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

By Dominic Joyce, Yinan Song

This publication 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 needs to be counted with complex rational weights. The $\bar{DT}{}^\alpha(\tau)$ are outlined for all periods $\alpha$, and are equivalent to $DT^\alpha(\tau)$ while it really is outlined. they're unchanged below deformations of $X$, and rework via a wall-crossing formulation less than swap of balance situation $\tau$. To turn out all this, the authors learn the neighborhood constitution of the moduli stack $\mathfrak M$ of coherent sheaves on $X$. They convey that an atlas for $\mathfrak M$ might be written in the community as $\mathrm{Crit}(f)$ for $f:U\to{\mathbb C}$ holomorphic and $U$ tender, 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 houses. additionally they expand the speculation 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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2 to study the Behrend function νM . However, as Behrend says [3, p. 5]: ‘We do not know if every scheme admitting a symmetric obstruction theory can locally be written as the critical locus of a regular function on a smooth scheme. 3). But here we sketch an alternative approach due to Behrend [3], which could perhaps be used to give a strictly algebraic proof of the same identities. 15. Let K be an algebraically closed field, and M a smooth Kscheme. Let ω be a 1-form on M , that is, ω ∈ H 0 (T ∗ M ).

2] proves 1 α1 ¯ (τ ) ∗ ¯α2 (τ ) ∗ · · · ∗ ¯αn (τ ). n! 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 ).

But any constructible subset S of X(K) can be covered by finitely many such subsets ϕ∗ (W (K)), so νX |S is constructible, and thus νX is locally constructible. 5. 3 also holds for Artin K-stacks X, Y locally of finite type. 2. Milnor fibres and vanishing cycles We define Milnor fibres for holomorphic functions on complex analytic spaces. 6. Let U be a complex analytic space, f : U → C a holomorphic function, and x ∈ U . Let d( , ) be a metric on U near x induced by a local embedding of U in some CN .

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