Algebraic Geometry

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

By Dominic Joyce, Yinan Song

This publication 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 $DT^\alpha(\tau)$ while it's outlined. they're unchanged below deformations of $X$, and remodel by means of a wall-crossing formulation lower than switch of balance situation $\tau$. To end up 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 $\mathfrak M$ can be written in the neighborhood as $\mathrm{Crit}(f)$ for $f:U\to{\mathbb C}$ holomorphic and $U$ delicate, 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 kin $I$ coming from a superpotential $W$ on $Q

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Extra info for A theory of generalized Donaldson-Thomas invariants

Example text

4. 15) is given by . 16) defining Donaldson–Thomas invariants. 15) is non-local, and non-motivic, and makes sense only if Mα st (τ ) is a proper K-scheme. 16) is local, and (in a sense) motivic, and makes sense for arbitrary finite type K-schemes Mα st (τ ). 5 that this is not a α good idea, as then DT (τ ) would not be unchanged under deformations of X. 16) was the inspiration for this book. It shows that Donaldson– Thomas invariants DT α (τ ) can be written as motivic invariants, like those studied in [51, 52, 53, 54, 55], and so it raises the possibility that we can extend the results of [51, 52, 53, 54, 55] to Donaldson–Thomas invariants by including Behrend functions as weights.

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.

4, with K of characteristic zero and X a Calabi–Yau 3-fold over K. t. 9. 6] we define invariants J α (τ ) ∈ Q for all α ∈ C(coh(X)) by Ψ ¯α (τ ) = J α (τ )λα . 11. These J (τ ) are rational numbers ‘counting’ τ α α semistable sheaves E in class α. When Mα ss (τ ) = Mst (τ ) we have J (τ ) = α α χ(Mst (τ )), that is, J (τ ) is the Euler characteristic of the moduli space Mα st (τ ). As we explain in chapter 4, this is not weighted by the Behrend function νMαst (τ ) , and is not the Donaldson–Thomas invariant DT α (τ ).

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