## A Theory of Generalized Donaldson-Thomas Invariants (Memoirs by Dominic Joyce, Yinan Song

By Dominic Joyce, Yinan Song

This e-book stories 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 needs 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 truly is outlined. they're unchanged below deformations of $X$, and rework by way of a wall-crossing formulation lower than swap of balance situation $\tau$. To turn out all this, the authors research 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. in addition 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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In ); τ, τ˜). 26) 2 n! orderings i1 , . . , in of I: ia ib edge • → • in Γ implies a < b Then as in [54, Th. 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.

4. SF k [(U × [Spec K/Gm ], ρ)] with algebra stabilizers, for U a quasiprojective K-variety ¯ ind (MA , χ, Q) is spanned over Q by [(U × [Spec K/Gm ], ρ)] with and k 0. Also SF al algebra stabilizers, for U a quasiprojective K-variety. 2. (Weak) stability conditions on A Next we discuss material in [53] on stability conditions. 5. 2). Suppose (T, ) is a totally ordered set, and τ : C(A) → T a map. We call (τ, T, ) a stability condition on A if whenever α, β, γ ∈ C(A) with β = α + γ then either τ (α) < τ (β) < τ (γ), or τ (α) > τ (β) > τ (γ), or τ (α) = τ (β) = τ (γ).

Let K be an algebraically closed ﬁeld of characteristic zero, and X be a K-scheme, algebraic K-space, or Artin K-stack, locally of ﬁnite type. Then there is a well-deﬁned Behrend function νX , a Z-valued locally constructible function on X, which is characterized uniquely by the property that if W is a ﬁnite type K-scheme and ϕ : W → X is a 1-morphism of Artin stacks that is smooth of relative dimension n then ϕ∗ (νX ) = (−1)n νW in CF(W ). Proof. As Artin K-stacks include K-schemes and algebraic K-spaces, it is enough to do the Artin stack case.