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Polynomial Bridgeland stability conditions and the large volume limit. (English) Zbl 1171.14011

The author introduces the notion of polynomial stability conditions on triangulated categories, which generalize Bridgeland’s stability conditions. This is done by allowing the central charge to have values in complex-valued polynomials instead of complex numbers.
The main result is to describe a family of polynomial stability conditions on any normal projective variety. Remark that only few examples of Bridgeland stability conditions are so far described, almost always in dimension \(\leq 2\). The family of polynomial stability condition described here is associated to a bounded t-structure of perverse coherent sheaves. Such family contains stability conditions corresponding to Simpson stability and stability conditions that should correspond to the large volume limit of Bridgeland stability conditions. In the case of surfaces, the author describes precisely the polynomial stability arising as limit of Bridgeland ones. He finally remarks how polynomial stability conditions could be helpful in the description of Bridgeland ones, since they give rise to new t-structures. As a nice geometrical application, the author shows that a normal projective variety \(X\) can be reconstructed, up to isomorphism, by its bounded derived category, a polynomial stability condition and the class of the structure sheaf of a point \(O_x \in N(X)\) in the numerical Grothendieck group. At the end of the paper, the author introduces a natural topology on the space of locally finite polynomial stability conditions, which is compatible with the furgetful map. Under some finiteness assumption, he shows that the map is a local homeomorphism (which is always the case for locally finite Bridgeland stability conditions).
The main application is an intepretation, in terms of wall crossing phenomena in the family of polynomial stablity conditions, of the PT/DT correspondence, conjectured in [R. Pandharipande, R. Thomas, Curve counting via stable pairs in the derived category, arxiv:0707.2348], and of the relation between stable pairs and one dimensional torsion free sheaves on a Calabi-Yau threefold described in [R. Pandharipande, R. Thomas, Stable pairs and BPS invariants, arxiv:0711.3899].
The paper is clearly written and also contains some figures to support the text, as for examples in the description of the PT/DT correspondence in terms of wall crossings.

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
18E30 Derived categories, triangulated categories (MSC2010)
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14D20 Algebraic moduli problems, moduli of vector bundles
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
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References:

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