×

Curve counting theories via stable objects. I: DT/PT correspondence. (English) Zbl 1207.14020

The author studies the DT/PT correspondence via stable objects in a triangulated category wih respect to a weak stability condition. If \(X\) is a complex Calabi-Yau 3-fold, \(n\in\mathbb{Z}\) and \(\beta\in H_{2}(X,\mathbb{Z})\), the DT-invariants of \(X\) are defined as \(I_{n,\beta}:=\int_{[I_{n}(X,\beta)]^{\mathrm{vir}}}1\in\mathbb{Z}\), where \(I_{n}(X,\beta)\) is the moduli space of torsion coherent sheaves \(\mathcal{I}\) of rank 1 with trivial determinant and \(\mathrm{ch}(\mathcal{I})=(1,0,-\beta,-n)\). Their generating series is \(DT(X):=\sum_{n,\beta}I_{n,\beta}x^{n}y^{\beta}\). Letting \(DT_{0}(X):=\sum_{n}I_{n,0}x^{n}\), define \(DT'(X):=DT(X)/DT_{0}(X)=\sum_{\beta}DT'_{\beta}(X)y^{\beta}\). The PT-invariants are defined as \(P_{n,\beta}:=\int_{[P_{n}(X,\beta)]^{\mathrm{vir}}}1\in\mathbb{Z}\), where \(P_{n}(X,\beta)\) is the moduli space of stable pairs \((F,s)\), where \(F\) is a pure \(1-\)dimensional coherent sheaf such that \([F]=\beta\) and \(\chi(F)=n\). Their generating series is \(PT(X):=\sum_{n,\beta}P_{n,\beta}x^{n}y^{\beta}\). The PT-Conjecture states that \(DT'(X)=PT(X)\) [see R. Pandharipande and R. Thomas, Invent. Math. 178, No. 2, 407–447 (2009; Zbl 1204.14026)]).
Let \(\widehat{DT}(X):=\sum_{n,\beta}\chi(I_{n}(X,\beta))x^{n}y^{\beta}\), where \(\chi(.)\) is the Euler characteristic, and similarily define \(\widehat{DT}_{0}(X)\), \(\widehat{DT}'(X)\) and \(\widehat{PT}(X)\): these are all related to \(DT(X)\), \(DT_{0}(X)\), \(DT'(X)\), \(PT(X)\). The main result of the paper is that \(\widehat{DT}'(X)=\widehat{PT}(X)\). Using a result of Y. Toda [Generating functions of stable pair invariants via wall-crossing in derived categories, arXiv:0806.0062], the author proves that \(\widehat{DT}'_{\beta}(X)\) is the Laurent expansion of a rational function in \(x\) which is invariant under changing \(x\) with \(1/x\), showing a conjecture of W.-P. Li and Z. Qin [Commun. Anal. Geom. 14, No. 2, 387–410 (2006; Zbl 1111.14002)].
The method used to prove this result is based on stable objects with respect to a weak stability condition on a triangulated category \(\mathcal{D}\), with respect to a finitely generated free abelian group \(\Gamma\) with an increasing filtration \(\Gamma_{\bullet}:=\{\Gamma_{i}\}_{i=0}^{N}\) such that \(\Gamma_{i}/\Gamma_{i-1}\) is a free abelian group for every \(i\). The notion of weak stability condition the author proposes is a generalization of stability conditions of Bridgeland. An advantage of this generalization is that even if no example of Bridgeland’s stability conditions on a Calabi Yau 3-fold \(X\) is known, it is easy to produce examples of weak stability conditions on \(X\). Moreover, the set \(\mathrm{Stab}_{\Gamma_{\bullet}}(\mathcal{D})\) is easily shown to be a topological space, and each of its connected components is a complex manifold.
A weak stability condition can be defined using the heart \(\mathcal{A}\) of a bounded t-structure on \(\mathcal{D}\), and a weak stability function \(Z\). Using this formulation one can define easily the notion of \(Z-\)(semi)stable object in \(\mathcal{A}\). The notion of weak stability condition is strictly related to the polynomial stability conditions introduced by A. Bayer [Geom. Topol. 13, No. 4, 2389–2425 (2009; Zbl 1171.14011)] (the relation being similar to the one between \(\mu-\)stability and Gieseker-Maruyama stability on a smooth projective variety).
In section 3 the author presents the proof the main theorem: first, he defines a bounded t-structure, whose heart is denoted \(\mathcal{A}_{X}\), on a triangulated subcategory \(\mathcal{D}_{X}\) of \(D^{b}(\mathrm{Coh}(X))\). Then let \(N_{1}(X)\) be the group of numerical classes of curves on \(X\), and let \(\Gamma:=\mathbb{Z}\oplus N_{1}(X)\oplus\mathbb{Z}\), with filtration given by \(\Gamma_{0}:=\mathbb{Z}\), \(\Gamma_{1}:=\mathbb{Z}\oplus N_{1}(X)\) and \(\Gamma_{2}:=\Gamma\). Let \(\mathfrak{h}\) be the set of those complex numbers of the form \(re^{i\pi\phi}\), where \(r>0\) and \(0<\phi\leq 1\). Fix \(z_{0},z_{1}\in\mathfrak{h}\) and \(\omega\) an ample line bundle on \(X\). Let \(\xi:=(-z_{0},-i\omega,z_{1})\in\Gamma\otimes\mathbb{C}\): using \(\xi\) the author defines a weak stability function \(Z_{\xi}\), so that one has a weak stability condition \(\sigma_{\xi}:=(Z_{\xi},\mathcal{A}_{X})\). The set of these stability conditions is denoted \(\mathcal{V}_{X}\), and it is shown to verify a technical condition (Assumption 4.1). If \(v=(-n,-\beta,1)\in\Gamma\), then the moduli stack \(M^{v}(\sigma_{\xi})\) of \(\sigma_{\xi}-\)semistable objects \(E\) in \(\mathcal{A}_{X}\) such that \(cl(E)=v\) is shown to be either \([I_{n}(X,\beta)/\mathbb{G}_{m}]\) (if \(\mathrm{arg}(z_{0})<\mathrm{arg}(z_{1})\)) or \([P_{n}(X,\beta)/\mathbb{G}_{m}]\) (if \(\mathrm{arg}(z_{0})>\mathrm{arg}(z_{1})\)), where in both cases \(\mathbb{G}_{m}\) acts trivially. This, together with some general results shown in sections 4 and 5, is one of the key points on the proof of the main theorem.
In section 4 the author shows that there are invariants \(\widehat{DT}_{n,\beta}(\sigma)\) related to the Joyce invariant introduced in D. Joyce [Adv. Math. 217, No. 1, 125–204 (2008; Zbl 1134.14008)], and here recalled by the author. In particular, if \(M^{v}(\sigma)=[M/\mathbb{G}_{m}]\), where \(M\) is a scheme on which \(\mathbb{G}_{m}\) acts trivially, we have that \(\widehat{DT}_{n,\beta}(\sigma)=\chi(M)\). In section 5 the author recalls a wall-crossing formula of D. Joyce [Zbl 1134.14008], which is used to show that the invariants \(\widehat{DT}_{n,\beta}(\sigma)\) do not depend on \(\sigma\), whenever \(\sigma\) is chosen to be generic in \(\mathcal{V}_{X}\). The last sections of the paper are devoted to prove technical results.
The author points out some recent developments about the PT-Conjecture: first, using some conjectural results of M. Kontsevitch and Y. Soibelman [Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:math.AG/0605196], the same method of the present paper should give a proof of the PT-Conjecture. This was proved by T. Bridgeland in [Hall algebras and curve-counting invariants, preprint] using a different method. Another possible proof of the PT-Conjecture would be given by the same method of the present paper, once the results of D. Joyce and Y. Song [A theory of generalized Donaldson-Thomas invariants, arXiv:0810.5645] were extended to derived categories. The required elements seem to be solved by K. Behrend and E. Getzler in a paper in preparation. Finally, an independent proof of the main theorem of the present paper is given by J. Stoppa and R. Thomas [Hilbert schemes and stable pairs: GIT and derived category wall crossing, arXiv:0903.1444].

MSC:

14D20 Algebraic moduli problems, moduli of vector bundles
18E30 Derived categories, triangulated categories (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[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. · Zbl 1191.14050
[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. · Zbl 1170.14004
[4] K. Behrend and E. Getzler. Chern-Simons functional. In preparation.
[5] R. Bezrukavnikov. Perverse coherent sheaves (after Deligne). math.AG/0005152. · Zbl 1205.18010
[6] T. Bridgeland. Hall algebras and curve-counting invariants. Preprint. http://www. tombridgeland.staff.shef.ac.uk/papers/dtpt.pdf. · Zbl 1234.14039
[7] Tom Bridgeland, Stability conditions on triangulated categories, Ann. of Math. (2) 166 (2007), no. 2, 317 – 345. · Zbl 1137.18008
[8] Tom Bridgeland, Stability conditions on \?3 surfaces, Duke Math. J. 141 (2008), no. 2, 241 – 291. · Zbl 1138.14022
[9] Jan Cheah, On the cohomology of Hilbert schemes of points, J. Algebraic Geom. 5 (1996), no. 3, 479 – 511. · Zbl 0889.14001
[10] Michel Van den Bergh, Three-dimensional flops and noncommutative rings, Duke Math. J. 122 (2004), no. 3, 423 – 455. · Zbl 1074.14013
[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. · Zbl 0849.16011
[12] Dominic Joyce, Configurations in abelian categories. I. Basic properties and moduli stacks, Adv. Math. 203 (2006), no. 1, 194 – 255. · Zbl 1102.14009
[13] Dominic Joyce, Configurations in abelian categories. II. Ringel-Hall algebras, Adv. Math. 210 (2007), no. 2, 635 – 706. · Zbl 1119.14005
[14] Dominic Joyce, Configurations in abelian categories. III. Stability conditions and identities, Adv. Math. 215 (2007), no. 1, 153 – 219. , https://doi.org/10.1016/j.aim.2007.04.002 Dominic Joyce, Configurations in abelian categories. IV. Invariants and changing stability conditions, Adv. Math. 217 (2008), no. 1, 125 – 204. · Zbl 1134.14008
[15] Dominic Joyce, Motivic invariants of Artin stacks and ’stack functions’, Q. J. Math. 58 (2007), no. 3, 345 – 392. · Zbl 1131.14005
[16] Dominic Joyce, Configurations in abelian categories. III. Stability conditions and identities, Adv. Math. 215 (2007), no. 1, 153 – 219. , https://doi.org/10.1016/j.aim.2007.04.002 Dominic Joyce, Configurations in abelian categories. IV. Invariants and changing stability conditions, Adv. Math. 217 (2008), no. 1, 125 – 204. · Zbl 1134.14008
[17] D. Joyce and Y. Song. A theory of generalized Donaldson-Thomas invariants. math.AG/ 0810.5645. · Zbl 1259.14054
[18] Masaki Kashiwara, \?-structures on the derived categories of holonomic \?-modules and coherent \?-modules, Mosc. Math. J. 4 (2004), no. 4, 847 – 868, 981 (English, with English and Russian summaries). · Zbl 1073.14023
[19] M. Kontsevich and Y. Soibelman. Stability structures, motivic Donaldson-Thomas invariants and cluster transformations. math.AG/0811.2435. · Zbl 1248.14060
[20] M. Levine and R. Pandharipande, Algebraic cobordism revisited, Invent. Math. 176 (2009), no. 1, 63 – 130. · Zbl 1210.14025
[21] Jun Li, Zero dimensional Donaldson-Thomas invariants of threefolds, Geom. Topol. 10 (2006), 2117 – 2171. · Zbl 1140.14012
[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. · Zbl 1111.14002
[23] Max Lieblich, Moduli of complexes on a proper morphism, J. Algebraic Geom. 15 (2006), no. 1, 175 – 206. · Zbl 1085.14015
[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. · Zbl 1108.14046
[25] R. Pandharipande and R. P. Thomas, Curve counting via stable pairs in the derived category, Invent. Math. 178 (2009), no. 2, 407 – 447. · Zbl 1204.14026
[26] J. Stoppa and R. P. Thomas. Hilbert schemes and stable pairs: GIT and derived category wall crossings. math.AG/0903.1444. · Zbl 1243.14009
[27] R. P. Thomas, A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on \?3 fibrations, J. Differential Geom. 54 (2000), no. 2, 367 – 438. · Zbl 1034.14015
[28] Y. Toda. Curve counting theories via stable objects II. DT/ncDT/flop formula. In preparation. · Zbl 1267.14049
[29] Y. Toda. Generating functions of stable pair invariants via wall-crossings in derived categories. math.AG/0806.0062. · Zbl 1216.14009
[30] Yukinobu Toda, Limit stable objects on Calabi-Yau 3-folds, Duke Math. J. 149 (2009), no. 1, 157 – 208. · Zbl 1172.14007
[31] Yukinobu Toda, Moduli stacks and invariants of semistable objects on \?3 surfaces, Adv. Math. 217 (2008), no. 6, 2736 – 2781. · Zbl 1136.14007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.