Theory of Generalized Donaldson-Thomas Invariants by Dominic Joyce (PDF)

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This book studies generalized Donaldson-Thomas invariants $ ar{DT}{}^ lpha( au)$. They are rational numbers which `count’ both $ au$-stable and $ au$-semistable coherent sheaves with Chern character $ lpha$ on $X$; strictly $ au$-semistable sheaves must be counted with complicated rational weights. The $ ar{DT}{}^ lpha( au)$ are defined for all classes $ lpha$, and are equal to $DT^ lpha( au)$ when it is defined. They are unchanged under deformations of $X$, and transform by a wall-crossing formula under change of stability condition $ au$. 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 $f:U o{mathbb C}$ holomorphic and $U$ smooth, and use this to deduce identities on the Behrend function $ u_mathfrak M$. They compute the invariants $ ar{DT}{}^ lpha( au)$ in examples, and make a conjecture about their integrality properties. They also extend the theory to abelian categories $mathrm{mod}$-$mathbb{C}Q ackslash I$ of representations of a quiver $Q$ with relations $I$ coming from a superpotential $W$ on $Q$.

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