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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.

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.

Reviewer: P. E. Newstead (Liverpool)

### 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 |