By Dominic Joyce, Yinan Song

ISBN-10: 0821852795

ISBN-13: 9780821852798

This booklet experiences 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 sessions $\alpha$, and are equivalent to $DT^\alpha(\tau)$ whilst it's outlined. they're unchanged less than deformations of $X$, and remodel via a wall-crossing formulation below 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 exhibit that an atlas for $\mathfrak M$ might 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. in addition they expand the idea to abelian different types $\mathrm{mod}$-$\mathbb{C}Q\backslash I$ of representations of a quiver $Q$ with kinfolk $I$ coming from a superpotential $W$ on $Q

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**Example text**

N − 1 we have either (a) τ (αi ) τ (αi+1 ) and τ˜(α1 + · · · + αi ) > τ˜(αi+1 + · · · + αn ) or (b) τ (αi ) > τ (αi+1 ) and τ˜(α1 + · · · + αi ) τ˜(αi+1 + · · · + αn ), then deﬁne S(α1 , . . , αn ; τ, τ˜) = (−1)r , where r is the number of i = 1, . . , n − 1 satisfying (a). Otherwise deﬁne S(α1 , . . , αn ; τ, τ˜) = 0. Now deﬁne U (α1 , . . 8) l−1 · l i=1 S(βbi−1 +1 , βbi−1 +2 , . . , βbi ; τ, τ˜) 1 l m n, 0=a0

25) yields a transformation law for the J α (τ ) under change of stability condition: J α (˜ τ)= V (I, Γ, κ; τ, τ˜) · iso. 27) (τ ). As in [54, Rem. 29], V (I, Γ, κ; τ, τ˜) depends on the orientation of Γ only up to sign: changing the directions of k edges multiplies V (I, Γ, κ; τ, τ˜) by (−1)k . 27) is χ ¯ is antisymmetric, it follows that V (I, Γ, κ; τ, τ˜) · •→ • χ(κ(i), independent of the orientation of Γ. CHAPTER 4 Behrend functions and Donaldson–Thomas theory We now discuss Behrend functions of schemes and stacks, and their application to Donaldson–Thomas invariants.

M, ai−1 < j and τ˜(γi ) = τ˜(α1 + · · · + αn ), i = 1, . . , l m · 1 . (a − ai−1 )! 13. 2 hold, and (τ, T, ), (˜ τ , T˜ , ), (ˆ τ , Tˆ , ) be permissible weak stability conditions on A with (ˆ τ , Tˆ , ) dominating (τ, T, ) and (˜ τ , T˜, ). Then for all α ∈ C(A) we have α δ¯ss (˜ τ) = S(α1 , . . ,αn ∈C(A): α1 +···+αn =α ¯α (˜ τ) = ss ss U (α1 , . . 10). Here the third stability condition (ˆ τ , Tˆ, ) may be thought of as lying on a ‘wall’ ˜ separating (τ, T, ) and (˜ τ , T , ) in the space of stability conditions.

### A theory of generalized Donaldson-Thomas invariants by Dominic Joyce, Yinan Song

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