Flat surfaces and stability structures

Haiden, F.; Katzarkov, L.; Kontsevich, M.

doi:10.1007/s10240-017-0095-y

Haiden, F.¹ ; Katzarkov, L.^2,³ ; Kontsevich, M.⁴

¹ Department of Mathematics, Science Center, Harvard University One Oxford Street 02138 Cambridge MA USA
² Fakultät für Mathematik, Universität Wien Oskar-Morgenstern-Platz 1 1090 Wien Austria
³ HSE Moscow Moscow Russia
⁴ Institut des Hautes Études Scientifiques 35 route de Chartres 91440 Bures-sur-Yvette France

Publications Mathématiques de l'IHÉS, Tome 126 (2017), pp. 247-318.

MR Zbl

DOI : 10.1007/s10240-017-0095-y

Affiliations des auteurs :

Haiden, F. ¹ ; Katzarkov, L. ^{2, 3} ; Kontsevich, M. ⁴

@article{PMIHES_2017__126__247_0,
     author = {Haiden, F. and Katzarkov, L. and Kontsevich, M.},
     title = {Flat surfaces and stability structures},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {247--318},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {126},
     year = {2017},
     doi = {10.1007/s10240-017-0095-y},
     mrnumber = {3735868},
     zbl = {1390.32010},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1007/s10240-017-0095-y/}
}

TY  - JOUR
AU  - Haiden, F.
AU  - Katzarkov, L.
AU  - Kontsevich, M.
TI  - Flat surfaces and stability structures
JO  - Publications Mathématiques de l'IHÉS
PY  - 2017
SP  - 247
EP  - 318
VL  - 126
PB  - Springer Berlin Heidelberg
PP  - Berlin/Heidelberg
UR  - http://archive.numdam.org/articles/10.1007/s10240-017-0095-y/
DO  - 10.1007/s10240-017-0095-y
LA  - en
ID  - PMIHES_2017__126__247_0
ER  -

%0 Journal Article
%A Haiden, F.
%A Katzarkov, L.
%A Kontsevich, M.
%T Flat surfaces and stability structures
%J Publications Mathématiques de l'IHÉS
%D 2017
%P 247-318
%V 126
%I Springer Berlin Heidelberg
%C Berlin/Heidelberg
%U http://archive.numdam.org/articles/10.1007/s10240-017-0095-y/
%R 10.1007/s10240-017-0095-y
%G en
%F PMIHES_2017__126__247_0

Haiden, F.; Katzarkov, L.; Kontsevich, M. Flat surfaces and stability structures. Publications Mathématiques de l'IHÉS, Tome 126 (2017), pp. 247-318. doi : 10.1007/s10240-017-0095-y. http://archive.numdam.org/articles/10.1007/s10240-017-0095-y/

Bibliographie
Cité par

[1.] Abouzaid, M. On the Fukaya categories of higher genus surfaces, Adv. Math., Volume 217 (2008), pp. 1192-1235 | DOI | MR | Zbl

[2.] Abouzaid, M.; Seidel, P. An open string analogue of Viterbo functoriality, Geom. Topol., Volume 14 (2010), pp. 627-718 | DOI | MR | Zbl

[3.] Assem, I.; Skowroński, A. Iterated tilted algebras of type ${\tilde{A}}_{n}$ , Math. Z., Volume 195 (1987), pp. 269-290 | DOI | MR | Zbl

[4.] Atiyah, M. F. Vector bundles over an elliptic curve, Proc. Lond. Math. Soc., Volume 3 (1957), pp. 414-452 | DOI | MR | Zbl

[5.] Auroux, D. Fukaya categories and bordered Heegaard–Floer homology, Proceedings of the International Congress of Mathematicians, vol. II (2010), pp. 917-941

[6.] Bardzell, M. The alternating syzygy behavior of monomial algebras, J. Algebra, Volume 188 (1997), pp. 69-89 | DOI | MR | Zbl

[7.] Bayer, A.; Macrì, E.; Toda, Y. Bridgeland stability conditions on threefolds I: Bogomolov-Gieseker type inequalities, J. Algebraic Geom., Volume 23 (2014), pp. 117-163 | DOI | MR | Zbl

[8.] Bekkert, V.; Merklen, H. A. Indecomposables in derived categories of gentle algebras, Algebr. Represent. Theory, Volume 6 (2003), pp. 285-302 | DOI | MR | Zbl

[9.] Bocklandt, R. Noncommutative mirror symmetry for punctured surfaces, Trans. Am. Math. Soc., Volume 368 (2016), pp. 429-469 | DOI | MR | Zbl

[10.] Bowman, J. P.; Valdez, F. Wild singularities of flat surfaces, Isr. J. Math., Volume 197 (2013), pp. 69-97 | DOI | MR | Zbl

[11.] Bridgeland, T. Stability conditions on triangulated categories, Ann. Math., Volume 166 (2007), pp. 317-345 | DOI | MR | Zbl

[12.] Bridgeland, T. Stability conditions on $K 3$ surfaces, Duke Math. J., Volume 141 (2008), pp. 241-291 | DOI | MR | Zbl

[13.] Bridgeland, T. Stability conditions and Kleinian singularities, Int. Math. Res. Not., Volume 21 (2009), pp. 4142-4157 | MR | Zbl

[14.] T. Bridgeland, Y. Qiu and T. Sutherland, Stability conditions on the $A_{2}$ quiver, | arXiv

[15.] Bridgeland, T.; Smith, I. Quadratic differentials as stability conditions, Publ. Math. IHÉS, Volume 121 (2015), pp. 155-278 | DOI | MR | Zbl

[16.] Burban, I.; Drozd, Y. On derived categories of certain associative algebras, Representations of Algebras and Related Topics (2005), pp. 109-128

[17.] Burban, I.; Kreußler, B. Derived categories of irreducible projective curves of arithmetic genus one, Compos. Math., Volume 142 (2006), pp. 1231-1262 | DOI | MR | Zbl

[18.] Dimitrov, G.; Haiden, F.; Katzarkov, L.; Kontsevich, M. Dynamical systems and categories, The Influence of Solomon Lefschetz in Geometry and Topology: 50 Years of Mathematics at CINVESTAV (2014), pp. 133-170

[19.] G. Dimitrov and L. Katzarkov, Stability conditions on the acyclic triangular quiver, | arXiv

[20.] T. Dyckerhoff, A1-homotopy invariants of topological Fukaya categories of surfaces, | arXiv

[21.] T. Dyckerhoff and M. Kapranov, Triangulated surfaces in triangulated categories, | arXiv

[22.] Fukaya, K.; Oh, Y.-G.; Ohta, H.; Ono, K. Lagrangian Intersection Floer Theory, Anomaly and Obstruction, Parts I, II (2009) | Zbl

[23.] Gaiotto, D.; Moore, G. W.; Neitzke, A. Wall-crossing, Hitchin systems, and the WKB approximation, Adv. Math., Volume 234 (2013), pp. 239-403 | DOI | MR | Zbl

[24.] Grothendieck, A. Groupes de classes des categories abeliennes et triangulees. Complexes parfaits (Redige par I. Bucur), Semin. Geom. Algebr. Bois-Marie 1965–1966, SGA 5 (1977), pp. 351-371

[25.] Harer, J. L. Stability of the homology of the mapping class groups of orientable surfaces, Ann. Math. (2), Volume 121 (1985), pp. 215-249 | DOI | MR | Zbl

[26.] Harer, J. L. The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math., Volume 84 (1986), pp. 157-176 | DOI | MR | Zbl

[27.] A. Ikeda, Stability conditions on ${CY}_{n}$ categories associated to $A_{n}$ -quivers and period maps, | arXiv

[28.] D. Joyce, Conjectures on Bridgeland stability for Fukaya categories of Calabi–Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow, | arXiv

[29.] Kajiura, H.; Saito, K.; Takahashi, A. Matrix factorization and representations of quivers. II. Type $A D E$ case, Adv. Math., Volume 211 (2007), pp. 327-362 | DOI | MR | Zbl

[30.] Kontsevich, M. Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians, vols. 1, 2 (1995), pp. 120-139

[31.] M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, | arXiv

[32.] Kontsevich, M.; Soibelman, Y. Notes on $A_{\infty}$ -algebras, $A_{\infty}$ -categories and non-commutative geometry, Homological Mirror Symmetry (2009), pp. 153-219 | DOI

[33.] Lyubashenko, V.; Ovsienko, S. A construction of quotient $A_{\infty}$ -categories, Homol. Homotopy Appl., Volume 8 (2006), pp. 157-203 | DOI | MR | Zbl

[34.] Macrì, E. Stability conditions on curves, Math. Res. Lett., Volume 14 (2007), pp. 657-672 | DOI | MR | Zbl

[35.] Masur, H.; Smillie, J. Hausdorff dimension of sets of nonergodic measured foliations, Ann. Math. (2), Volume 134 (1991), pp. 455-543 | DOI | MR | Zbl

[36.] D. Nadler, Cyclic symmetries of $A_{n}$ -quiver representations, | arXiv

[37.] Nazarova, L. A.; Roĭter, A. V. A certain problem of I.M. Gel’fand, Funkc. Anal. Prilozh., Volume 7 (1973), pp. 54-69

[38.] Nevanlinna, R. Über Riemannsche Flächen mit endlich vielen Windungspunkten, Acta Math., Volume 58 (1932), pp. 295-373 | DOI | MR | Zbl

[39.] Okada, S. Stability manifold of $P^{1}$ , J. Algebraic Geom., Volume 15 (2006), pp. 487-505 | DOI | Zbl

[40.] J. Pascaleff and N. Sibilla, Topological fukaya category and mirror symmetry for punctured surfaces, | arXiv

[41.] Polishchuk, A.; Zaslow, E. Categorical mirror symmetry: the elliptic curve, Adv. Theor. Math. Phys., Volume 2 (1998), pp. 443-470 | DOI | MR | Zbl

[42.] P. Seidel, Fukaya $A_{\infty}$ structures associated to Lefschetz fibrations. II, | arXiv

[43.] Seidel, P. Graded Lagrangian submanifolds, Bull. Soc. Math. Fr., Volume 128 (2000), pp. 103-149 | DOI | Numdam | MR | Zbl

[44.] Seidel, P. Fukaya Categories and Picard-Lefschetz Theory (2008) | DOI | Zbl

[45.] Sibilla, N.; Treumann, D.; Zaslow, E. Ribbon graphs and mirror symmetry, Sel. Math. New Ser., Volume 20 (2014), pp. 979-1002 | DOI | MR | Zbl

[46.] I. Smith, Quiver algebras as Fukaya categories, | arXiv

[47.] Strebel, K. Quadratic Differentials (1984) [Results in Mathematics and Related Areas (3)] | DOI | Zbl

[48.] Thomas, R. P. Moment maps, monodromy and mirror manifolds, Symplectic Geometry and Mirror Symmetry (2001), pp. 467-498 | DOI

[49.] Thomas, R. P.; Yau, S.-T. Special Lagrangians, stable bundles and mean curvature flow, Commun. Anal. Geom., Volume 10 (2002), pp. 1075-1113 | DOI | MR | Zbl

[50.] Veech, W. A. Flat surfaces, Am. J. Math., Volume 115 (1993), pp. 589-689 | DOI | MR | Zbl

Cité par Sources :