By Dominic Joyce, Yinan Song

This booklet reviews generalized Donaldson-Thomas invariants $\bar{DT}{}^\alpha(\tau)$. they're rational numbers which 'count' either $\tau$-stable and $\tau$-semistable coherent sheaves with Chern personality $\alpha$ on $X$; strictly $\tau$-semistable sheaves has to be counted with complex rational weights. The $\bar{DT}{}^\alpha(\tau)$ are outlined for all periods $\alpha$, and are equivalent to $DT^\alpha(\tau)$ whilst it's outlined. they're unchanged less than deformations of $X$, and remodel through a wall-crossing formulation lower than switch of balance situation $\tau$. To end up all this, the authors examine the neighborhood constitution of the moduli stack $\mathfrak M$ of coherent sheaves on $X$. They convey that an atlas for $\mathfrak M$ could be written in the neighborhood as $\mathrm{Crit}(f)$ for $f:U\to{\mathbb C}$ holomorphic and $U$ delicate, and use this to infer identities at the Behrend functionality $\nu_\mathfrak M$. They compute the invariants $\bar{DT}{}^\alpha(\tau)$ in examples, and make a conjecture approximately their integrality homes. additionally they expand the idea to abelian different types $\mathrm{mod}$-$\mathbb{C}Q\backslash I$ of representations of a quiver $Q$ with family members $I$ coming from a superpotential $W$ on $Q

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1] proves that given a ﬁnite type K-scheme X, there exists a unique cycle cX ∈ Z∗ (X), such that for any ´etale map ϕ : U → X for a Kscheme U and any closed embedding U → M into a smooth K-scheme M , we have ϕ∗ (cX ) = cU/M in Z∗ (U ). If X is a subscheme of a smooth M we take U = X and get cX = cX/M . Behrend calls cX the signed support of the intrinsic normal cone, or the distinguished cycle of X. Write CFZ (X) for the group of Z-valued constructible functions on X. The local Euler obstruction is a group isomorphism Eu : Z∗ (X) → CFZ (X).

5), because as the family of τ -semistable sheaves in class α is bounded, there are only ﬁnitely ways to write α = α1 + · · · + αn with τ -semistable sheaves in class αi for all i. 7) α∈C(A):τ (α)=t where δ¯0 is the identity 1 in SFal (MA ). For α ∈ C(A) and t = τ (α), using the n−1 1 n xn and exp(x) = 1 + n 1 n! 7) to MA . 5) are inverse, since log and exp are inverse. Thus, knowing the ¯α (τ ) α (τ ). is equivalent to knowing the δ¯ss α α α (τ ). The diﬀerence between ¯α (τ ) and If Mss (τ ) = Mst (τ ) then ¯α (τ ) = δ¯ss α α ¯ δss (τ ) is that ¯ (τ ) ‘counts’ strictly semistable sheaves in a special, complicated way.

A) If K is an algebraically closed ﬁeld of characteristic zero, and X is a ﬁnite type K-scheme, then the Behrend function νX is a well-deﬁned Z-valued constructible function on X, in the Zariski topology. (b) If Y is a complex analytic space then the Behrend function νY is a well-deﬁned Z-valued locally constructible function on Y, in the analytic topology. (c) If X is a ﬁnite type C-scheme, with underlying complex analytic space Xan , then the algebraic Behrend function νX in (a) and the analytic Behrend function νXan in (b) coincide.