From real affine geometry to complex geometry

Gross, Mark; Siebert, Bernd

doi:10.4007/annals.2011.174.3.1

Cited by 244 publications

(633 citation statements)

References 24 publications

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“…§1 is devoted to the construction of the fundamental tool of the paper, scattering diagrams. While [GS11] defined these in much greater generality, here they are collections of walls living in a vector space with attached functions constructed canonically from a choice of seed data. A precise definition can be found in §1.1.…”

Section: Note This Implies Mid(v ) = Up(v ) = Can(v )mentioning

confidence: 99%

“…For the conjecture to hold as stated, one needs further affineness assumptions. Here we apply methods developed in the study of mirror symmetry, in particular scattering diagrams, introduced by Kontsevich and Soibelman in [KS06] for two dimensions and by Gross and Siebert in [GS11] for all dimensions, broken lines, introduced by Gross in [G09] and developed further by Carl, Pumperla and Siebert in [CPS], and theta functions, introduced by Gross, Hacking, Keel and Siebert, see [GHK11], [CPS], [GS12], and [GHKS], to prove the conjecture in this corrected form. We give in addition a formula for the structure constants in this basis, non-negative integers given by counts of broken lines.…”

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confidence: 99%

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Canonical bases for cluster algebras

Gross¹,

Hacking²,

Keel³

et al. 2017

J. Amer. Math. Soc.

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356

968

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Abstract. In [GHK11], Conjecture 0.6, the first three authors conjectured that the ring of regular functions on a natural class of affine log Calabi-Yau varieties (those with maximal boundary) has a canonical vector space basis parameterized by the integral tropical points of the mirror. Further, the structure constants for the multiplication rule in this basis should be given by counting broken lines (certain combinatorial objects, morally the tropicalisations of holomorphic discs).Here we prove the conjecture in the case of cluster varieties, where the statement is a more precise form of the Fock-Goncharov dual basis conjecture, [FG06], Conjecture 4.3. In particular, under suitable hypotheses, for each Y the partial compactification of an affine cluster variety U given by allowing some frozen variables to vanish, we obtain canonical bases for H 0 (Y, O Y ) extending to a basis of H 0 (U, O U ). Each choice of seed canonically identifies the parameterizing sets of these bases with integral points in a polyhedral cone. These results specialize to basis results of combinatorial representation theory. For example, by considering the open double Bruhat cell U in the basic affine space Y we obtain a canonical basis of each irreducible representation of SL r , parameterized by a set which each choice of seed identifies with integral points of a lattice polytope. These bases and polytopes are all constructed essentially without representation theoretic considerations. Along the way, our methods prove a number of conjectures in cluster theory, including positivity of the Laurent phenomenon for cluster algebras of geometric type.

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Section: Note This Implies Mid(v ) = Up(v ) = Can(v )mentioning

confidence: 99%

mentioning

confidence: 99%

Canonical bases for cluster algebras

Gross¹,

Hacking²,

Keel³

et al. 2017

J. Amer. Math. Soc.

Self Cite

356

968

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show abstract

“…The reference is [35, §8], see also the generalizations by Gross and Siebert [23,24]. We compute that…”

Section: The Example Of a Del Pezzo Surfacementioning

confidence: 99%

“…[19,Remark 0.21]). We refer to [35,22,23] for the notion of scattering diagram. However, the definition of broken line in [19] is based on the scattering diagram and is combinatoric in nature.…”

Section: Introductionmentioning

confidence: 99%

Enumeration of holomorphic cylinders in log Calabi–Yau surfaces. I

2016

Math. Ann.

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Abstract. We define the counting of holomorphic cylinders in log CalabiYau surfaces. Although we start with a complex log Calabi-Yau surface, the counting is achieved by applying methods from non-archimedean geometry. This gives rise to new geometric invariants. Moreover, we prove that the counting satisfies a property of symmetry. Explicit calculations are given for a del Pezzo surface in detail, which verify the conjectured wall-crossing formula for the focusfocus singularity. Our holomorphic cylinders are expected to give a geometric understanding of the combinatorial notion of broken line by Gross, Hacking, Keel and Siebert. Our tools include Berkovich spaces, tropical geometry, GromovWitten theory and the GAGA theorem for non-archimedean analytic stacks.

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“…Gross [Gro01] gave a verification of the above picture in the topological level. The Gross-Siebert program [GS11] gave a sophisticated algebraic formulation of the SYZ construction. Moreover, the SYZ program was successfully carried out in various situations for Kähler manifolds [LYZ00, Leu05,Aur07,CO06,CL10,FOOO10,CLL12,AAK].…”

Section: Introductionmentioning

confidence: 99%

Non-Kähler SYZ Mirror Symmetry

Lau

Tseng

Yau

2015

Commun. Math. Phys.

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Abstract. We study SYZ mirror symmetry in the context of non-Kähler Calabi-Yau manifolds. In particular, we study the six-dimensional Type II supersymmetric SU (3) systems with Ramond-Ramond fluxes, and generalize them to higher dimensions. We show that Fourier-Mukai transform provides the mirror map between these Type IIA and Type IIB supersymmetric systems in the semi-flat setting. This is concretely exhibited by nilmanifolds.

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From real affine geometry to complex geometry

Cited by 244 publications

References 24 publications

Canonical bases for cluster algebras

Canonical bases for cluster algebras

Enumeration of holomorphic cylinders in log Calabi–Yau surfaces. I

Non-Kähler SYZ Mirror Symmetry

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