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Perverse coherent sheaves on blowup. III: Blow-up formula from wall-crossing. (English) Zbl 1220.14012

In their previous work [“Perverse coherent sheaves on blow-up. I: A quiver description”, arXiv:0802.3120] and [J. Algebr. Geom. 20, No. 1, 47–100 (2011; Zbl 1208.32013)], the authors have constructed a zig-zag of birational morphisms between the moduli spaces of stable torsion-free sheaves on smooth projective surfaces (over \(\mathbb{C}\)) and their blow-ups. This can be understood as a variation of GIT quotients [M. Thaddeus, J. Am. Math. Soc. 9, No. 3, 691–723 (1996; Zbl 0874.14042)]. Here, it allows them to closely follow T. Mochizuki’s arguments in [Lect. Notes Math. 1972. Berlin: Springer (2009; Zbl 1177.14003)] to compare Donaldson-type invariants. The authors only deal with the case \(\mathbb{P}^2\) and moduli of framed sheaves, but they conjecture that their result is universal. The algorithm itself is difficult, and no explicit formula is written down, but the authors succeed to deduce vanishing results. This allows them to give applications to Nekrasov-type partition functions for theories with \(5\)-dimensional Chern-Simons terms and Casimir operators.

MSC:

14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
16G20 Representations of quivers and partially ordered sets
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