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Wall crossing for derived categories of moduli spaces of sheaves on rational surfaces. (English) Zbl 1405.14110

Variation of stability for a moduli problem often leads to birational moduli spaces. It is then natural to study the derived categories. In this paper, the author works in the situation of torsion-free sheaves of rank \(2\) on a rationsl surface and obtains a result which can be regarded as a categorification of the wall-crossing formula for Donaldson invariants. More precisely, let \(S\) be a smooth rational surface over \({\mathbb C}\) with \(K_S<0\) and \(L_-\), \(L_+\) ample line bundles on \(S\) separated by a single wall defined by a divisor \(\xi\) such that \(L_-\cdot\xi<0<L_+\cdot\xi\) and \(0\leq\omega_S^{-1}\cdot\xi\). Let \({\mathcal M}_{L_\pm}(\Delta,c)\) be the \({\mathbb G}_m\)-rigidified moduli stacks of Gieseker \(L_\pm\)-semi-stable torsion-free sheaves of rank \(2\) with Chern classes \(c_1\), \(c_2\). Then (Theorem 1) there is a semi-orthogonal decomposition of \(\text{D}^b(\text{coh}{\mathcal M}_{L_+}(\Delta,c))\) as \[ \left\langle\text{D}^b(\text{coh}H^{l_\xi}),\cdots,\text{D}^b(\text{coh}H^{l_\xi}),\cdots,\text{D}^b(\text{coh}H^0),\cdots,\text{D}^b(\text{coh}H^0),\text{D}^b(\text{coh}{\mathcal M}_{L_-}(\Delta,c))\right\rangle, \] where \(l_\xi:=(4c_2-c_1^2+\xi^2)/4\), \(H^l:=\text{Hilb}^l(S)\times\text{Hilb}^{l_\xi-l}(S)\) and each \(\text{D}^b(\text{coh}H^l)\) is repeated \(\mu_\xi:=\omega_S^{-1}\cdot\xi\) times, with the convention that \(\text{Hilb}^0(S):=\text{Spec}{\mathbb C}\).
As well as Theorem 1, the method of proof is of interest and has other potential applications. The method comes under the heading of “windows”, which are a machine for constructing semi-orthogonal decompositions of \(\text{D}^b(\text{coh}X)\). Previous constructions involved an Artin stack \({\mathcal X}\) with a global quotient presentation \({\mathcal X}=[X/G]\). The author provides a definition of window which avoids the use of a quotient presentation and is more intrinsic to \({\mathcal X}\). This definition (Definition 2.1) uses a type of groupoid in Bialynicki-Birula strata. Roughly half of the paper (section 2) is taken up with a description of the method, culminating in Theorem 2.29, which describes a semi-orthogonal decomposition associated to an elementary wall-crossing.

MSC:

14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J26 Rational and ruled surfaces
18E30 Derived categories, triangulated categories (MSC2010)
14D20 Algebraic moduli problems, moduli of vector bundles
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