Curve counting theories via stable objects I. DT/PT correspondence

Toda, Yukinobu

doi:10.1090/S0894-0347-10-00670-3

Curve counting theories via stable objects I. DT/PT correspondence

Author: Yukinobu Toda
Journal: J. Amer. Math. Soc. 23 (2010), 1119-1157
MSC (2010): Primary 14D20; Secondary 18E30
DOI: https://doi.org/10.1090/S0894-0347-10-00670-3
Published electronically: April 16, 2010
MathSciNet review: 2669709
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The Donaldson-Thomas invariant is a curve counting invariant on Calabi-Yau 3-folds via ideal sheaves. Another counting invariant via stable pairs is introduced by Pandharipande and Thomas, which counts pairs of curves and divisors on them. These two theories are conjecturally equivalent via generating functions, called DT/PT correspondence. In this paper, we show the Euler characteristic version of DT/PT correspondence, using the notion of weak stability conditions and the wall-crossing formula.

1. A. Bayer.
Polynomial Bridgeland stability conditions and the large volume limit.
math.AG/ 0712.1083.
2. K. Behrend.
Donaldson-Thomas invariants via microlocal geometry.
Ann. of Math (to appear).
math.AG/0507523.
3. Kai Behrend and Barbara Fantechi, Symmetric obstruction theories and Hilbert schemes of points on threefolds, Algebra Number Theory 2 (2008), no. 3, 313–345. MR 2407118, https://doi.org/10.2140/ant.2008.2.313
4. K. Behrend and E. Getzler.
Chern-Simons functional.
In preparation.
5. R. Bezrukavnikov.
Perverse coherent sheaves (after Deligne).
math.AG/0005152.
6. T. Bridgeland.
Hall algebras and curve-counting invariants.
Preprint.
http://www. tombridgeland.staff.shef.ac.uk/papers/dtpt.pdf.
7. Tom Bridgeland, Stability conditions on triangulated categories, Ann. of Math. (2) 166 (2007), no. 2, 317–345. MR 2373143, https://doi.org/10.4007/annals.2007.166.317
8. Tom Bridgeland, Stability conditions on 𝐾3 surfaces, Duke Math. J. 141 (2008), no. 2, 241–291. MR 2376815, https://doi.org/10.1215/S0012-7094-08-14122-5
9. Jan Cheah, On the cohomology of Hilbert schemes of points, J. Algebraic Geom. 5 (1996), no. 3, 479–511. MR 1382733
10. Michel Van den Bergh, Three-dimensional flops and noncommutative rings, Duke Math. J. 122 (2004), no. 3, 423–455. MR 2057015, https://doi.org/10.1215/S0012-7094-04-12231-6
11. Dieter Happel, Idun Reiten, and Sverre O. Smalø, Tilting in abelian categories and quasitilted algebras, Mem. Amer. Math. Soc. 120 (1996), no. 575, viii+ 88. MR 1327209, https://doi.org/10.1090/memo/0575
12. Dominic Joyce, Configurations in abelian categories. I. Basic properties and moduli stacks, Adv. Math. 203 (2006), no. 1, 194–255. MR 2231046, https://doi.org/10.1016/j.aim.2005.04.008
13. Dominic Joyce, Configurations in abelian categories. II. Ringel-Hall algebras, Adv. Math. 210 (2007), no. 2, 635–706. MR 2303235, https://doi.org/10.1016/j.aim.2006.07.006
14. Dominic Joyce, Configurations in abelian categories. III. Stability conditions and identities, Adv. Math. 215 (2007), no. 1, 153–219. MR 2354988, https://doi.org/10.1016/j.aim.2007.04.002
15. Dominic Joyce, Motivic invariants of Artin stacks and ‘stack functions’, Q. J. Math. 58 (2007), no. 3, 345–392. MR 2354923, https://doi.org/10.1093/qmath/ham019
16. Dominic Joyce, Configurations in abelian categories. III. Stability conditions and identities, Adv. Math. 215 (2007), no. 1, 153–219. MR 2354988, https://doi.org/10.1016/j.aim.2007.04.002
17. D. Joyce and Y. Song.
A theory of generalized Donaldson-Thomas invariants.
math.AG/ 0810.5645.
18. Masaki Kashiwara, 𝑡-structures on the derived categories of holonomic 𝒟-modules and coherent 𝒪-modules, Mosc. Math. J. 4 (2004), no. 4, 847–868, 981 (English, with English and Russian summaries). MR 2124169, https://doi.org/10.17323/1609-4514-2004-4-4-847-868
19. M. Kontsevich and Y. Soibelman.
Stability structures, motivic Donaldson-Thomas invariants and cluster transformations.
math.AG/0811.2435.
20. M. Levine and R. Pandharipande, Algebraic cobordism revisited, Invent. Math. 176 (2009), no. 1, 63–130. MR 2485880, https://doi.org/10.1007/s00222-008-0160-8
21. Jun Li, Zero dimensional Donaldson-Thomas invariants of threefolds, Geom. Topol. 10 (2006), 2117–2171. MR 2284053, https://doi.org/10.2140/gt.2006.10.2117
22. Wei-Ping Li and Zhenbo Qin, On the Euler numbers of certain moduli spaces of curves and points, Comm. Anal. Geom. 14 (2006), no. 2, 387–410. MR 2255015
23. Max Lieblich, Moduli of complexes on a proper morphism, J. Algebraic Geom. 15 (2006), no. 1, 175–206. MR 2177199, https://doi.org/10.1090/S1056-3911-05-00418-2
24. D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory. I, Compos. Math. 142 (2006), no. 5, 1263–1285. MR 2264664, https://doi.org/10.1112/S0010437X06002302
25. R. Pandharipande and R. P. Thomas, Curve counting via stable pairs in the derived category, Invent. Math. 178 (2009), no. 2, 407–447. MR 2545686, https://doi.org/10.1007/s00222-009-0203-9
26. J. Stoppa and R. P. Thomas.
Hilbert schemes and stable pairs: GIT and derived category wall crossings.
math.AG/0903.1444.
27. R. P. Thomas, A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on 𝐾3 fibrations, J. Differential Geom. 54 (2000), no. 2, 367–438. MR 1818182
28. Y. Toda.
Curve counting theories via stable objects II. DT/ncDT/flop formula.
In preparation.
29. Y. Toda.
Generating functions of stable pair invariants via wall-crossings in derived categories.
math.AG/0806.0062.
30. Yukinobu Toda, Limit stable objects on Calabi-Yau 3-folds, Duke Math. J. 149 (2009), no. 1, 157–208. MR 2541209, https://doi.org/10.1215/00127094-2009-038
31. Yukinobu Toda, Moduli stacks and invariants of semistable objects on 𝐾3 surfaces, Adv. Math. 217 (2008), no. 6, 2736–2781. MR 2397465, https://doi.org/10.1016/j.aim.2007.11.010