By Dominic Joyce, Yinan Song

ISBN-10: 0821852795

ISBN-13: 9780821852798

This publication 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 complex rational weights. The $\bar{DT}{}^\alpha(\tau)$ are outlined for all sessions $\alpha$, and are equivalent to $DT^\alpha(\tau)$ whilst it truly is outlined. they're unchanged lower than deformations of $X$, and remodel by means of a wall-crossing formulation below switch 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 exhibit that an atlas for $\mathfrak M$ might be written in the neighborhood as $\mathrm{Crit}(f)$ 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 $\mathrm{mod}$-$\mathbb{C}Q\backslash I$ of representations of a quiver $Q$ with family $I$ coming from a superpotential $W$ on $Q

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**Extra info for A theory of generalized Donaldson-Thomas invariants**

**Example text**

3 ) ∈ ΛX then (λ2 + λ2 ) − 12 (λ1 + λ1 )2 = λ2 − 12 λ21 + λ2 − 12 (λ1 )2 + λ1 λ1 , 42 4. BEHREND FUNCTIONS AND DONALDSON–THOMAS THEORY and the right hand side is the sum of three terms in H 4 (X; Z). So (λ0 +λ0 , . . , λ3 + λ3 ) ∈ ΛX . Also (−λ2 ) − 12 (−λ1 )2 = − λ2 − 12 λ21 − λ21 , with the right hand the sum of two terms in H 4 (X; Z). So (−λ0 , . . , −λ3 ) ∈ ΛX , and ΛX is a subgroup of H even (X; Q). We have 1 6 H even (X; Z)/torsion ⊆ ΛX ⊆ H even (X; Z)/torsion ⊆ H even (X; Q), 3 so ΛX is a lattice of rank i=0 b2i (X) as H even (X; Z)/torsion is.

23) taken over all Σ generate the subgroup of (0, 0, λ2 , λ3 ) ∈ ΛX with λ2 ∈ H 4 (X; Z) and λ3 ∈ H 6 (X; Z). 21) for all β ∈ H 2 (X; Z), these generate ΛX . 20. (a) Our proof used the Hodge Conjecture over Z for Calabi– Yau 3-folds, proved by Voisin [103]. But the Hodge Conjecture over Z is false in general, so the theorem may not generalize to other classes of varieties. 5. CHARACTERIZING K num (coh(X)) FOR CALABI–YAU 3-FOLDS 43 (b) In fact ΛX is a subring of H even (X; Q). Also, K0 (coh(X)), K num (coh(X)) naturally have the structure of rings, with multiplication ‘ · ’ characterized by [E] · [F ] = [E ⊗ F ] for E, F locally free.

Note that these are not coherent sheaves, which are sheaves of OX -modules. A sheaf C is called constructible if there is a locally ﬁnite stratiﬁcation X = j∈J Xj of X in the complex analytic topology, such that C|Xj is a Q-local system for all j ∈ J, and all the stalks Cx for x ∈ X are ﬁnite-dimensional Q-vector spaces. A complex C • of sheaves of Q-modules on X is called constructible if all its cohomology sheaves H i (C • ) for i ∈ Z are constructible. b (X) for the bounded derived category of constructible complexes Write DCon b on X.

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

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