Enumerative Geometry of Del Pezzo Surfaces

Lin, Yu-Shen

doi:10.48550/arxiv.2005.08681

Cited by 3 publications

(9 citation statements)

References 42 publications

(67 reference statements)

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“…Moreover, assume that the there is certain compatibility between the compactification and the asymptotic of the metric behaviour. Then one can follow the similar argument in the work of the second author [32] and prove that the counting of the Maslov index two discs with Lagrangian fibre boundary conditions can be computed by the weighted count of broken lines. However, the authors are unaware of such asymptotic estimates of the metrics in the literature.…”

Section: Further Remarksmentioning

confidence: 68%

“…The coefficients of log f γ (u) have enumerative meanings: counting of the Maslov index zero discs bounded by the 1-parameter family of Lagrangians [32] or counting of rational curves with certain tangency conditions [18]. This motivates the following definition.…”

Section: Fukaya's Trick and Open Gromov-witten Invariantsmentioning

confidence: 99%

“…Under the identification (Λ * ) 2 ∼ = (G an m ) 2 , Φ i,i+1 is simply the restriction of the map (G an m ) 2 → (G an m ) 2 with the same equation as in (29). Therefore, we have the same commutative diagram as in (32) with V i , V i+1 replaced by U + i , U i+1 for any open subset U + i ⊆ R 2 such that ω(γ i+1 ) > 0 on U + i , which we will choose it explicitly later.…”

Section: Construction Of Family Floer Mirror Of X Iimentioning

confidence: 99%

“…When the 2-dimensional Calabi-Yau admits a special Lagrangian fibration, it is well-known that the special Lagrangian torus fibres bounding holomorphic discs supports along affine lines with respect to the complex affine coordinates. Using the Fukaya's trick, the second author identified a version of open Gromov-Witten invariants with tropical discs counting [35,32], which lays out a foundation to the connection between family Floer mirror and Gross-Siebert/Gross-Hacking-Keel mirror. The examples are engineer such that all the wall functions are polynomials.…”

Section: Introductionmentioning

confidence: 99%

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Some Examples of Family Floer Mirrors

Cheung¹,

Lin²

2021

Preprint

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In this article, we give explicit calculations for the family Floer mirrors of some non-compact Calabi-Yau surfaces. We compare it with the mirror construction of Gross-Hacking-Keel for suitably chosen log Calabi-Yau pairs and the rank two cluster varieties of finite type. In particular, the analytifications of the later two give partial compactifications of the family Floer mirrors that we computed.

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Section: Further Remarksmentioning

confidence: 68%

Section: Fukaya's Trick and Open Gromov-witten Invariantsmentioning

confidence: 99%

Section: Construction Of Family Floer Mirror Of X Iimentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Some Examples of Family Floer Mirrors

Cheung¹,

Lin²

2021

Preprint

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“…Remark 4. The expectation regarding the correspondence between the count of Maslov index 2 disks with boundary on a SYZ fibre and its tropical counterpart was proven in [24] for a SYZ fibration on the complement of an smooth anti-canonical divisor in a del Pezzo surface. More precisely, given a del Pezzo surface Y and a smooth anti-canonical divisor D, there exists a special Lagrangian fibration on Y \ D with respect to the complete Ricci-flat Tian-Yau metric [9].…”

Section: Wall-crossingmentioning

confidence: 99%

Algebraic and symplectic viewpoint on compactifications of two-dimensional cluster varieties of finite type

Cheung,

Vianna

2020

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In this article we explore compactifications of cluster varieties of finite type in complex dimension two. Cluster varieties can be viewed as the spec of a ring generated by theta functions and a compactification of such varieties can be given by a grading on that ring, which can be described by positive polytopes [17].In the examples we exploit, the cluster variety can be interpreted as the complement of certain divisors in del Pezzo surfaces. In the symplectic viewpoint, they can be described via almost toric fibrations over R 2 (after completion). Once identifying them as almost toric manifolds, one can symplectically view them inside other del Pezzo surfaces. So we can identify other symplectic compactifications of the same cluster variety, which we expect should also correspond to different algebraic compactifications. Both viewpoints are presented here and several compactifications have their corresponding polytopes compared. The finiteness of the cluster mutations are explored to provide cycles in the graph describing monotone Lagrangian tori in del Pezzo surfaces connected via almost toric mutation [34].

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The SYZ mirror symmetry conjecture for del Pezzo surfaces and rational elliptic surfaces

Collins¹,

Jacob²,

Lin³

2020

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We prove the Strominger-Yau-Zaslow mirror symmetry conjecture for non-compact Calabi-Yau surfaces arising from, on the one hand, pairs ( Y , Ď) of a del Pezzo surface Y and Ď a smooth anticanonical divisor and, on the other hand, pairs (Y, D) of a rational elliptic surface Y , and D a singular fiber of Kodaira type I k . Three main results are established concerning the latter pairs (Y, D). First, adapting work of Hein [25], we prove the existence of a complete Calabi-Yau metric on Y \D asymptotic to a (generically non-standard) semi-flat metric in every Kähler class. Secondly, we prove an optimal uniqueness theorem to the effect that, modulo automorphisms, every Kähler class on Y \ D admits a unique asymptotically semi-flat Calabi-Yau metric. This result yields a finite dimensional Kähler moduli space of Calabi-Yau metrics on Y \ D. Further, this result answers a question of Tian-Yau [40] and settles a folklore conjecture of Yau [43] in this setting. Thirdly, building on the authors' previous work [9], we prove that Y \D equipped with an asymptotically semi-flat Calabi-Yau metric ωCY admits a special Lagrangian fibration whenever the de Rham cohomology class of ωCY is not topologically obstructed. Combining these results we define a mirror map from the moduli space of del Pezzo pairs ( Y , Ď) to the complexified Kähler moduli of (Y, D) and prove that the special Lagrangian fibration on (Y, D) is T -dual to the special Lagrangian fibration on ( Y , Ď) previously constructed by the authors in [9]. We give some applications of these results, including to the study of automorphisms of del Pezzo surfaces fixing an anti-canonical divisor.

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Enumerative Geometry of Del Pezzo Surfaces

Cited by 3 publications

References 42 publications

Some Examples of Family Floer Mirrors

Some Examples of Family Floer Mirrors

Algebraic and symplectic viewpoint on compactifications of two-dimensional cluster varieties of finite type

The SYZ mirror symmetry conjecture for del Pezzo surfaces and rational elliptic surfaces

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