Legendrian fronts for affine varieties

Casals, Roger; Murphy, Emmy

doi:10.1215/00127094-2018-0055

Cited by 28 publications

(89 citation statements)

References 72 publications

(123 reference statements)

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“…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 recall how to obtain the Legendrian front description of a Weinstein manifold

M

starting from a symplectic Lefschetz fibration. This is written down systematically by Casals–Murphy in .…”

Section: From Lefschetz Fibrations To Legendrian Frontsmentioning

confidence: 99%

“…Let

π : M \to C

be a Lefschetz fibration with smooth fiber

F_{T}

which is a plumbing of

T^{*} S^{n - 1}

according to some tree

T

. Given these data, Casals and Murphy suggest in the following procedure to obtain a Legendrian handle body decomposition of

M

. Draw

r

(n - 1)

‐handles which correspond to the zero sections

L_{1}, \dots, L_{r}

T^{*} S^{n - 1}

in the plumbing

F_{T}

. Find a Lefschetz fibration

π_{F} : F_{T} \to C

so that the Lagrangian spheres

{L_{i}}

appear as matching cycles of

π_{F}

with matching paths

γ_{1}, \dots, γ_{r} \subset C

. For any vanishing cycle

V_{j} \subset F_{T}

π

, draw the embedded path

β_{j} \subset C

under the projection of

π_{F}

. Express each matching path

β_{j}

V_{j}

as a word in half‐twists along the paths in

{γ_{i}}

.…”

Section: From Lefschetz Fibrations To Legendrian Frontsmentioning

confidence: 99%

“…From [, Section 4.1], we see that

M_{1, 1, 0}

is obtained by attaching a Weinstein 3‐handle to

D^{6}

along the Legendrian surface

Λ_{1, 1, 0}

depicted in Figure . Although in , the Legendrian front

Λ_{1, 1, 0}

M_{1, 1, 0}

is obtained using a different Lefschetz fibration, one will end up with the same front by starting from our general framework in Section , and cancelling the 2‐handle

R

with the 3‐handle corresponding to the Legendrian sphere

Λ_{A}

(see Figure ).…”

Section: Combinatorial Computationsmentioning

confidence: 99%

“…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. Using an algorithm due to Casals–Murphy , this Lefschetz fibration on

M_{p, q, r}

can then be translated to produce a Legendrian frontal description of

M_{p, q, r}

which involves both 2‐handles and 3‐handles. After a standard procedure of handle cancellations and Reidemeister moves, we can simplify the front diagram so as to obtain the attaching link

{normalΛ}_{p, q, r} \subset S^{5}

for

M_{p, q, r}

.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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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“…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 recall how to obtain the Legendrian front description of a Weinstein manifold

M

starting from a symplectic Lefschetz fibration. This is written down systematically by Casals–Murphy in .…”

Section: From Lefschetz Fibrations To Legendrian Frontsmentioning

confidence: 99%

“…Let

π : M \to C

be a Lefschetz fibration with smooth fiber

F_{T}

which is a plumbing of

T^{*} S^{n - 1}

according to some tree

T

. Given these data, Casals and Murphy suggest in the following procedure to obtain a Legendrian handle body decomposition of

M

. Draw

r

(n - 1)

‐handles which correspond to the zero sections

L_{1}, \dots, L_{r}

T^{*} S^{n - 1}

in the plumbing

F_{T}

. Find a Lefschetz fibration

π_{F} : F_{T} \to C

so that the Lagrangian spheres

{L_{i}}

appear as matching cycles of

π_{F}

with matching paths

γ_{1}, \dots, γ_{r} \subset C

. For any vanishing cycle

V_{j} \subset F_{T}

π

, draw the embedded path

β_{j} \subset C

under the projection of

π_{F}

. Express each matching path

β_{j}

V_{j}

as a word in half‐twists along the paths in

{γ_{i}}

.…”

Section: From Lefschetz Fibrations To Legendrian Frontsmentioning

confidence: 99%

“…From [, Section 4.1], we see that

M_{1, 1, 0}

is obtained by attaching a Weinstein 3‐handle to

D^{6}

along the Legendrian surface

Λ_{1, 1, 0}

depicted in Figure . Although in , the Legendrian front

Λ_{1, 1, 0}

M_{1, 1, 0}

is obtained using a different Lefschetz fibration, one will end up with the same front by starting from our general framework in Section , and cancelling the 2‐handle

R

with the 3‐handle corresponding to the Legendrian sphere

Λ_{A}

(see Figure ).…”

Section: Combinatorial Computationsmentioning

confidence: 99%

“…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. Using an algorithm due to Casals–Murphy , this Lefschetz fibration on

M_{p, q, r}

can then be translated to produce a Legendrian frontal description of

M_{p, q, r}

which involves both 2‐handles and 3‐handles. After a standard procedure of handle cancellations and Reidemeister moves, we can simplify the front diagram so as to obtain the attaching link