long as the homology of the tropical toric variety is also torsion free.…”

Section: Introductionmentioning

confidence: 82%

“…The star of a face τ of X is a basic open subset and satisfies Poincaré duality from [JRS18]. Therefore, we have rank F p pτ q " rank H 0 pstarpτ q; F p q " rank H n c pstarpτ q; F n´p q " ÿ σ Ą sτ dim σ"q p´1q n´q rank F n´p pσq.…”

Section: Betti Numbers Of Tropical Homology and Hodge Numbersmentioning

confidence: 99%

“…nd there are isomorphisms of the corresponding homology groups. By [JRS18], the space R kˆTq´k satisfies Poincaré duality for tropical homology and the Borel-Moore tropical homology groups of R q´kˆTk are zero except in degree q " dim γ. The statement of the Lemma 3.6 follows.…”

Section: Tropical Lefschetz Section Theoremmentioning

confidence: 99%

“…When q ă p we have H q pY ; F Y p q " 0, so there is the isomorphism H q`1 pY ; F p Y q -HompH q`1 pY ; F Y p q, Zq. The tropical toric variety Y is a tropical manifold, thus Poincaré duality for tropical homology with integral coefficients from [JRS18] states that…”

mentioning

confidence: 99%

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Lefschetz section theorems for tropical hypersurfaces

Arnal¹,

Renaudineau²,

Shaw³

2021

Annales Henri Lebesgue

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Nous démontrons des analogues au théorème de la section hyperplane de Lefschetz pour l'homologie tropicale entière d'hypersurfaces tropicales dans des variétés toriques tropicales. Nous en déduisons que les groupes d'homologie des hypersurfaces tropicales nonsingulières compactes (ou contenues dans R n ) sont sans torsion. Nous en déduisons également une relation entre les coefficients du genre χ y des hypersurfaces complexes dans les variétés toriques et les caractéristiques d'Euler des complexes de chaînes cellulaires tropicales des hypersurfaces tropicales. Il s'ensuit que les groupes d'homologies tropicales à coefficient entier ont pour rang les nombres de Hodges d'hypersurfaces compactes non-singulières dans des variétés toriques complexes. Finalement pour les hypersurfaces tropicales dans certaines variétés toriques affines, nous relions les rangs de leurs groupes d'homologie tropicale aux nombres de Hodge-Deligne des hypersurfaces complexes correspondantes.

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Section: Introductionmentioning

confidence: 82%

Section: Betti Numbers Of Tropical Homology and Hodge Numbersmentioning

confidence: 99%

Section: Tropical Lefschetz Section Theoremmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Lefschetz section theorems for tropical hypersurfaces

Arnal¹,

Renaudineau²,

Shaw³

2021

Annales Henri Lebesgue

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“…Another theme we take up is that of extending tropical intersection theory from R r to more general situations. Such generalizations have been considered before, for example by K. Shaw and François-Rau [9,27] for matroidal fans or by Jell-Shaw-Smacka and Jell-Rau-Shaw [17,18] for partial compactifications of R r . Our theory of δ-forms on tropical spaces presents a different approach to this question.…”

Section: Introductionmentioning

confidence: 99%

$δ$-Forms on Lubin--Tate Space

Andreas¹

2021

Preprint

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We extend Gubler-Künnemann's theory of δ-forms from algebraic varieties to good Berkovich spaces. This is based on the observation that skeletons in such spaces satisfy a tropical balance condition. Our main result is that complete intersection formal models of cycles give rise to Green δ-forms for their generic fibers. We moreover show that, in certain situations, intersection numbers on formal models are given by the ⋆-product of their Green currents. In this way, we generalize some results for divisor intersection to higher codimension situations. We illustrate the mentioned results in the context of an intersection problem for Lubin-Tate spaces.

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Constructing smooth and fully faithful tropicalizations for Mumford curves

Jell

2020

Sel. Math. New Ser.

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The tropicalization of an algebraic variety X is a combinatorial shadow of X, which is sensitive to a closed embedding of X into a toric variety. Given a good embedding, the tropicalization can provide a lot of information about X. We construct two types of these good embeddings for Mumford curves: fully faithful tropicalizations, which are embeddings such that the tropicalization admits a continuous section to the associated Berkovich space $$X^{{{\,\mathrm{an}\,}}}$$ X an of X, and smooth tropicalizations. We also show that a smooth curve that admits a smooth tropicalization is necessarily a Mumford curve. Our key tool is a variant of a lifting theorem for rational functions on metric graphs.

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Lefschetz (1,1)-theorem in tropical geometry

Cited by 29 publications

References 15 publications

Lefschetz section theorems for tropical hypersurfaces

Lefschetz section theorems for tropical hypersurfaces

$δ$-Forms on Lubin--Tate Space

Constructing smooth and fully faithful tropicalizations for Mumford curves

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