Volume 20, issue 1 (2016)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 27
Issue 8, 2937–3385
Issue 7, 2497–2936
Issue 6, 2049–2496
Issue 5, 1657–2048
Issue 4, 1273–1655
Issue 3, 823–1272
Issue 2, 417–821
Issue 1, 1–415

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Author Index
To Appear
Other MSP Journals
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