By Dominic Joyce, Yinan Song
ISBN-10: 0821852795
ISBN-13: 9780821852798
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 complex rational weights. The $\bar{DT}{}^\alpha(\tau)$ are outlined for all periods $\alpha$, and are equivalent to $DT^\alpha(\tau)$ whilst it truly is outlined. they're unchanged lower than deformations of $X$, and remodel by way of a wall-crossing formulation lower than switch of balance situation $\tau$. To turn out all this, the authors examine the neighborhood constitution of the moduli stack $\mathfrak M$ of coherent sheaves on $X$. They express that an atlas for $\mathfrak M$ could 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 relatives $I$ coming from a superpotential $W$ on $Q
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Extra info for A theory of generalized Donaldson-Thomas invariants
Sample text
From now on we will neglect torsion in H even (X; Z), so by an abuse of notation, when we say that an element λi of H 2i (X; Q) lies in H 2i (X; Z), we really mean that λi lies in the image of H 2i (X; Z) in H 2i (X; Q). By the Hirzebruch–Riemann–Roch Theorem [40, Th. 19) 1 where td(T X) is the Todd class of T X, which is 1+ 12 c2 (T X) as X is a Calabi–Yau ∨ 3-fold, and (λ0 , λ1 , λ2 , λ3 ) = (λ0 , −λ1 , λ2 , −λ3 ), writing elements of H even (X; Q) as (λ0 , . . , λ3 ) with λi ∈ H 2i (X; Q). The Chern character is additive over short exact sequences.
Thus there exists y ∈ Y (K) with (π1 )∗ (y) = w and (π2 )∗ (y) = w . 1) thus gives (−1)n νW (w) = νY (y) = (−1)n νW (w ), so that (−1)n νW (w) = (−1)n νW (w ). Hence νX (x) is well-defined. Therefore there exists a unique function νX : X(K) → Z with the property in the proposition. It remains only to show that νX is locally constructible. For ϕ, W, n as above, ϕ∗ (νX ) = (−1)n νW and νW constructible imply that νX is constructible on the constructible set ϕ∗ (W (K)) ⊆ X(K). But any constructible subset S of X(K) can be covered by finitely many such subsets ϕ∗ (W (K)), so νX |S is constructible, and thus νX is locally constructible.
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. It is a triangulated category.
A theory of generalized Donaldson-Thomas invariants by Dominic Joyce, Yinan Song
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