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
HTML articles powered by AMS MathViewer

by Yukinobu Toda

J. Amer. Math. Soc. 23 (2010), 1119-1157

DOI: https://doi.org/10.1090/S0894-0347-10-00670-3

Published electronically: April 16, 2010

PDF | Request permission

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.

References

A. Bayer. Polynomial Bridgeland stability conditions and the large volume limit. math.AG/ 0712.1083.
K. Behrend. Donaldson-Thomas invariants via microlocal geometry. Ann. of Math (to appear). math.AG/0507523.
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, DOI 10.2140/ant.2008.2.313
K. Behrend and E. Getzler. Chern-Simons functional. In preparation.
R. Bezrukavnikov. Perverse coherent sheaves (after Deligne). math.AG/0005152.
T. Bridgeland. Hall algebras and curve-counting invariants. Preprint. http://www. tombridgeland.staff.shef.ac.uk/papers/dtpt.pdf.
Tom Bridgeland, Stability conditions on triangulated categories, Ann. of Math. (2) 166 (2007), no. 2, 317–345. MR 2373143, DOI 10.4007/annals.2007.166.317
Tom Bridgeland, Stability conditions on $K3$ surfaces, Duke Math. J. 141 (2008), no. 2, 241–291. MR 2376815, DOI 10.1215/S0012-7094-08-14122-5
Jan Cheah, On the cohomology of Hilbert schemes of points, J. Algebraic Geom. 5 (1996), no. 3, 479–511. MR 1382733
Michel Van den Bergh, Three-dimensional flops and noncommutative rings, Duke Math. J. 122 (2004), no. 3, 423–455. MR 2057015, DOI 10.1215/S0012-7094-04-12231-6
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, DOI 10.1090/memo/0575
Dominic Joyce, Configurations in abelian categories. I. Basic properties and moduli stacks, Adv. Math. 203 (2006), no. 1, 194–255. MR 2231046, DOI 10.1016/j.aim.2005.04.008
Dominic Joyce, Configurations in abelian categories. II. Ringel-Hall algebras, Adv. Math. 210 (2007), no. 2, 635–706. MR 2303235, DOI 10.1016/j.aim.2006.07.006
Dominic Joyce, Configurations in abelian categories. III. Stability conditions and identities, Adv. Math. 215 (2007), no. 1, 153–219. MR 2354988, DOI 10.1016/j.aim.2007.04.002
Dominic Joyce, Motivic invariants of Artin stacks and ‘stack functions’, Q. J. Math. 58 (2007), no. 3, 345–392. MR 2354923, DOI 10.1093/qmath/ham019
Dominic Joyce, Configurations in abelian categories. IV. Invariants and changing stability conditions, Adv. Math. 217 (2008), no. 1, 125–204. MR 2357325, DOI 10.1016/j.aim.2007.06.011
D. Joyce and Y. Song. A theory of generalized Donaldson-Thomas invariants. math.AG/ 0810.5645.
Masaki Kashiwara, $t$-structures on the derived categories of holonomic $\scr D$-modules and coherent $\scr O$-modules, Mosc. Math. J. 4 (2004), no. 4, 847–868, 981 (English, with English and Russian summaries). MR 2124169, DOI 10.17323/1609-4514-2004-4-4-847-868
M. Kontsevich and Y. Soibelman. Stability structures, motivic Donaldson-Thomas invariants and cluster transformations. math.AG/0811.2435.
M. Levine and R. Pandharipande, Algebraic cobordism revisited, Invent. Math. 176 (2009), no. 1, 63–130. MR 2485880, DOI 10.1007/s00222-008-0160-8
Jun Li, Zero dimensional Donaldson-Thomas invariants of threefolds, Geom. Topol. 10 (2006), 2117–2171. MR 2284053, DOI 10.2140/gt.2006.10.2117
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, DOI 10.4310/CAG.2006.v14.n2.a6
Max Lieblich, Moduli of complexes on a proper morphism, J. Algebraic Geom. 15 (2006), no. 1, 175–206. MR 2177199, DOI 10.1090/S1056-3911-05-00418-2
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, DOI 10.1112/S0010437X06002302
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, DOI 10.1007/s00222-009-0203-9
J. Stoppa and R. P. Thomas. Hilbert schemes and stable pairs: GIT and derived category wall crossings. math.AG/0903.1444.
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, DOI 10.4310/jdg/1214341649
Y. Toda. Curve counting theories via stable objects II. DT/ncDT/flop formula. In preparation.
Y. Toda. Generating functions of stable pair invariants via wall-crossings in derived categories. math.AG/0806.0062.
Yukinobu Toda, Limit stable objects on Calabi-Yau 3-folds, Duke Math. J. 149 (2009), no. 1, 157–208. MR 2541209, DOI 10.1215/00127094-2009-038
Yukinobu Toda, Moduli stacks and invariants of semistable objects on $K3$ surfaces, Adv. Math. 217 (2008), no. 6, 2736–2781. MR 2397465, DOI 10.1016/j.aim.2007.11.010