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Generating functions of stable pair invariants via wall-crossings in derived categories. (English) Zbl 1216.14009

Saito, Masa-Hiko (ed.) et al., New developments in algebraic geometry, integrable systems and mirror symmetry. Papers based on the conference “New developments in algebraic geometry, integrable systems and mirror symmetry”, Kyoto, Japan, January 7–11, 2008, and the workshop “Quantum cohomology and mirror symmetry”, Kobe, Japan, January 4–5, 2008. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-62-4/hbk). Advanced Studies in Pure Mathematics 59, 389-434 (2010).
The purpose of the article under review is to interpret the so-called rationality conjecture for PT-theory as a wall-crossing formula in the bounded derived category of a smooth projective Calabi-Yau threefold \(X\).
Given such an \(X\) over \(\mathbb{C}\) there are several curve counting invariants which one can consider. The basic approach is very roughly the following: One considers an appropriate moduli problem, shows that it admits an obstruction theory, and integrates over the associated virtual class.
Firstly, there are Gromov-Witten invariants: Given a class \(\beta \in H^2(X,\mathbb{Z})\) and \(g\geq 0\), one considers the moduli stack of stable maps \(f: C\rightarrow X\), with \(g(C)=g\) and \(f_*[C]=\beta\), and obtains Gromov-Witten invariants \(N_{g,\beta}\).
Secondly, Donaldson-Thomas invariants \(I_{n,\beta}\) arise if one considers the Hilbert scheme of 1-dimensional subschemes \(Z\subset X\) satisfying \([Z]=\beta\) and \(\chi(\mathcal{O}_Z)=n\) for \(n\in \mathbb{Z}\). One can also consider the ideal sheaves associated to the subschemes.
Thirdly, R. Pandharipande and R. P. Thomas introduced the following notion in [Invent. Math. 178, No. 2, 407–447 (2009; Zbl 1204.14026)]: A stable pair consists of a pure one-dimensional sheaf \(F\) on \(X\) and a morphism \(s: \mathcal{O}_X\rightarrow F\) with zero-dimensional cokernel. If one fixes \(\beta\) as above and \(n\in \mathbb{Z}\), the moduli space \(P_n(X,\beta)\) of stable pairs satisfying \([F]=\beta\) and \(\chi(F)=n\) exists, and one then obtains the PT-invariants \(P_{n,\beta}\).
In all three cases one puts the invariants into a generating function and, conjecturally, these functions are equal after a suitable change of variables. This conjecture is based on the rationality conjecture: For a fixed \(\beta\), the generating series \(I_\beta(q)=\sum_{n\in \mathbb{Z}} I_{n,\beta}q^n/ \sum_{n\in\mathbb{Z}} I_{n,0}q^n\) and \(P_{\beta}(q)=\sum_{n\in \mathbb{Z}}P_{n,\beta}q^n\) should be Laurent expansions of rational functions of \(q\), invariant under \(q \leftrightarrow 1/q\).
Note that stable pairs and ideal sheaves can be interpreted as objects in the bounded derived category \({\text D}^{\text b}(X)\). It is believed that it is possible to interpret the equality of the associated generating series as wall-crossing formulas for counting invariants in \({\text D}^{\text b}(X)\).
In the present article a motivic version of the rationality conjecture for the PT-invariants is established. More precisely, consider \(P^{eu}_{n,\beta}\), the Euler characteristic of \(P_n(X,\beta)\), which (up to sign) coincides with \(P_{n,\beta}\) if \(P_n(X,\beta)\) is non-singular. Then the author proves that the rationality conjecture holds for the generating series \(P^{eu}_\beta(q)=\sum_{n\in \mathbb{Z}} P^{eu}_{n,\beta}q^n\).
The idea is to use the notion of limit stability introduced by the author in [Duke Math. J. 149, No. 1, 157–208 (2009; Zbl 1172.14007)] and D. Joyce’s work on counting invariants of semistable objects in abelian categories (see, in particular, [Adv. Math. 210, No. 2, 635–706 (2007; Zbl 1119.14005)] and [Adv. Math. 217, No. 1, 125–204 (2008; Zbl 1134.14008)]). In fact, limit stability (which is roughly a stability condition on a certain abelian category of perverse coherent sheaves) does not combine well with Joyce’s work, so the author introduces a coarser version of it to prove the main result.
For the entire collection see [Zbl 1200.14002].

MSC:

14D20 Algebraic moduli problems, moduli of vector bundles
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
18E30 Derived categories, triangulated categories (MSC2010)
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