Homological mirror symmetry for hypersurface cusp singularities

Keating, Ailsa

doi:10.1007/s00029-017-0334-6

Cited by 19 publications

(21 citation statements)

References 41 publications

(72 reference statements)

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“…Note that many of the known examples (say those studied in ) of the directed Fukaya categories

A (π)

associated to Lefschetz fibrations

π

can actually be identified with one‐way algebras. In particular, the directed

A_{\infty}

‐algebras

A_{p, q, r}

encountered in Section are one‐way algebras.…”

Section: Split‐generation and Quasi‐dilationsmentioning

confidence: 99%

“…We describe a Lefschetz fibration, whose construction is essentially due to Keating , on

M_{p, q, r}

. With Casals–Murphy recipe , we are able to get the Legendrian front presentation of this fibration, which, after simplifications, gives us the Legendrian attaching link

{normalΛ}_{p, q, r} \subset S^{5}

of the Weinstein manifold

M_{p, q, r}

.…”

Section: From Lefschetz Fibrations To Legendrian Frontsmentioning

confidence: 99%

“…We start by recalling a Lefschetz fibration on the Milnor fibers

T_{p, q, r} \subset {double-struckC}^{3}

(defined by

(16)

) constructed by Keating in . The construction is divided into three steps.…”

Section: From Lefschetz Fibrations To Legendrian Frontsmentioning

confidence: 99%

“…It seems to be appropriate to say here a few words about the method we use to prove Theorem . In order to get the Legendrian frontal description of the Milnor fibers

M_{p, q, r}

, we start with a Lefschetz fibration on

T_{p, q, r}

constructed by Keating in , whose smooth fiber is symplectomorphic to a 3‐punctured torus. This fibration induces naturally a Lefschetz fibration on the stabilization

M_{p, q, r}

, whose smooth fiber is symplectomorphic to a four‐dimensional

D_{4}

Milnor fiber.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, new examples of Liouville manifolds which admit quasi-dilations in the sense of Seidel-Solomon are obtained.1 p + 1 q + 1 r = 1. Changing the value of λ will not change the symplectic structure on M p,q,r up to exact symplectomorphism.Note that the above manifolds are stabilizations of the Milnor fibers T p,q,r ⊂ C 3 associated to the non-simple isolated singularity of modality one t p,q,r (x, y, z) := x p + y q + z r + λxyz = 0 (16) studied by Keating in [37,38]. Equivalently, they can be realized as smooth fibers of the suspension of the Lefschetz fibrationt p,q,r : C 3 → C, witht p,q,r being a morsification of t p,q,r .…”

mentioning

confidence: 99%

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Koszul duality via suspending Lefschetz fibrations

2019

Journal of Topology

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Let M be a Liouville 6‐manifold which is the smooth fiber of a Lefschetz fibration on C4 constructed by suspending a Lefschetz fibration on C3. We prove that for many examples including stabilizations of Milnor fibers of hypersurface cusp singularities, the compact Fukaya category F(M) and the wrapped Fukaya category W(M) are related through A∞‐Koszul duality, by identifying them with cyclic and Calabi–Yau completions of the same quiver algebra. This implies the split‐generation of the compact Fukaya category F(M) by vanishing cycles. Moreover, new examples of Liouville manifolds which admit quasi‐dilations in the sense of Seidel–Solomon are obtained.

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“…Note that many of the known examples (say those studied in ) of the directed Fukaya categories

A (π)

associated to Lefschetz fibrations

π

can actually be identified with one‐way algebras. In particular, the directed

A_{\infty}

‐algebras

A_{p, q, r}

encountered in Section are one‐way algebras.…”

Section: Split‐generation and Quasi‐dilationsmentioning

confidence: 99%

“…We describe a Lefschetz fibration, whose construction is essentially due to Keating , on

M_{p, q, r}

. With Casals–Murphy recipe , we are able to get the Legendrian front presentation of this fibration, which, after simplifications, gives us the Legendrian attaching link

{normalΛ}_{p, q, r} \subset S^{5}

of the Weinstein manifold

M_{p, q, r}

.…”

Section: From Lefschetz Fibrations To Legendrian Frontsmentioning

confidence: 99%

“…We start by recalling a Lefschetz fibration on the Milnor fibers

T_{p, q, r} \subset {double-struckC}^{3}

(defined by

(16)

) constructed by Keating in . The construction is divided into three steps.…”

Section: From Lefschetz Fibrations To Legendrian Frontsmentioning

confidence: 99%

“…It seems to be appropriate to say here a few words about the method we use to prove Theorem . In order to get the Legendrian frontal description of the Milnor fibers

M_{p, q, r}

, we start with a Lefschetz fibration on

T_{p, q, r}

constructed by Keating in , whose smooth fiber is symplectomorphic to a 3‐punctured torus. This fibration induces naturally a Lefschetz fibration on the stabilization

M_{p, q, r}

, whose smooth fiber is symplectomorphic to a four‐dimensional

D_{4}

Milnor fiber.…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Koszul duality via suspending Lefschetz fibrations

2019

Journal of Topology

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Arithmetic mirror symmetry for genus 1 curves with n marked points

Lekili

Polishchuk

2016

Sel. Math. New Ser.

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Abstract. We establish a Z[[t 1 , . . . , t n ]]-linear derived equivalence between the relative Fukaya category of the 2-torus with n distinct marked points and the derived category of perfect complexes on the n-Tate curve. Specialising to t 1 = . . . = t n = 0 gives a Z-linear derived equivalence between the Fukaya category of the n-punctured torus and the derived category of perfect complexes on the standard (Néron) n-gon. We prove that this equivalence extends to a Z-linear derived equivalence between the wrapped Fukaya category of the n-punctured torus and the derived category of coherent sheaves on the standard n-gon. The corresponding results for n = 1 were established in [31]. IntroductionOver the past few decades, Kontsevich's Homological Mirror Symmetry (HMS) conjecture ([28]), which predicts a remarkable equivalence between an A-model category associated to a symplectic manifold and a B -model category associated to a complex manifold, has been studied extensively in many instances: abelian varieties, toric varieties, hypersurfaces in projective spaces, etc. From the beginning, the case of the symplectic 2-torus played a special role as this is the simplest instance of a Calabi-Yau manifold where concrete computations in the Fukaya category can be made, and the non-trivial nature of the HMS conjecture becomes manifest. Calculations provided in [38]

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Monodromy of monomially admissible Fukaya-Seidel categories mirror to toric varieties

Hanlon

2019

Advances in Mathematics

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Mirror symmetry for a toric variety involves Laurent polynomials whose symplectic topology is related to the algebraic geometry of the toric variety. We show that there is a monodromy action on the monomially admissible Fukaya-Seidel categories of these Laurent polynomials as the arguments of their coefficients vary that corresponds under homological mirror symmetry to tensoring by a line bundle naturally associated to the monomials whose coefficients are rotated. In the process, we introduce the monomially admissible Fukaya-Seidel category as a new interpretation of the Fukaya-Seidel category of a Laurent polynomial on (C * ) n , which has other potential applications, and give evidence of homological mirror symmetry for non-compact toric varieties.

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Homological mirror symmetry for hypersurface cusp singularities

Cited by 19 publications

References 41 publications

Koszul duality via suspending Lefschetz fibrations

Koszul duality via suspending Lefschetz fibrations

Arithmetic mirror symmetry for genus 1 curves with n marked points

Monodromy of monomially admissible Fukaya-Seidel categories mirror to toric varieties

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