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Virtual signed Euler characteristics

2016; American Mathematical Society; Volume: 26; Issue: 2 Linguagem: Inglês

10.1090/jag/690

1534-7486

Yunfeng Jiang, Richard P. Thomas,

Geometry and complex manifolds

Roughly speaking, to any space M with perfect obstruction theory we associate a space N with symmetric perfect obstruction theory.It is a cone over M given by the dual of the obstruction sheaf of M , and contains M as its zero section.It is locally the critical locus of a function.More precisely, in the language of derived algebraic geometry, to any quasi-smooth space M we associate its (-1)-shifted cotangent bundle N .By localising from N to its C * -fixed locus M this gives five notions of virtual signed Euler characteristic of M :(1) The Ciocan-Fontanine-Kapranov/Fantechi-Göttsche signed virtual Euler characteristic of M defined using its own obstruction theory, (2) Graber-Pandharipande's virtual Atiyah-Bott localisation of the virtual cycle of N to M , (3) Behrend's Kai-weighted Euler characteristic localisation of the virtual cycle of N to M , (4) Kiem-Li's cosection localisation of the virtual cycle of N to M , (5) (-1) vd times by the topological Euler characteristic of M .Our main result is that (1)=(2) and (3)=(4)=(5).The first two are deformation invariant while the last three are not.