By Dominic Joyce, Yinan Song

ISBN-10: 0821852795

ISBN-13: 9780821852798

This ebook stories 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 really is outlined. they're unchanged less than deformations of $X$, and remodel by means of a wall-crossing formulation less than switch of balance $\tau$. To turn out all this, the authors examine the neighborhood constitution of the moduli stack $\mathfrak M$ of coherent sheaves on $X$. They convey that an atlas for $\mathfrak M$ can be written in the community as $\mathrm{Crit}(f)$ for $f:U\to{\mathbb C}$ holomorphic and $U$ gentle, 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 homes. 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 members $I$ coming from a superpotential $W$ on \$Q

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Extra resources for A theory of generalized Donaldson-Thomas invariants

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13 applies. 4. 5]. We restrict to K of characteristic zero so that Euler characteristics over K are well-behaved. 15. Let K be an algebraically closed ﬁeld of characteristic zero. A Calabi–Yau 3-fold is a smooth projective 3-fold X over K, with trivial canonical bundle KX . From chapter 5 onwards we will also assume that H 1 (OX ) = 0, but this is not needed for the results of [51, 52, 53, 54]. Take A to be coh(X) and K(coh(X)) to be K num (coh(X)). 1). As X is a Calabi–Yau 3-fold, Serre duality gives Exti (F, E) ∼ = Ext3−i (E, F )∗ , so dim Exti (F, E) = dim Ext3−i (E, F ) for all E, F ∈ coh(X).

Suppose X → M is an embedding of X as a closed subscheme of a smooth K-scheme M . Let CX M be the normal cone of X in M , as in [28, p. 73], and π : CX M → X the projection. 1], deﬁne a cycle cX/M ∈ Z∗ (X) by cX/M = C (−1)dim π(C ) mult(C )π(C ), where the sum is over all irreducible components C of CX M . It turns out that cX/M depends only on X, and not on the embedding X → M . Behrend [3, Prop. 1] proves that given a ﬁnite type K-scheme X, there exists a unique cycle cX ∈ Z∗ (X), such that for any ´etale map ϕ : U → X for a Kscheme U and any closed embedding U → M into a smooth K-scheme M , we have ϕ∗ (cX ) = cU/M in Z∗ (U ).

Deﬁne a total order ‘ ’ on G by p p for p, p ∈ G if either (a) deg p > deg p , or (b) deg p = deg p and p(t) p (t) for all t 0. We write p < q if p q and p = q. 5 to hold. The eﬀect of (a) is that τ -semistable sheaves are automatically pure, because if 0 = S ⊂ E with dim S < dim E then S destabilizes E. Fix a very ample line bundle OX (1) on X. For E ∈ coh(X), the Hilbert polynomial PE is the unique polynomial in Q[t] such that PE (n) = dim H 0 (E(n)) for all n 0. Equivalently, PE (n) = χ ¯ [OX (−n)], [E] for all n ∈ Z.