Flat surfaces and stability structures

Haiden, Fabian; Katzarkov, Ludmil; Kontsevich, Maxim

doi:10.1007/s10240-017-0095-y

Cited by 136 publications

(293 citation statements)

References 51 publications

Supporting

Mentioning

289

Contrasting

Order By: Relevance

“…Remark Our conventions are slightly different from those in : their markings

M

correspond to the subsets

\partial S ∖ M

. Also, their arc systems allow faces of arbitrary genus.…”

Section: Curved Complexes For Marked Surfacesmentioning

confidence: 99%

“…Remark In , Haiden, Katzarkov and Kontsevich associate with the triple

(S, M, A)

the path algebra

{scriptA}^{\infty} : = {double-struckF}_{2} Q false(S, M, A false) / false{p_{1} p_{2} = 0 = q_{2} q_{1} ∣ arcs a \in A false} .

Note that the relations they impose on the free quiver algebra

{double-struckF}_{2} Q false(S, M, A false)

are in some sense ‘dual’ to ours. Haiden, Katzarkov and Kontsevich define an

A_{\infty}

‐structure on

A^{\infty}

, which then gives rise to the notion of twisted complexes using an

A_{\infty}

‐version of

{prefixCx}^{0} false(prefixMat (\cdot) false)

.…”

Section: Curved Complexes For Marked Surfacesmentioning

confidence: 99%

“…Haiden, Katzarkov and Kontsevich define an

A_{\infty}

‐structure on

A^{\infty}

, which then gives rise to the notion of twisted complexes using an

A_{\infty}

‐version of

{prefixCx}^{0} false(prefixMat (\cdot) false)

. In [, Theorem 4.3], they show that twisted complexes over

A^{\infty}

are classified by immersed curves on

S

with local systems. Curved complexes over

scriptA

and twisted complexes over

A^{\infty}

are closely related.…”

Section: Curved Complexes For Marked Surfacesmentioning

confidence: 99%

See 2 more Smart Citations

Peculiar modules for 4‐ended tangles

Zibrowius

2020

Journal of Topology

View full text Add to dashboard Cite

With a 4‐ended tangle T, we associate a Heegaard Floer invariant prefixCFT∂false(Tfalse), the peculiar module of T. Based on Zarev's bordered sutured Heegaard Floer theory (Zarev, PhD Thesis, Columbia University, 2011), we prove a glueing formula for this invariant which recovers link Floer homology HFL̂. Moreover, we classify peculiar modules in terms of immersed curves on the 4‐punctured sphere. In fact, based on an algorithm of Hanselman, Rasmussen and Watson (Preprint, 2016, arXiv:1604.03466v2), we prove general classification results for the category of curved complexes over a marked surface with arc system. This allows us to reinterpret the glueing formula for peculiar modules in terms of Lagrangian intersection Floer theory on the 4‐punctured sphere. We then study some applications: Firstly, we show that peculiar modules detect rational tangles. Secondly, we give short proofs of various skein exact triangles. Thirdly, we compute the peculiar modules of the 2‐stranded pretzel tangles T2n,−false(2m+1false) for n,m>0 using nice diagrams. We then observe that these peculiar modules enjoy certain symmetries which imply that mutation of the tangles T2n,−false(2m+1false) preserves δ‐graded, and for some orientations even bigraded link Floer homology.

show abstract

“…Remark Our conventions are slightly different from those in : their markings

M

correspond to the subsets

\partial S ∖ M

. Also, their arc systems allow faces of arbitrary genus.…”

Section: Curved Complexes For Marked Surfacesmentioning

confidence: 99%

“…Remark In , Haiden, Katzarkov and Kontsevich associate with the triple

(S, M, A)

the path algebra

{scriptA}^{\infty} : = {double-struckF}_{2} Q false(S, M, A false) / false{p_{1} p_{2} = 0 = q_{2} q_{1} ∣ arcs a \in A false} .

Note that the relations they impose on the free quiver algebra

{double-struckF}_{2} Q false(S, M, A false)

are in some sense ‘dual’ to ours. Haiden, Katzarkov and Kontsevich define an

A_{\infty}

‐structure on

A^{\infty}

, which then gives rise to the notion of twisted complexes using an

A_{\infty}

‐version of

{prefixCx}^{0} false(prefixMat (\cdot) false)

.…”

Section: Curved Complexes For Marked Surfacesmentioning

confidence: 99%

“…Haiden, Katzarkov and Kontsevich define an

A_{\infty}

‐structure on

A^{\infty}

, which then gives rise to the notion of twisted complexes using an

A_{\infty}

‐version of

{prefixCx}^{0} false(prefixMat (\cdot) false)

. In [, Theorem 4.3], they show that twisted complexes over

A^{\infty}

are classified by immersed curves on

S

with local systems. Curved complexes over

scriptA

and twisted complexes over

A^{\infty}

are closely related.…”

Section: Curved Complexes For Marked Surfacesmentioning

confidence: 99%

See 1 more Smart Citation

Peculiar modules for 4‐ended tangles

Zibrowius

2020

Journal of Topology

View full text Add to dashboard Cite

show abstract

“…Definition 2.7. pFpS, Aqq If S is an oriented graded marked surface and A is an arc system then [HKK17] defines an A 8 -category FpS, Aq with objects ObpFpS, Aqq " A given by the set of graded arcs in A. The morphisms in FpS, Aq are k -linear combinations of boundary paths.…”

Section: Fukaya Categoriesmentioning

confidence: 99%

The Hall Algebras of Surfaces I

Cooper¹,

Samuelson²

2018

J. Inst. Math. Jussieu

View full text Add to dashboard Cite

show abstract

“…One of the main topics of current interest we do not cover in these notes is the "local case" (i.e., stability conditions on CY3 triangulated categories defined using quivers with potential; see, e.g., [Bri06b,KS08,BS15]) or more generally stability conditions on Fukaya categories (see, e.g., [DHKK14,HKK14,Joy15]). Another fundamental topic is the connection with counting invariants; for a survey we refer to [Tod12a,Tod12b,Tod14].…”

Section: Introductionmentioning

confidence: 99%

Lectures on Bridgeland Stability

Macrì

Schmidt

2017

Lecture Notes of the Unione Matematica Italiana

View full text Add to dashboard Cite

Abstract. In these lecture notes we give an introduction to Bridgeland stability conditions on smooth complex projective varieties with a particular focus on the case of surfaces. This includes basic definitions of stability conditions on derived categories, basics on moduli spaces of stable objects and variation of stability. These notes originated from lecture series by the first author at the summer school Recent advances in algebraic and arithmetic geometry, Siena, Italy, August 24-28, 2015 and at the school Moduli of Curves, CIMAT, Guanajuato, Mexico, February 22 -March 4, 2016.

show abstract

Flat surfaces and stability structures

Cited by 136 publications

References 51 publications

Peculiar modules for 4‐ended tangles

Peculiar modules for 4‐ended tangles

The Hall Algebras of Surfaces I

Lectures on Bridgeland Stability

Contact Info

Product

Resources

About