This ebook reports 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 advanced 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 less than deformations of $X$, and remodel by way of a wall-crossing formulation lower than switch of balance $\tau$. To end up 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 $\mathfrak M$ might be written in the community as $\mat{Crit}(f)$ for $f:U\to{\mathbb C}$ holomorphic and $U$ soft, 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. in addition they expand the idea to abelian different types $\mat{mod}$-$\mathbb{C}Q\backslash I$ of representations of a quiver $Q$ with family $I$ coming from a superpotential $W$ on $Q$.

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