The local Donaldson–Thomas theory of curves

Okounkov, A¹ ; Pandharipande, R¹

¹ Department of Mathematics, Princeton University, Princeton NJ 08544, USA

Geometry & topology, Tome 14 (2010) no. 3, pp. 1503-1567.

Résumé

The local Donaldson–Thomas theory of curves is solved by localization and degeneration methods. The results complete a triangle of equivalences relating Gromov–Witten theory, Donaldson–Thomas theory, and the quantum cohomology of the Hilbert scheme of points of the plane.

Détail

DOI : 10.2140/gt.2010.14.1503

Keywords: Gromov–Witten, Donaldson–Thomas, Hilbert scheme

Bibliographie

[1] R Bott, A residue formula for holomorphic vector-fields, J. Differential Geometry 1 (1967) 311

[2] J Bryan, T Graber, The crepant resolution conjecture, from: "Algebraic geometry – Seattle 2005. Part 1", Proc. Sympos. Pure Math. 80, Amer. Math. Soc. (2009) 23

[3] J Bryan, R Pandharipande, Curves in Calabi–Yau threefolds and topological quantum field theory, Duke Math. J. 126 (2005) 369

[4] J Bryan, R Pandharipande, The local Gromov–Witten theory of curves, J. Amer. Math. Soc. 21 (2008) 101

[5] S K Donaldson, R P Thomas, Gauge theory in higher dimensions, from: "The geometric universe (Oxford, 1996)", Oxford Univ. Press (1998) 31

[6] Y Eliashberg, A Givental, H Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. (2000) 560

[7] A Gholampour, On the equivariant Gromov–Witten theory of $\mathbb{P}^2$–bundles over curves, Comm. Anal. Geom. 14 (2006) 633

[8] A Gholampour, Y Song, Evidence for the Gromov–Witten/Donaldson–Thomas correspondence, Math. Res. Lett. 13 (2006) 623

[9] T Graber, R Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999) 487

[10] I Grojnowski, Instantons and affine algebras I: The Hilbert scheme and vertex operators, Math. Res. Lett. 3 (1996) 275

[11] M Haiman, Notes on Macdonald polynomials and the geometry of Hilbert schemes, from: "Symmetric functions 2001: surveys of developments and perspectives", NATO Sci. Ser. II Math. Phys. Chem. 74, Kluwer Acad. Publ. (2002) 1

[12] E N Ionel, T H Parker, Relative Gromov–Witten invariants, Ann. of Math. $(2)$ 157 (2003) 45

[13] Y H Kiem, J Li, Gromov–Witten invariants of varieties with holomorphic 2–forms

[14] M Lehn, Chern classes of tautological sheaves on Hilbert schemes of points on surfaces, Invent. Math. 136 (1999) 157

[15] M Lehn, C Sorger, Symmetric groups and the cup product on the cohomology of Hilbert schemes, Duke Math. J. 110 (2001) 345

[16] A M Li, Y Ruan, Symplectic surgery and Gromov–Witten invariants of Calabi–Yau 3–folds, Invent. Math. 145 (2001) 151

[17] J Li, A degeneration formula of GW-invariants, J. Differential Geom. 60 (2002) 199

[18] J Li, private communication (2004)

[19] J Li, B Wu, Degeneration of Donaldson–Thomas invariants, preprint (2009)

[20] W P Li, Z Qin, W Wang, Vertex algebras and the cohomology ring structure of Hilbert schemes of points on surfaces, Math. Ann. 324 (2002) 105

[21] C C Liu, K Liu, J Zhou, A proof of a conjecture of Mariño–Vafa on Hodge integrals

[22] I G Macdonald, Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press (1995)

[23] M Mariño, C Vafa, Framed knots at large $N$

[24] D Maulik, N Nekrasov, A Okounkov, R Pandharipande, Gromov–Witten theory and Donaldson–Thomas theory I, Compos. Math. 142 (2006) 1263

[25] D Maulik, N Nekrasov, A Okounkov, R Pandharipande, Gromov–Witten theory and Donaldson–Thomas theory II, Compos. Math. 142 (2006) 1286

[26] D Maulik, R Pandharipande, A topological view of Gromov–Witten theory, Topology 45 (2006) 887

[27] D Maulik, R Pandharipande, R Thomas, Curves on $K3$ surfaces and modular forms

[28] H Nakajima, Lectures on Hilbert schemes of points on surfaces, University Lecture Series 18, American Mathematical Society (1999)

[29] A Okounkov, R Pandharipande, Hodge integrals and invariants of the unknot, Geom. Topol. 8 (2004) 675

[30] A Okounkov, R Pandharipande, Virasoro constraints for target curves, Invent. Math. 163 (2006) 47

[31] A Okounkov, R Pandharipande, Quantum cohomology of the Hilbert scheme of points in the plane, Invent. Math. 179 (2010) 523

[32] A Okounkov, N Reshetikhin, Correlation function of Schur process with application to local geometry of a random 3–dimensional Young diagram, J. Amer. Math. Soc. 16 (2003) 581

[33] A Okounkov, N Reshetikhin, C Vafa, Quantum Calabi–Yau and classical crystals

[34] R P Thomas, A holomorphic Casson invariant for Calabi–Yau 3–folds, and bundles on $K3$ fibrations, J. Differential Geom. 54 (2000) 367

[35] E Vasserot, Sur l'anneau de cohomologie du schéma de Hilbert de $\mathbb{C}^2$, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) 7