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Polynomial Bridgeland stability conditions for the derived category of sheaves on surfaces. (English) Zbl 1231.14012

A Bridgeland stability condition on a triangulated category \(\mathcal{T}\) is given, roughly speaking, by the heart \(\mathcal{A}\) of a bounded t-structure on \(\mathcal{T}\) and a homomorphism from the Grothendieck group of \(\mathcal{A}\) to the complex numbers which satisfies some conditions. Given a smooth projective variety \(Y\), examples of stability conditions on the bounded derived category of sheaves on \(Y\) have been constructed in several cases, but the existence of a stability condition on an arbitrary variety is an open question.
Polynomial Bridgeland stability conditions (the main idea is to substitute \(\mathbb{C}\) in the above construction by \(\mathbb{C}[m]\)) were later defined by A. Bayer in [Geom.Topol.13, No. 4, 2389–2425 (2009; Zbl 1171.14011)]. They are related to the “large volume limits” of Bridgeland stability conditions and have the advantage that they exist on the bounded derived category over any normal projective variety.
The purpose of the paper under review is to study polynomial Bridgeland stability conditions for a smooth projective surface \(X\). Starting with some data, in particular an ample divisor \(L\) on X, there exists a polynomial Bridgeland stability condition \((Z_{\Omega_L}, \mathcal{P}_{\Omega_L})\). The main results can be roughly summarised as follows. Firstly, under some numerical assumptions on the above data, the Bogomolov inequality \(2rc_2\geq (r-1)c_1^2\) holds for any \(Z_{\Omega_L}\)-semistable object \(E\) (in the bounded derived category of \(X\)) with \(\text{rk}(E)=r\), \(c_1(E)=c_1\) and \(c_2(E)=c_2\). Secondly, the semistability of an object remains the same when the ample divisor varies within a chamber in the Kähler cone. Lastly, the moduli space of semistable objects \(\mathcal{M}_{\Omega_L}(r,c_1,c_2)\) of type \((r,c_1,c_2)\) can be identified, again under some numerical assumptions on the above stability data, with the moduli space of torsion free sheaves which are of the same type and are Gieseker-semistable with respect to \(L\). This last statement has to be read as an equality of sets, since it is not known in general whether \(\mathcal{M}_{\Omega_L}(r,c_1,c_2)\) exists as a scheme.
The paper is organized as follows. After recalling the main definitions from Bayer’s article mentioned above in section 2 and devoting section 3 to a generalization of basic definitions and results concerning walls, chambers and variations of \(\mu\)-stability as the polarization changes, the authors prove their main results in section 4.

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli

Citations:

Zbl 1171.14011
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