Symmetric Semi-perfect Obstruction Theory Revisited

In this paper we survey some results on the symmetric semi-perfect obstruction theory on a Deligne-Mumford stack X constructed by Chang-Li,and Behrend’s theorem equating the weighted Euler characteristic of X and the virtual count of X by symmetric semi-perfect obstruction theories.As an application,we prove that Joyce’s d-critical scheme admits a symmetric semi-perfect obstruction theory,which can be applied to the virtual Euler characteristics by Jiang-Thomas.

Motivic Milnor Fibre of Cyclic L∞-algebras

Abstract We define the motivic Milnor fiber of cyclic L∞-algebras of dimension three using the method of Denef and Loeser of motivic integration. It is proved by Nicaise and Sebag that the topo- logical Euler characteristic of the motivic Milnor fiber is equal to the Euler characteristic of the étale cohomology of the analytic Milnor fiber. We prove that the value of Behrend function on the germ moduli space determined by a cyclic L∞-algebra L is equal to the Euler characteristic of the analytic Milnor fiber. Thus we prove that the Behrend function depends only on the formal neighborhood of the moduli space.

The Chen-Ruan Cohomology of Almost Contact Orbifolds

Comparing to the Ch-~Ruan cohomology theory for the almost complex orbifolds, we study the orbifold cohomology theory for almost contact orbifolds. We define the Chen-Ruan cohomology group of any almost contact orbifold. Using the methods for almost complex orbifolds, we define the obstruction bundle for any 3-multisector of the almost contact orbifolds and the Chen^Ruan cup product for the Che-Ruan cohomology. We also prove that under this cup product the direct sum of all dimensional orbifold cohomology groups constitutes a cohomological ring. Finally we calculate two examples.

The Moduli Space of Stable Coherent Sheaves via Non-archimedean Geometry

We provide a construction of the moduli space of stable coherent sheaves in the world of non-archimedean geometry,where we use the notion of Berkovich non-archimedean analytic spaces.The motivation for our construction is Tony Yue Yu’s non-archimedean enumerative geometry in Gromov-Witten theory.The construction of the moduli space of stable sheaves using Berkovich analytic spaces will give rise to the non-archimedean version of Donaldson-Thomas invariants.In this paper we give the moduli construction over a non-archimedean field K.We use the machinery of formal schemes,that is,we define and construct the formal moduli stack of(semi)-stable coherent sheaves over a discrete valuation ring R,and taking generic fiber we get the non-archimedean analytic moduli space of semistable coherent sheaves over the fractional non-archimedean field K.We generalize Joyce’s dcritical scheme structure in[37]or Kiem-Li’s virtual critical manifolds in[38]to the world of formal schemes,and Berkovich non-archimedean analytic spaces.As an application,we provide a proof for the motivic localization formula for a d-critical non-archimedean K-analytic space using global motive of vanishing cycles and motivic integration on oriented formal d-critical schemes.This generalizes Maulik’s motivic localization formula for the motivic Donaldson-Thomas invariants.