Tropical descendant Gromov–Witten invariants

Markwig, Hannah; Rau, Johannes

doi:10.1007/s00229-009-0256-5

Cited by 34 publications

(107 citation statements)

References 9 publications

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Mentioning

103

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“…The modular description of M 0,n endows it with n-divisor classes ψ i ∈ Pic(M 0,n ). We will show that the tropicalization of these ψ-classes in our sense recovers the definition given in [44] (which was based on Mikhalkin's definition [48]). Invoking the machinery that we develop in Sections 3 and 4 gives a very conceptual proof of the fact that top-dimensional intersections of the ψ i have the same degrees as top-dimensional intersections of the tropical ψ-classes.…”

Section: Introductionsupporting

confidence: 59%

“…Remark In case we take top‐dimensional intersections with

Y = P^{r}

, the tropical degree

normalΔ

containing each of the vectors

0true e_{1}, \dots, e_{r}, - \sum e_{i}

exactly

d

times, and all the

D_{i j}

equal to classes of lines, we obtain the tropical descendant Gromov–Witten invariants of and up to factor

{false(d! false)}^{r + 1}

. In particular, slight variations of the recursion formulas of also hold for the corresponding algebraic invariants.…”

Section: Applicationsmentioning

confidence: 99%

“…In , Markwig and Rau defined genus 0 tropical descendant Gromov–Witten invariants of

{double-struckR}^{r}

by substituting tropical for algebraic objects in the classical definition of genus 0 descendant Gromov–Witten invariants of toric varieties. In this definition, the tropical moduli space chosen to replace the moduli space of rational stable maps is the moduli space of tropical stable maps, which has been introduced in .…”

Section: Applicationsmentioning

confidence: 99%

“…Similarly, the tropicalization

Trop ({prefixev}_{i})

of the

i

th evaluation map is equal to the

i

th tropical evaluation map

{ev}_{i}^{trop}

by [, Proposition 5.1.2]. In accordance with , we define the

k

ψ

‐class

{\hat{ψ}}_{k}

(respectively,

{\hat{ψ}}_{k}^{trop}

) on

LSM (Y, Δ)

(respectively,

TSM (R^{r}, Δ)

) as the pullback via

μ

(respectively,

μ^{trop}

) of the

k

ψ

‐class

ψ_{k}

(respectively,

ψ_{k}^{trop}

) on

{\bar{M}}_{0, n + m}

(respectively,

{\bar{M}}_{0, n + m}^{trop}

). As a direct consequence of our previous results, we obtain the following correspondence theorem.…”

Section: Applicationsmentioning

confidence: 99%

“…Remark We defined the

ψ

‐classes

{\hat{ψ}}_{k}

LSM (Y, Δ)

to be the analogs of the tropical

ψ

‐classes on

{prefixTSM}^{\circ} false({double-struckR}^{r}, normalΔ false)

of , that is, as pullbacks of

ψ

‐classes on

{\bar{M}}_{0, n + m}

. It has been shown in that these are equal to the classes on

LSM (Y, Δ)

obtained by taking Chern classes of cotangent line bundles.…”

Section: Applicationsmentioning

confidence: 99%

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Intersection theory on tropicalizations of toroidal embeddings

Groß

2018

Proc. London Math. Soc.

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We show how to equip the cone complexes of toroidal embeddings with additional structure that allows to define a balancing condition for weighted subcomplexes. We then proceed to develop the foundations of an intersection theory on cone complexes including push-forwards, intersections with tropical divisors, and rational equivalence. These constructions are shown to have an algebraic interpretation: Ulirsch's tropicalizations of subvarieties of toroidal embeddings carry natural multiplicities making them tropical cycles, and the induced tropicalization map for cycles respects push-forwards, intersections with boundary divisors, and rational equivalence. As an application we prove a correspondence between the genus 0 tropical descendant Gromov-Witten invariants introduced by Markwig and Rau and the genus 0 logarithmic descendant Gromov-Witten invariants of toric varieties.

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Section: Introductionsupporting

confidence: 59%

“…Remark In case we take top‐dimensional intersections with

Y = P^{r}

, the tropical degree

normalΔ

containing each of the vectors

0true e_{1}, \dots, e_{r}, - \sum e_{i}

exactly

d

times, and all the

D_{i j}

equal to classes of lines, we obtain the tropical descendant Gromov–Witten invariants of and up to factor

{false(d! false)}^{r + 1}

. In particular, slight variations of the recursion formulas of also hold for the corresponding algebraic invariants.…”

Section: Applicationsmentioning

confidence: 99%

“…In , Markwig and Rau defined genus 0 tropical descendant Gromov–Witten invariants of

{double-struckR}^{r}

Section: Applicationsmentioning

confidence: 99%

“…Similarly, the tropicalization

Trop ({prefixev}_{i})

of the

i

th evaluation map is equal to the

i

th tropical evaluation map

{ev}_{i}^{trop}

by [, Proposition 5.1.2]. In accordance with , we define the

k

ψ

‐class

{\hat{ψ}}_{k}

(respectively,

{\hat{ψ}}_{k}^{trop}

) on

LSM (Y, Δ)

(respectively,

TSM (R^{r}, Δ)

) as the pullback via

μ

(respectively,

μ^{trop}

) of the

k

ψ

‐class

ψ_{k}

(respectively,

ψ_{k}^{trop}

) on

{\bar{M}}_{0, n + m}

(respectively,

{\bar{M}}_{0, n + m}^{trop}

). As a direct consequence of our previous results, we obtain the following correspondence theorem.…”

Section: Applicationsmentioning

confidence: 99%

“…Remark We defined the

ψ

‐classes

{\hat{ψ}}_{k}

LSM (Y, Δ)

to be the analogs of the tropical

ψ

‐classes on

{prefixTSM}^{\circ} false({double-struckR}^{r}, normalΔ false)

of , that is, as pullbacks of

ψ

‐classes on

{\bar{M}}_{0, n + m}

. It has been shown in that these are equal to the classes on

LSM (Y, Δ)

obtained by taking Chern classes of cotangent line bundles.…”

Section: Applicationsmentioning

confidence: 99%

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Intersection theory on tropicalizations of toroidal embeddings

Groß

2018

Proc. London Math. Soc.

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Enumerative Aspects of the Gross-Siebert Program

Garrel¹,

Peter²,

Ruddat³

2015

Calabi-Yau Varieties: Arithmetic, Geometry and Physics

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We present enumerative aspects of the Gross-Siebert program in this introductory survey. After sketching the program's main themes and goals, we review the basic definitions and results of logarithmic and tropical geometry. We give examples and a proof for counting algebraic curves via tropical curves. To illustrate an application of tropical geometry and the Gross-Siebert program to mirror symmetry, we discuss the mirror symmetry of the projective plane.Michel van Garrel KIAS,discretizing Hitchin's Legendre duality.Enumerative aspects of the Gross-Siebert program 3 Kontsevich and Soibelman [32] demonstrated how one could reconstruct a K3 surface from an affine structure with singularities on S 2 . Using logarithmic geometry, Gross and Siebert were able to solve the reconstruction problem [20] in any dimension, obtaining a degenerating family of Calabi-Yau manifolds X → D over a holomorphic disk from the information of (B, P, ϕ) and a log structure. Furthermore, this family is parametrized by a canonical coordinate (in the usual sense in mirror symmetry). The construction features wall-crossings and scatterings, structures that encode enumerative information linking symplectic with complex geometry via tropical geometry. As will be hinted at in this exposition, Gromov-Witten theory [21] can also be incorporated in this framework. Toric conventionsWe assume familiarity with toric geometry. The interested reader is referred to the excellent exposition of Fulton [10]. As the following story is closely tied to toric geometry, it is convenient to begin by making a few conventions regarding notation.SetFor n ∈ N, set n, m to be the evaluation of n on m. Set a toric fan Σ in M R . Let Σ [n] signify the set of n dimensional cones of Σ . Let X Σ be the toric variety defined by Σ .Denote by T Σ the free abelian group generated by Σ [1] . For ρ ∈ Σ [1] , denote by v ρ the corresponding generator in T Σ . We will need the mapwhereρ is the integral vector generating ρ, that is ρ ∩ M = Z ≥0ρ .

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