Geometry & Topology Volume 28, issue 8 (2024)

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

Submission form

Editorial board

Subscriptions

ISSN (electronic): 1364-0380

ISSN (print): 1465-3060

Author index

To appear

Other MSP journals

Holomorphic anomaly equations for the Hilbert scheme of points of a K3 surface

Georg Oberdieck

Geometry & Topology 28 (2024) 3779–3868

DOI: 10.2140/gt.2024.28.3779

Bibliography

1	D Abramovich, C Cadman, J Wise, Relative and orbifold Gromov–Witten invariants, Algebr. Geom. 4 (2017) 472 MR3683504
2	M Alim, H Movasati, E Scheidegger, S T Yau, Gauss–Manin connection in disguise : Calabi–Yau threefolds, Comm. Math. Phys. 344 (2016) 889 MR3508164
3	H Argüz, P Bousseau, R Pandharipande, D Zvonkine, Gromov–Witten theory of complete intersections via nodal invariants, J. Topol. 16 (2023) 264 MR4575868
4	Y Bae, T H Buelles, Curves on K3 surfaces in divisibility 2, Forum Math. Sigma 9 (2021) MR4211942
5	Y Bae, J Schmitt, Chow rings of stacks of prestable curves, I, Forum Math. Sigma 10 (2022) MR4430955
6	Y Bae, D Holmes, R Pandharipande, J Schmitt, R Schwarz, Pixton’s formula and Abel–Jacobi theory on the Picard stack, Acta Math. 230 (2023) 205 MR4628606
7	A Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom. 18 (1983) 755 MR0730926
8	K Behrend, Gromov–Witten invariants in algebraic geometry, Invent. Math. 127 (1997) 601 MR1431140
9	K Behrend, The product formula for Gromov–Witten invariants, J. Algebraic Geom. 8 (1999) 529 MR1689355
10	K Behrend, B Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997) 45 MR1437495
11	M Bershadsky, S Cecotti, H Ooguri, C Vafa, Holomorphic anomalies in topological field theories, Nuclear Phys. B 405 (1993) 279 MR1240687
12	S Bloch, A Okounkov, The character of the infinite wedge representation, Adv. Math. 149 (2000) 1 MR1742353
13	P Bousseau, H Fan, S Guo, L Wu, Holomorphic anomaly equation for (ℙ²,E) and the Nekrasov–Shatashvili limit of local ℙ², Forum Math. Pi 9 (2021) MR4254953
14	J Bryan, N C Leung, The enumerative geometry of K3 surfaces and modular forms, J. Amer. Math. Soc. 13 (2000) 371 MR1750955
15	J Bryan, G Oberdieck, CHL Calabi–Yau threefolds : curve counting, Mathieu moonshine and Siegel modular forms, Commun. Number Theory Phys. 14 (2020) 785 MR4164175
16	J Bryan, R Pandharipande, The local Gromov–Witten theory of curves, J. Amer. Math. Soc. 21 (2008) 101 MR2350052
17	J Bryan, G Oberdieck, R Pandharipande, Q Yin, Curve counting on abelian surfaces and threefolds, Algebr. Geom. 5 (2018) 398 MR3813750
18	Y Cao, G Oberdieck, Y Toda, Gopakumar–Vafa type invariants of holomorphic symplectic 4–folds, Comm. Math. Phys. 405 (2024) 26 MR4698661
19	H L Chang, S Guo, J Li, BCOV’s Feynman rule of quintic 3–folds, preprint (2018) arXiv:1810.00394
20	T Coates, H Iritani, Gromov–Witten invariants of local ℙ² and modular forms, Kyoto J. Math. 61 (2021) 543 MR4301054
21	M Eichler, D Zagier, The theory of Jacobi forms, 55, Birkhäuser (1985) MR0781735
22	B Eynard, N Orantin, Computation of open Gromov–Witten invariants for toric Calabi–Yau 3–folds by topological recursion : a proof of the BKMP conjecture, Comm. Math. Phys. 337 (2015) 483 MR3339157
23	B Eynard, N Orantin, M Mariño, Holomorphic anomaly and matrix models, J. High Energy Phys. 2007 (2007) 058 MR2326593
24	C Faber, R Pandharipande, Relative maps and tautological classes, J. Eur. Math. Soc. 7 (2005) 13 MR2120989
25	B Fang, Y Ruan, Y Zhang, J Zhou, Open Gromov–Witten theory of K_ℙ², K_ℙ¹×ℙ¹, K_{W ℙ[1,1,2]}, K_𝔽₁ and Jacobi forms, Comm. Math. Phys. 369 (2019) 675 MR3962006
26	B Fang, C C M Liu, Z Zong, On the remodeling conjecture for toric Calabi–Yau 3–orbifolds, J. Amer. Math. Soc. 33 (2020) 135 MR4066474
27	A Fujiki, On the de Rham cohomology group of a compact Kähler symplectic manifold, from: "Algebraic geometry" (editor T Oda), Adv. Stud. Pure Math. 10, North-Holland (1987) 105 MR0946237
28	I Grojnowski, Instantons and affine algebras, I : The Hilbert scheme and vertex operators, Math. Res. Lett. 3 (1996) 275 MR1386846
29	S Guo, F Janda, Y Ruan, Structure of higher genus Gromov–Witten invariants of quintic 3–folds, preprint (2018) arXiv:1812.11908
30	D Huybrechts, A global Torelli theorem for hyperkähler manifolds (after M Verbitsky), from: "Séminaire Bourbaki, 2010/2011", Astérisque 348, Soc. Math. France (2012) MR3051203
31	J W van Ittersum, G Oberdieck, A Pixton, Gromov–Witten theory of K3 surfaces and a Kaneko–Zagier equation for Jacobi forms, Selecta Math. 27 (2021) 64 MR4281261
32	M Kaneko, D Zagier, A generalized Jacobi theta function and quasimodular forms, from: "The moduli space of curves" (editors R Dijkgraaf, C Faber, G van der Geer), Progr. Math. 129, Birkhäuser (1995) 165 MR1363056
33	Y H Kiem, J Li, Localizing virtual cycles by cosections, J. Amer. Math. Soc. 26 (2013) 1025 MR3073883
34	B Kim, F Sato, A generalization of Fulton–MacPherson configuration spaces, Selecta Math. 15 (2009) 435 MR2551189
35	N Koblitz, Introduction to elliptic curves and modular forms, 97, Springer (1993) MR1216136
36	M Kool, R Thomas, Reduced classes and curve counting on surfaces, I : Theory, Algebr. Geom. 1 (2014) 334 MR3238154
37	A Kresch, Cycle groups for Artin stacks, Invent. Math. 138 (1999) 495 MR1719823
38	Y P Lee, F Qu, A product formula for log Gromov–Witten invariants, J. Math. Soc. Japan 70 (2018) 229 MR3750275
39	M Lehn, Chern classes of tautological sheaves on Hilbert schemes of points on surfaces, Invent. Math. 136 (1999) 157 MR1681097
40	H Lho, Gromov–Witten invariants of Calabi–Yau fibrations, preprint (2019) arXiv:1904.10315
41	H Lho, Gromov–Witten invariants of Calabi–Yau manifolds with two Kähler parameters, Int. Math. Res. Not. 2021 (2021) 7552 MR4259155
42	H Lho, R Pandharipande, Stable quotients and the holomorphic anomaly equation, Adv. Math. 332 (2018) 349 MR3810256
43	H Lho, R Pandharipande, Crepant resolution and the holomorphic anomaly equation for [ℂ³∕ℤ₃], Proc. Lond. Math. Soc. 119 (2019) 781 MR3960668
44	H Lho, R Pandharipande, Holomorphic anomaly equations for the formal quintic, Peking Math. J. 2 (2019) 1 MR4059998
45	J Li, Stable morphisms to singular schemes and relative stable morphisms, J. Differential Geom. 57 (2001) 509 MR1882667
46	J Li, A degeneration formula of GW-invariants, J. Differential Geom. 60 (2002) 199 MR1938113
47	J Li, Lecture notes on relative GW-invariants, from: "Intersection theory and moduli" (editors E Arbarello, G Ellingsrud, L Goettsche), ICTP Lect. Notes 19, Abdus Salam Int. Cent. Theoret. Phys. (2004) 41 MR2172495
48	J Li, G Tian, Virtual moduli cycles and Gromov–Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998) 119 MR1467172
49	A Libgober, Elliptic genera, real algebraic varieties and quasi-Jacobi forms, from: "Topology of stratified spaces" (editors G Friedman, E Hunsicker, A Libgober, L Maxim), Math. Sci. Res. Inst. Publ. 58, Cambridge Univ. Press (2011) 95 MR2796409
50	H Liu, Quasimaps and stable pairs, Forum Math. Sigma 9 (2021) MR4247638
51	E Looijenga, V A Lunts, A Lie algebra attached to a projective variety, Invent. Math. 129 (1997) 361 MR1465328
52	E Markman, On the monodromy of moduli spaces of sheaves on K3 surfaces, J. Algebraic Geom. 17 (2008) 29 MR2357680
53	E Markman, A survey of Torelli and monodromy results for holomorphic-symplectic varieties, from: "Complex and differential geometry" (editors W Ebeling, K Hulek, K Smoczyk), Springer Proc. Math. 8, Springer (2011) 257 MR2964480
54	E Markman, On the existence of universal families of marked irreducible holomorphic symplectic manifolds, Kyoto J. Math. 61 (2021) 207 MR4260432
55	D Maulik, Gromov–Witten theory of 𝒜_n–resolutions, Geom. Topol. 13 (2009) 1729 MR2496055
56	D Maulik, A Oblomkov, Donaldson–Thomas theory of 𝒜_n × ℙ¹, Compos. Math. 145 (2009) 1249 MR2551996
57	D Maulik, A Oblomkov, Quantum cohomology of the Hilbert scheme of points on 𝒜_n–resolutions, J. Amer. Math. Soc. 22 (2009) 1055 MR2525779
58	D Maulik, R Pandharipande, A topological view of Gromov–Witten theory, Topology 45 (2006) 887 MR2248516
59	D Maulik, R Pandharipande, Gromov–Witten theory and Noether–Lefschetz theory, from: "A celebration of algebraic geometry" (editors B Hassett, J McKernan, J Starr, R Vakil), Clay Math. Proc. 18, Amer. Math. Soc. (2013) 469 MR3114953
60	D Maulik, N Nekrasov, A Okounkov, R Pandharipande, Gromov–Witten theory and Donaldson–Thomas theory, I, Compos. Math. 142 (2006) 1263 MR2264664
61	D Maulik, N Nekrasov, A Okounkov, R Pandharipande, Gromov–Witten theory and Donaldson–Thomas theory, II, Compos. Math. 142 (2006) 1286 MR2264665
62	D Maulik, R Pandharipande, R P Thomas, Curves on K3 surfaces and modular forms, J. Topol. 3 (2010) 937 MR2746343
63	D Maulik, A Oblomkov, A Okounkov, R Pandharipande, Gromov–Witten/Donaldson–Thomas correspondence for toric 3–folds, Invent. Math. 186 (2011) 435 MR2845622
64	T Milanov, Y Ruan, Y Shen, Gromov–Witten theory and cycle-valued modular forms, J. Reine Angew. Math. 735 (2018) 287 MR3757479
65	H Nakajima, Heisenberg algebra and Hilbert schemes of points on projective surfaces, Ann. of Math. 145 (1997) 379 MR1441880
66	A Neguţ, G Oberdieck, Q Yin, Motivic decompositions for the Hilbert scheme of points of a K3 surface, J. Reine Angew. Math. 778 (2021) 65 MR4308613
67	D Nesterov, Quasimaps to moduli spaces of sheaves, preprint (2021) arXiv:2111.11417
68	D Nesterov, Quasimaps to moduli spaces of sheaves on a K3 surface, Forum Math. Sigma 12 (2024) MR4746880
69	G Oberdieck, A Serre derivative for even weight Jacobi forms, preprint (2012) arXiv:1209.5628
70	G Oberdieck, The enumerative geometry of the Hilbert schemes of points of a K3 surface, PhD thesis, ETH Zürich (2015)
71	G Oberdieck, Gromov–Witten invariants of the Hilbert schemes of points of a K3 surface, Geom. Topol. 22 (2018) 323 MR3720346
72	G Oberdieck, On reduced stable pair invariants, Math. Z. 289 (2018) 323 MR3803792
73	G Oberdieck, Gromov–Witten theory of K3 × ℙ¹ and quasi-Jacobi forms, Int. Math. Res. Not. 2019 (2019) 4966 MR4009529
74	G Oberdieck, A Lie algebra action on the Chow ring of the Hilbert scheme of points of a K3 surface, Comment. Math. Helv. 96 (2021) 65 MR4228614
75	G Oberdieck, Gromov–Witten theory and Noether–Lefschetz theory for holomorphic-symplectic varieties, Forum Math. Sigma 10 (2022) MR4404412
76	G Oberdieck, Marked relative invariants and GW/PT correspondences, Adv. Math. 439 (2024) 109472 MR4687909
77	G Oberdieck, Multiple cover formulas for K3 geometries, wall-crossing, and Quot schemes, Geom. Topol. 28 (2024) 3221
78	G Oberdieck, R Pandharipande, Curve counting on K3 ×E, the Igusa cusp form χ₁₀, and descendent integration, from: "K3 surfaces and their moduli" (editors C Faber, G Farkas, G van der Geer), Progr. Math. 315, Birkhäuser (2016) 245 MR3524171
79	G Oberdieck, A Pixton, Holomorphic anomaly equations and the Igusa cusp form conjecture, Invent. Math. 213 (2018) 507 MR3827207
80	G Oberdieck, A Pixton, Gromov–Witten theory of elliptic fibrations : Jacobi forms and holomorphic anomaly equations, Geom. Topol. 23 (2019) 1415 MR3956895
81	G Oberdieck, A Pixton, Quantum cohomology of the Hilbert scheme of points on an elliptic surface, (2023) arXiv:2312.13188
82	G Oberdieck, J Shen, Curve counting on elliptic Calabi–Yau threefolds via derived categories, J. Eur. Math. Soc. 22 (2020) 967 MR4055994
83	J Oesinghaus, Quasisymmetric functions and the Chow ring of the stack of expanded pairs, Res. Math. Sci. 6 (2019) 5 MR3911800
84	A Okounkov, R Pandharipande, Virasoro constraints for target curves, Invent. Math. 163 (2006) 47 MR2208418
85	A Okounkov, R Pandharipande, The local Donaldson–Thomas theory of curves, Geom. Topol. 14 (2010) 1503 MR2679579
86	A Okounkov, R Pandharipande, Quantum cohomology of the Hilbert scheme of points in the plane, Invent. Math. 179 (2010) 523 MR2587340
87	R Pandharipande, A Pixton, Gromov–Witten/pairs descendent correspondence for toric 3–folds, Geom. Topol. 18 (2014) 2747 MR3285224
88	R Pandharipande, A Pixton, Gromov–Witten/pairs correspondence for the quintic 3–fold, J. Amer. Math. Soc. 30 (2017) 389 MR3600040
89	, The Stacks project, electronic reference (2005–)
90	M Verbitsky, Cohomology of compact hyper-Kähler manifolds and its applications, Geom. Funct. Anal. 6 (1996) 601 MR1406664
91	M Verbitsky, Mapping class group and a global Torelli theorem for hyperkähler manifolds, Duke Math. J. 162 (2013) 2929 MR3161308
92	X Wang, Quasi-modularity and holomorphic anomaly equation for the twisted Gromov–Witten theory : 𝒪(3) over ℙ², Acta Math. Sin. (Engl. Ser.) 35 (2019) 1945 MR4033592
93	X Wang, Finite generation and holomorphic anomaly equation for equivariant Gromov–Witten invariants of K_ℙ¹×ℙ¹, Front. Math. 18 (2023) 17 MR4610301
94	A Weil, Elliptic functions according to Eisenstein and Kronecker, 88, Springer (1976) MR0562289
95	D Zagier, Periods of modular forms and Jacobi theta functions, Invent. Math. 104 (1991) 449 MR1106744
96	C Ziegler, Jacobi forms of higher degree, Abh. Math. Sem. Univ. Hamburg 59 (1989) 191 MR1049896
97	D Zvonkine, Intersection of double loci with boundary strata, unpublished note (2015)