Seifert form of chain-type invertible singularities. (English) Zbl 1512.53082

Varolgunes, Umut

Kyoto J. Math. 63, No. 1, 155-193 (2023).

Summary: In this paper, we confirm a conjecture of P. Orlik and R. Randell [Invent. Math. 39, 199–211 (1977; Zbl 0341.14001)] from 1977 on the Seifert form of chain-type invertible singularities. We use Lefschetz bifibration techniques as developed by P. Seidel [Fukaya categories and Picard-Lefschetz theory. Zürich: European Mathematical Society (EMS) (2008; Zbl 1159.53001)] (inspired by Arnold and Donaldson) and take advantage of the symmetries at hand. We believe that our method will be useful in understanding the homological/categorical version of Berglund-Hübsch mirror conjecture for invertible singularities.

MSC:

53D37	Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category
14J33	Mirror symmetry (algebro-geometric aspects)
14D05	Structure of families (Picard-Lefschetz, monodromy, etc.)
53D40	Symplectic aspects of Floer homology and cohomology

Keywords:

invertible singularities; mirror symmetry; Seifert form

Citations:

Zbl 0341.14001; Zbl 1159.53001

References:

[1]	S. Akbulut and B. Ozbagci, Lefschetz fibrations on compact Stein surfaces, Geom. Topol. 5 (2001), 319-334. · Zbl 1002.57062 · doi:10.2140/gt.2001.5.319
[2]	D. Aramaki and A. Takahashi, Maximally-graded matrix factorizations for an invertible polynomial of chain type, Adv. Math. 373 (2020), article no. 107320. · Zbl 1455.14033 · doi:10.1016/j.aim.2020.107320
[3]	S. Balnojan and C. Hertling, Conjectures on spectral numbers for upper triangular matrices and for singularities, Math. Phys. Anal. Geom. 23 (2020), no. 1, paper no. 5. · Zbl 1439.32073 · doi:10.1007/s11040-019-9327-3
[4]	P. Berglund and T. Hübsch, A generalized construction of mirror manifolds, Nuclear Phys. B 393 (1993), nos. 1-2, 377-391. · Zbl 1245.14039 · doi:10.1016/0550-3213(93)90250-S
[5]	S. A. Broughton, Milnor numbers and the topology of polynomial hypersurfaces, Invent. Math. 92 (1988), no. 2, 217-241. · Zbl 0658.32005 · doi:10.1007/BF01404452
[6]	K. Dilcher, J. D. Nulton, and K. B Stolarsky, The zeros of a certain family of trinomials, Glasgow Math. J. 34 (1992), no. 1, 55-74. · Zbl 0751.30007 · doi:10.1017/S0017089500008545
[7]	A. Dimca, Monodromy and betti numbers of weighted complete intersections, Topology 24 (1985), no. 3, 369-374. · Zbl 0595.32011 · doi:10.1016/0040-9383(85)90009-6
[8]	A. Dimca, Singularities and Topology of Hypersurfaces, Springer, New York, 1992. · Zbl 0753.57001
[9]	A. Dold, Lectures on Algebraic Topology, Grundlehren Math. Wiss. 200, Springer, Berlin, 1972.
[10]	A. H Durfee, Fibered knots and algebraic singularities, Topology 13 (1974), no. 1, 47-59. · Zbl 0275.57007 · doi:10.1016/0040-9383(74)90037-8
[11]	W. Ebeling, Functions of Several Complex Variables and Their Singularities, Grad. Studies in Math. 83, Amer. Math. Soc., Providence, 2007. · Zbl 1188.32001 · doi:10.1090/gsm/083
[12]	W. Ebeling and S. M. Gusein-Zade, Monodromy of dual invertible polynomials, Mosc. Math. J. 11 (2011), no. 3, 463-472. · Zbl 1257.32028
[13]	J. Egerváry, On the trinomial equation, Math. Phys. Lapok 37 (1930), 36-57. · JFM 57.1350.04
[14]	H. Fan, W. Jiang, and D. Yang, Fukaya category of Landau-Ginzburg model, preprint, arXiv:1812.11748 [math.SG].
[15]	S. Ganatra, J. Pardon, and V. Shende, Covariantly functorial wrapped Floer theory on Liouville sectors, Publ. Math. Inst. Hautes Études Sci. 131 (2020), 73-200. · Zbl 1508.53091 · doi:10.1007/s10240-019-00112-x
[16]	A. L. Gorodentsev and S. A. Kuleshov, Helix theory, Mosc. Math. J. 4 (2004), no. 2, 377-440. · Zbl 1072.14020 · doi:10.17323/1609-4514-2004-4-2-377-440
[17]	S. Harvey, K. Kawamuro, and O. Plamenevskaya, On transverse knots and branched covers, Int. Math. Res. Not. IMRN 2009, no. 3, 512-546. · Zbl 1160.57004 · doi:10.1093/imrn/rnn138
[18]	S. A. Horocholyn, On the stokes matrices of the \[tt\ast -toda equation \], Tokyo J. Math. 40 (2017), no. 1, 185-202. · Zbl 1373.81327 · doi:10.3836/tjm/1502179222
[19]	L. H. Kauffman, Branched coverings, open books and knot periodicity, Topology 13 (1974), no. 2, 143-160. · Zbl 0283.57011 · doi:10.1016/0040-9383(74)90005-6
[20]	A. Kuznetsov and M. Smirnov, On residual categories for Grassmannians, Proc. London Math. Soc. 120 (2020), no. 5, 617-641. · Zbl 1493.14078 · doi:10.1112/plms.12294
[21]	D. T. Lê, La monodromie n’a pas de points fixes, J. Fac. Sci. Univ. Tokyo Sect. IA Math 22 (1975), no. 3, 409-427. · Zbl 0355.32012
[22]	J. Milnor and P. Orlik, Isolated singularities defined by weighted homogeneous polynomials, Topology 9 (1970), no. 4, 385-393. · Zbl 0204.56503 · doi:10.1016/0040-9383(70)90061-3
[23]	P. Orlik and R. Randell, The monodromy of weighted homogeneous singularities, Invent. Math. 39 (1977), 99-211. · Zbl 0341.14001 · doi:10.1007/BF01402973
[24]	O. Plamenevskaya, Contact structures with distinct Heegaard Floer invariants, Math. Res. Lett. 11 (2004), no. 4, 547-561. · Zbl 1064.57031 · doi:10.4310/MRL.2004.v11.n4.a12
[25]	P. Seidel, Fukaya Categories and Picard-Lefschetz Theory, Zur. Lect. Adv. Math. 10, Eur. Math. Soc., Zurich, 2008. · Zbl 1159.53001
[26]	P. Seidel, Suspending Lefschetz fibrations, with an application to local mirror symmetry, Comm. Math. Phys. 297 (2010), no. 2, 515-528. · Zbl 1197.14044 · doi:10.1007/s00220-009-0944-8
[27]	P. Seidel, Picard-Lefschetz theory and dilating \[{\mathbb{C}^{\ast }} -actions \], J. Topol. 8 (2015), no. 4, 1167-1201. · Zbl 1332.53104 · doi:10.1112/jtopol/jtv029
[28]	J. Stallings, “Constructions of fibred knots and links” in Algebraic and Geometric Topology (Stanford, Calif., 1976), Part 2, Proc. Sympos. Pure Math. 32, Amer. Math. Soc., Providence, 1978, 55-60. · Zbl 0394.57007
[29]	P. G. Szabó, On the roots of the trinomial equation, CEJOR Cent. Eur. J. Oper. Res. 18 (2010), no. 1, 97-104. · Zbl 1204.65001 · doi:10.1007/s10100-009-0130-2
[30]	A. Takahashi, “Weighted projective lines associated to regular systems of weights of dual type” in New Developments in Algebraic Geometry, Integrable Systems and Mirror Symmetry (RIMS, Kyoto, 2008), Adv. Stud. Pure Math 59 (2010), Math. Soc. Japan, Tokyo, 2010, 371-388. · Zbl 1213.14075 · doi:10.2969/aspm/05910371
[31]	T. Theobald and T. De Wolff, Norms of roots of trinomials, Math. Ann. 366 (2016), nos. 1-2, 219-247. · Zbl 1379.12002 · doi:10.1007/s00208-015-1323-8
[32]	A. Weinstein, Symplectic v-manifolds, periodic orbits of Hamiltonian systems, and the volume of certain Riemannian manifolds, Comm. Pure Appl. Math. 30 (1977), no. 2, 265-271. · Zbl 0339.58007 · doi:10.1002/cpa.3160300207
[33]	J. H. Wilkinson, “The Perfidious polynomial” in Studies in Numerical Analysis, MAA Stud. Math. 24, Math. Assoc. Amer., Washington, DC, 1984, 1-28. · Zbl 0601.65028