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Quantum periods for $3$–dimensional Fano manifolds

Tom Coates, Alessio Corti, Sergey Galkin and Alexander Kasprzyk

Geometry & Topology 20 (2016) 103–256
1 M Akhtar, T Coates, S Galkin, A M Kasprzyk, Minkowski polynomials and mutations, SIGMA Symmetry Integrability Geom. Methods Appl. 8 (2012) MR3007265
2 D Auroux, Mirror symmetry and T–duality in the complement of an anticanonical divisor, J. Gökova Geom. Topol. GGT 1 (2007) 51 MR2386535
3 D Auroux, Special Lagrangian fibrations, wall-crossing, and mirror symmetry, from: "Geometry, analysis, and algebraic geometry : forty years of the Journal of Differential Geometry" (editors H D Cao, S T Yau), Surv. Differ. Geom. 13, International Press (2009) 1 MR2537081
4 V V Batyrev, Toric degenerations of Fano varieties and constructing mirror manifolds, from: "The Fano conference" (editors A Collino, A Conte, M Marchisio), Univ. Torino, Turin (2004) 109 MR2112571
5 A Beauville, Quantum cohomology of complete intersections, Mat. Fiz. Anal. Geom. 2 (1995) 384 MR1484335
6 K Behrend, The product formula for Gromov–Witten invariants, J. Algebraic Geom. 8 (1999) 529 MR1689355
7 K Behrend, B Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997) 45 MR1437495
8 A Bertram, I Ciocan-Fontanine, B Kim, Gromov–Witten invariants for abelian and nonabelian quotients, J. Algebraic Geom. 17 (2008) 275 MR2369087
9 I Ciocan-Fontanine, B Kim, C Sabbah, The abelian/nonabelian correspondence and Frobenius manifolds, Invent. Math. 171 (2008) 301 MR2367022
10 G Ciolli, Computing the quantum cohomology of some Fano threefolds and its semisimplicity, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 7 (2004) 511 MR2072951
11 T Coates, A Corti, S Galkin, V Golyshev, A Kasprzyk, Mirror symmetry and Fano manifolds, preprint (2012) arXiv:1212.1722
12 T Coates, A Corti, S Galkin, A Kasprzyk, Table of Laurent polynomial mirrors of 3–dimensional Fano manifolds (2016)
13 T Coates, A Corti, H Iritani, H H Tseng, Computing genus-zero twisted Gromov–Witten invariants, Duke Math. J. 147 (2009) 377 MR2510741
14 T Coates, S Galkin, A Kasprzyk, 3d Minkowski period sequences, online resource
15 T Coates, A Gholampour, H Iritani, Y Jiang, P Johnson, C Manolache, The quantum Lefschetz hyperplane principle can fail for positive orbifold hypersurfaces, Math. Res. Lett. 19 (2012) 997 MR3039825
16 T Coates, A Givental, Quantum Riemann–Roch, Lefschetz and Serre, Ann. of Math. 165 (2007) 15 MR2276766
17 T Coates, Y P Lee, A Corti, H H Tseng, The quantum orbifold cohomology of weighted projective spaces, Acta Math. 202 (2009) 139 MR2506749
18 D A Cox, S Katz, Mirror symmetry and algebraic geometry, 68, Amer. Math. Soc. (1999) MR1677117
19 B Fantechi, Stacks for everybody, from: "European Congress of Mathematics, I" (editors C Casacuberta, R M Miró-Roig, J Verdera, S Xambó-Descamps), Progr. Math. 201, Birkhäuser (2001) 349 MR1905329
20 S Galkin, Small toric degenerations of Fano threefolds, preprint (2007)
21 S Galkin, Toric degenerations of Fano manifolds, PhD thesis, Steklov Math. Institute (2008)
22 A Gathmann, Gromov–Witten invariants of blow-ups, J. Algebraic Geom. 10 (2001) 399 MR1832328
23 A B Givental, Equivariant Gromov–Witten invariants, Internat. Math. Res. Notices (1996) 613 MR1408320
24 A Givental, A mirror theorem for toric complete intersections, from: "Topological field theory, primitive forms and related topics" (editors M Kashiwara, A Matsuo, K Saito, I Satake), Progr. Math. 160, Birkhäuser (1998) 141 MR1653024
25 V V Golyshev, Classification problems and mirror duality, from: "Surveys in geometry and number theory : reports on contemporary Russian mathematics" (editor N Young), London Math. Soc. Lecture Note Ser. 338, Cambridge Univ. Press (2007) 88 MR2306141
26 M Gross, Toric degenerations and Batyrev–Borisov duality, Math. Ann. 333 (2005) 645 MR2198802
27 M Gross, B Siebert, Affine manifolds, log structures, and mirror symmetry, Turkish J. Math. 27 (2003) 33 MR1975331
28 M Gross, B Siebert, Mirror symmetry via logarithmic degeneration data, I, J. Differential Geom. 72 (2006) 169 MR2213573
29 M Gross, B Siebert, Mirror symmetry via logarithmic degeneration data, II, J. Algebraic Geom. 19 (2010) 679 MR2669728
30 N P Gushel’, Fano varieties of genus 8, Uspekhi Mat. Nauk 38 (1983) 163 MR693729
31 N P Gushel’, Fano 3–folds of genus 8, Algebra i Analiz 4 (1992) 120 MR1171957
32 J Harris, D Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982) 23 MR664324
33 J Hu, Gromov–Witten invariants of blow-ups along points and curves, Math. Z. 233 (2000) 709 MR1759269
34 J Hu, Gromov–Witten invariants of blow-ups along surfaces, Compositio Math. 125 (2001) 345 MR1818985
35 H Iritani, An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, Adv. Math. 222 (2009) 1016 MR2553377
36 V A Iskovskih, Fano threefolds, I, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977) 516 MR463151
37 V A Iskovskih, Fano threefolds, II, Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978) 506 MR503430
38 V A Iskovskih, Anticanonical models of three-dimensional algebraic varieties, from: "Current problems in mathematics, XII" (editor R V Gamkrelidze), VINITI (1979) 59 MR537685
39 L Katzarkov, M Kontsevich, T Pantev, Hodge theoretic aspects of mirror symmetry, from: "From Hodge theory to integrability and TQFT tt*-geometry" (editors R Y Donagi, K Wendland), Proc. Sympos. Pure Math. 78, Amer. Math. Soc. (2008) 87 MR2483750
40 M Kontsevich, Enumeration of rational curves via torus actions, from: "The moduli space of curves" (editors R Dijkgraaf, C Faber, G van der Geer), Progr. Math. 129, Birkhäuser (1995) 335 MR1363062
41 M Kontsevich, Homological algebra of mirror symmetry, from: "Proceedings of the International Congress of Mathematicians, I" (editor S D Chatterji), Birkhäuser (1995) 120 MR1403918
42 M Kontsevich, Y Manin, Gromov–Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994) 525 MR1291244
43 M Kontsevich, Y Soibelman, Homological mirror symmetry and torus fibrations, from: "Symplectic geometry and mirror symmetry" (editors K Fukaya, Y G Oh, K Ono, G Tian), World Sci. Publ. (2001) 203 MR1882331
44 M Kontsevich, Y Soibelman, Affine structures and non-Archimedean analytic spaces, from: "The unity of mathematics" (editors P Etingof, V Retakh, I M Singer), Progr. Math. 244, Birkhäuser (2006) 321 MR2181810
45 H H Lai, Gromov–Witten invariants of blow-ups along submanifolds with convex normal bundles, Geom. Topol. 13 (2009) 1 MR2469512
46 R Lazarsfeld, Positivity in algebraic geometry, II : Positivity for vector bundles, and multiplier ideals, 49, Springer, Berlin (2004) MR2095472
47 J Li, G Tian, Virtual moduli cycles and Gromov–Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998) 119 MR1467172
48 C Manolache, Virtual pull-backs, J. Algebraic Geom. 21 (2012) 201 MR2877433
49 K Matsuki, Weyl groups and birational transformations among minimal models, 557, Amer. Math. Soc. (1995) MR1259932
50 S Mori, S Mukai, Classification of Fano 3–folds with B2 2, Manuscripta Math. 36 (1981/82) 147 MR641971
51 S Mori, S Mukai, On Fano 3–folds with B2 2, from: "Algebraic varieties and analytic varieties" (editor S Iitaka), Adv. Stud. Pure Math. 1, North-Holland, Amsterdam (1983) 101 MR715648
52 S Mori, S Mukai, Classification of Fano 3–folds with B2 2, I, from: "Algebraic and topological theories" (editors M Nagata, S Araki, A Hattori), Kinokuniya, Tokyo (1986) 496 MR1102273
53 S Mori, S Mukai, Extremal rays and Fano 3–folds, from: "The Fano Conference" (editors A Collino, A Conte, M Marchisio), Univ. Torino, Turin (2004) 37 MR2112566
54 S Mukai, Biregular classification of Fano 3–folds and Fano manifolds of coindex 3, Proc. Nat. Acad. Sci. USA 86 (1989) 3000 MR995400
55 S Mukai, Curves and symmetric spaces, Proc. Japan Acad. Ser. A Math. Sci. 68 (1992) 7 MR1158012
56 S Mukai, Fano 3–folds, from: "Complex projective geometry" (editors G Ellingsrud, C Peskine, G Sacchiero, S A Strømme), London Math. Soc. Lecture Note Ser. 179, Cambridge Univ. Press (1992) 255 MR1201387
57 S Mukai, Curves and Grassmannians, from: "Algebraic geometry and related topics" (editors J H Yang, Y Namikawa), Conf. Proc. Lecture Notes Algebraic Geom. 1, Int. Press (1993) 19 MR1285374
58 S Mukai, Curves and symmetric spaces, I, Amer. J. Math. 117 (1995) 1627 MR1363081
59 S Mukai, New developments in Fano manifold theory related to the vector bundle method and moduli problems, Sūgaku 47 (1995) 125 MR1364825
60 S Mukai, Curves and symmetric spaces, II, Ann. of Math. 172 (2010) 1539 MR2726093
61 S Mukai, M Reid, H Takagi, Classification of indecomposable Gorenstein Fano 3–folds, unpublished manuscript
62 P E Newstead, Introduction to moduli problems and orbit spaces, 51, Narosa (1978) MR546290
63 R Pandharipande, Rational curves on hypersurfaces (after A Givental), from: "Séminaire Bourbaki 1997/98 (Exposé 848)", Astérisque 252, Soc. Math. France (1998) 307 MR1685628
64 V V Przhiyalkovskiĭ, Gromov–Witten invariants of Fano threefolds of genera 6 and 8, Mat. Sb. 198 (2007) 145 MR2354283
65 V V Przhiyalkovskiĭ, Quantum cohomology of smooth complete intersections in weighted projective spaces and in singular toric varieties, Mat. Sb. 198 (2007) 107 MR2360793
66 V Przyjalkowski, On Landau–Ginzburg models for Fano varieties, Commun. Number Theory Phys. 1 (2007) 713 MR2412270
67 M I Qureshi, Families of polarised varieties in weighted flag varieties, PhD thesis, University of Oxford (2011)
68 S Ramanan, The moduli spaces of vector bundles over an algebraic curve, Math. Ann. 200 (1973) 69 MR0325615
69 A Strangeway, A reconstruction theorem for quantum cohomology of Fano bundles on projective space, preprint (2013) arXiv:1302.5089v2
70 A Strominger, S T Yau, E Zaslow, Mirror symmetry is T–duality, Nuclear Phys. B 479 (1996) 243 MR1429831
71 A N Tjurin, Geometry of moduli of vector bundles, Uspehi Mat. Nauk 29 (1974) 59 MR0404261
72 A Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989) 613 MR1005008