This book studies generalized Donaldson-Thomas invariants$/bar{DT}{}^/alpha(/tau)$. They are rational numbers which ‚count‘ both $/tau$-stable and $/tau$-semistable coherent sheaves with Chern character $/alpha$ on $X$; strictly $/tau$-semistable sheaves must be counted with complicated rational weights. The $/bar{DT}{}^/alpha(/tau)$ are defined for all classes $/alpha$, and are equal to $DT^/alpha(/tau)$ when it is defined. They are unchanged under deformations of $X$, and transform by a wall-crossing formula under change of stability condition $/tau$. To prove all this, the authors study the local structure of the moduli stack $/mathfrak M$ of coherent sheaves on $X$. They show that an atlas for $/mathfrak M$ may be written locally as $/mathrm{Crit}(f)$ for $f:U/to{/mathbb C}$ holomorphic and $U$ smooth, and use this to deduce identities on the Behrend function $/nu_/mathfrak M$. They compute the invariants $/bar{DT}{}^/alpha(/tau)$ in examples, and make a conjecture about their integrality properties. They also extend the theory to abelian categories $/mathrm{mod}$-$/mathbb{C}Q/backslash I$ of representations of a quiver $Q$ with relations $I$ coming from a superpotential $W$ on $Q$.
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