Duke Mathematical Journal

The space of stability conditions on the local projective plane

Arend Bayer and Emanuele Macrì

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Abstract

We study the space of stability conditions on the total space of the canonical bundle over the projective plane. We explicitly describe a chamber of geometric stability conditions and show that its translates via autoequivalences cover a whole connected component. We prove that this connected component is simply connected. We determine the group of autoequivalences preserving this connected component, which turns out to be closely related to Γ1(3).

Finally, we show that there is a submanifold isomorphic to the universal covering of a moduli space of elliptic curves with Γ1(3)-level structure. The morphism is Γ1(3)-equivariant and is given by solutions of Picard-Fuchs equations. This result is motivated by the notion of Π-stability and by mirror symmetry.

Article information

Source
Duke Math. J., Volume 160, Number 2 (2011), 263-322.

Dates
First available in Project Euclid: 27 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1319721313

Digital Object Identifier
doi:10.1215/00127094-1444249

Mathematical Reviews number (MathSciNet)
MR2852118

Zentralblatt MATH identifier
1238.14014

Subjects
Primary: 14F05: Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20]
Secondary: 14J32: Calabi-Yau manifolds 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45] 18E30: Derived categories, triangulated categories

Citation

Bayer, Arend; Macrì, Emanuele. The space of stability conditions on the local projective plane. Duke Math. J. 160 (2011), no. 2, 263--322. doi:10.1215/00127094-1444249. https://projecteuclid.org/euclid.dmj/1319721313


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