Lagrangian tori in four-dimensional Milnor fibres

Keating, Ailsa

doi:10.1007/s00039-015-0353-4

Cited by 23 publications

(64 citation statements)

References 45 publications

(88 reference statements)

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“…When

\frac{1}{p} + \frac{1}{q} + \frac{1}{r} ⩽ 1

, the Weinstein manifold

M_{p, q, r}

is the Milnor fiber associated to the corresponding isolated singularity

t_{p, q, r} + w^{2} = 0

. However, in this paper we are also interested in the case when

\frac{1}{p} + \frac{1}{q} + \frac{1}{r} > 1

and

p, q, r ⩾ 2

, where

M_{p, q, r}

is no longer a Milnor fiber, but a generalized Milnor fiber in the sense of . More precisely, this means that the polynomial

t_{p, q, r} + w^{2}

C^{4}

has a finite number of isolated critical points, instead of a unique isolated critical point at the origin.…”

Section: Formality Of A∞‐structuresmentioning

confidence: 99%

“…In this paper, we consider the symplectic 6‐manifolds

M_{p, q, r} \subset {double-struckC}^{4}

which are Milnor fibers associated to the isolated singularities

x^{p} + y^{q} + z^{r} + λ x y z + w^{2} = 0,

where

\frac{1}{p} + \frac{1}{q} + \frac{1}{r} ⩽ 1 .

The constant

λ \in C

above is allowed to take all but finitely many values (see for details). To be explicit, we will take in this paper

λ = 1

when

\frac{1}{p} + \frac{1}{q} + \frac{1}{r} < 1

, and

λ = 0

when

\frac{1}{p} + \frac{1}{q} + \frac{1}{r} = 1

.…”

Section: Introductionmentioning

confidence: 99%

“…The corresponding Weinstein manifolds

M_{p, q, r}

are no longer Milnor fibers associated to any isolated singularity, but still share a lot of properties similar to Milnor fibers, for example, they are homotopy equivalent to wedges of Lagrangian spheres. For this reason, these Weinstein manifolds are called generalized Milnor fibers (see for a detailed discussion of their symplectic topology).…”

Section: Introductionmentioning

confidence: 99%

“…The detailed proof of the above corollary is given in Section . Note that when

r = 2

M_{p, q, r}

can be regarded as the smooth fiber of the double suspension of the Lefschetz fibration on

C^{2}

defined by the morsification of

false(x^{p - 2} - y^{2} false) false(x^{2} - λ y^{q - 2} false),

see [, Section 2.2.5]. In this case, the formality of

V_{p, q, 2}

follows from a result of Seidel .…”

Section: Introductionmentioning

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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“…When

\frac{1}{p} + \frac{1}{q} + \frac{1}{r} ⩽ 1

, the Weinstein manifold

M_{p, q, r}

is the Milnor fiber associated to the corresponding isolated singularity

t_{p, q, r} + w^{2} = 0

. However, in this paper we are also interested in the case when

\frac{1}{p} + \frac{1}{q} + \frac{1}{r} > 1

and

p, q, r ⩾ 2

, where

M_{p, q, r}

is no longer a Milnor fiber, but a generalized Milnor fiber in the sense of . More precisely, this means that the polynomial

t_{p, q, r} + w^{2}

C^{4}

has a finite number of isolated critical points, instead of a unique isolated critical point at the origin.…”

Section: Formality Of A∞‐structuresmentioning

confidence: 99%

“…In this paper, we consider the symplectic 6‐manifolds

M_{p, q, r} \subset {double-struckC}^{4}

which are Milnor fibers associated to the isolated singularities

x^{p} + y^{q} + z^{r} + λ x y z + w^{2} = 0,

where

\frac{1}{p} + \frac{1}{q} + \frac{1}{r} ⩽ 1 .

The constant

λ \in C

above is allowed to take all but finitely many values (see for details). To be explicit, we will take in this paper

λ = 1

when

\frac{1}{p} + \frac{1}{q} + \frac{1}{r} < 1

, and

λ = 0

when

\frac{1}{p} + \frac{1}{q} + \frac{1}{r} = 1

.…”

Section: Introductionmentioning

confidence: 99%

“…The corresponding Weinstein manifolds

M_{p, q, r}

Section: Introductionmentioning

confidence: 99%

“…The detailed proof of the above corollary is given in Section . Note that when

r = 2

M_{p, q, r}

can be regarded as the smooth fiber of the double suspension of the Lefschetz fibration on

C^{2}

defined by the morsification of

false(x^{p - 2} - y^{2} false) false(x^{2} - λ y^{q - 2} false),

see [, Section 2.2.5]. In this case, the formality of

V_{p, q, 2}

follows from a result of Seidel .…”

Section: Introductionmentioning

confidence: 99%

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Koszul duality via suspending Lefschetz fibrations

2019

Journal of Topology

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“…In particular, these tori are exact in the complement of . We also relate our work with [18] [30,Example 6.3]). In Sect.…”

Section: To Mutually Different Hamiltonian Isotopy Classesmentioning

confidence: 99%

Infinitely many monotone Lagrangian tori in del Pezzo surfaces

Vianna

2017

Sel. Math. New Ser.

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We construct almost toric fibrations (ATFs) on all del Pezzo surfaces, endowed with a monotone symplectic form. Except for CP 2 #CP 2 , CP 2 #2CP 2 , we are able to get almost toric base diagrams (ATBDs) of triangular shape and prove the existence of infinitely many symplectomorphism (in particular Hamiltonian isotopy) classes of monotone Lagrangian tori in CP 2 #kCP 2 for k = 0, 3, 4, 5, 6, 7, 8. We name these tori. Using the work of Karpov-Nogin, we are able to classify all ATBDs of triangular shape. We are able to prove that CP 2 #CP 2 also has infinitely many monotone Lagrangian tori up to symplectomorphism and we conjecture that the same holds for CP 2 #2CP 2 . Finally, the Lagrangian tori n 1 ,n 2 ,n 3 p,q,r

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Homological mirror symmetry for hypersurface cusp singularities

Keating¹

2017

Sel. Math. New Ser.

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We study versions of homological mirror symmetry for hypersurface cusp singularities and the three hypersurface simple elliptic singularities. We show that the Milnor fibres of each of these carries a distinguished Lefschetz fibration; its derived directed Fukaya category is equivalent to the derived category of coherent sheaves on a smooth rational surface Y p,q,r . By using localization techniques on both sides, we get an isomorphism between the derived wrapped Fukaya category of the Milnor fibre and the derived category of coherent sheaves on a quasi-projective surface given by deleting an anti-canonical divisor D from Y p,q,r . In the cusp case, the pair (Y p,q,r , D) is naturally associated to the dual cusp singularity, tying into Gross, Hacking and Keel's proof of Looijenga's conjecture.

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Lagrangian tori in four-dimensional Milnor fibres

Cited by 23 publications

References 45 publications

Koszul duality via suspending Lefschetz fibrations

Koszul duality via suspending Lefschetz fibrations

Infinitely many monotone Lagrangian tori in del Pezzo surfaces

Homological mirror symmetry for hypersurface cusp singularities

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