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Limit stable objects on Calabi-Yau 3-folds. (English) Zbl 1172.14007

In this paper, the author introduces the notion of limit stability condition and new enumerative invariants of curves on a Calabi-Yau 3-fold.
If \(X\) is a smooth Calabi-Yau 3-fold defined over \(\mathbb{C}\), R. Pandharipande and R. S. Thomas [Curve counting via stable pairs in the derived category, arXiv:0707.2348] define the \(PT\)-invariants for counting curves on \(X\). More precisely, one needs the notion of stable pair, which is a couple \((F,s)\), where \(F\) is a pure \(1\)-dimensional coherent sheaf on \(X\) and \(s\in \operatorname{Hom}(\mathcal{O}_{X},F)\) has \(0\)-dimensional cokernel. If \(\beta\in H^{4}(X,\mathbb{Z})\) and \(n\in\mathbb{Z}\), the moduli space \(P_{n}(X,\beta)\) of stable pairs \((F,s)\), where \(ch_{2}(F)=\beta\) and \(ch_{3}(F)=n\) is constructed as a projective variety, and the \(PT\)-invariant \(P_{n,\beta}\) is the integration over the virtual class of \(P_{n}(X,\beta)\).
In the paper, the author proposes new enumerative invariants of curves on \(X\) which are proven to be strictly related to the \(PT\)-invariants, but using stability conditions on the category of perverse sheaves on \(X\). Stability conditions are introduced by T. Bridgeland [Ann. Math. 166, No. 2, 317–345 (2007; Zbl 1137.18008)]: if \(\mathcal{D}\) is a triangulated category, a stability condition is a couple \((Z,\mathcal{A})\), where \(\mathcal{A}\) is an abelian category which is the heart of a \(t\)-structure on \(\mathcal{D}\) and \(Z:K(\mathcal{D})\longrightarrow\mathbb{C}\) is a group homomorphism satisfying two axioms: if \(E\neq 0\) and \(Z(E)=r(E)e^{i\pi\phi(E)}\), we need \(r(E)> 0\) and \(0<\phi(E)\leq\pi\); the second axiom is the existence of a Harder-Narasimhan filtration for every object. An object \(E\in\mathcal{A}\) is \(Z\)-stable if \(\phi(F)<\phi(E)\) for every \(F\subset E\). It is notoriusly difficult to produce stability conditions, and their existence is known only up to dimension 2.
The approach followed by the author is to define a limit stability condition: first, let \(\mathcal{A}^{p}\) be the heart of a perverse \(t\)-structure obtained as a tilting with respect to a well-chosen torsion pair (Definition 2.11). Then, choose \(\sigma=B+i\omega\in H^{2}(X,\mathbb{C})\) so that \(\omega\) is an ample class, and for every \(m\in\mathbb{Z}\) let \(\sigma_{m}:=B+im\omega\). For every \(E\in K(X)\) let \(Z_{\sigma_{m}}(E):=-\int_{X}exp(-\sigma_{m})ch(E)\sqrt(td(X))\). The couple \((Z_{\sigma_{m}},\mathcal{A}^{p})\) is not a stability condition, but for every object \(E\in\mathcal{A}^{p}\) there is \(m\gg 0\) such that \(0<\phi_{\sigma_{m}}(E)\leq\pi\) (here \(\phi_{\sigma_{m}}(E)\) is the phase of \(Z_{\sigma_{m}}(E)\)). Using this, the author defines an object \(E\in\mathcal{A}^{p}\) to be \(\sigma\)-limit stable if for every \(F\subset E\) we have \(\phi_{\sigma_{m}}(F)<\phi_{\sigma_{m}}(E)\) for \(m\gg 0\). The Harder-Narasimhan property (with respect to limit stability) is shown (Theorem 2.29). This notion of limit stability is included in the notion of polynomial stability introduced independently by A. Bayer [Geom. Topol. 13, No. 4, 2389–2425 (2009; Zbl 1171.14011)].
In Section 3 the author constructs the moduli space \(L_{n}^{\sigma}(X,\beta)\) of \(\sigma\)-limit stable objects in \(\mathcal{A}^{p}\) of rank \(-1\), trivial determinant, \(ch_{2}=\beta\in H^{4}(X,\mathbb{Z})\) and \(ch_{3}=n\in\mathbb{Z}\), as an algebraic space of finite type. This is elegantly obtained starting from the moduli space introduced by M. A. Inaba [J. Math. Kyoto Univ. 42, No. 2, 317–329 (2002; Zbl 1063.14013)], and it is one of the main results of the paper.
Section 4 consists of defining the new counting invariants as \(L_{n,\beta}(\sigma):=\sum_{p\in\mathbb{Z}}pe(\nu_{L}^{-1}(p))\), where \(\nu_{L}\) is the Behrend constructible function for \(L_{n}^{\sigma}(X,\beta)\) and \(e(.)\) is the Euler number. The other main result of the paper is about the relationship between \(L_{n,\beta}(\sigma)\) and the \(PT\)-invariant \(P_{n,\beta}\): more precisely, if \(k\in\mathbb{Z}\) and \(\sigma=k\omega+i\omega\), the author shows that \(L_{n}^{\sigma}(X,\beta)=P_{n}(X,\beta)\) and \(L_{n,\beta}(\sigma)=P_{n,\beta}\) if \(k\ll 0\), and that \(L_{n}^{\sigma}(X,\beta)=P_{-n}(X,\beta)\) and \(L_{n,\beta}(\sigma)=P_{-n,\beta}\) if \(k\gg 0\) (see Theorem 4.7 for precise bounds for \(k\)).
The conclusion of the paper deals with wall-crossing phenomena, aiming to the proof of the Pandharipande-Thomas conjecture about the rationality of the generating series of the \(PT\)-invariants. This conjecture is solved when \(\beta\) is the class of an irreducible curve by R. Pandharipande and R. P. Thomas [Stable pairs and BPS invariants, arXiv:0711.3899]. The author proposes a conjectural wall-crossing formula (Conjecture 4.11) to solve the Pandharipande-Thomas conjecture, and verifies that it holds true in three examples (described in Section 5). Further developments on this subjects are announced in another paper by the author [Generating functions of stable pair invariants via wall-crossings in derived categories, arXiv:0806.0062].

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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References:

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