Posted in Algebraic Geometry

By Dominic Joyce, Yinan Song

ISBN-10: 0821852795

ISBN-13: 9780821852798

This booklet 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 needs to be counted with advanced 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 less than deformations of $X$, and remodel by way of a wall-crossing formulation less than swap of balance $\tau$. To end up all this, the authors learn the neighborhood constitution of the moduli stack $\mathfrak M$ of coherent sheaves on $X$. They express that an atlas for $\mathfrak M$ might 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 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 kinfolk $I$ coming from a superpotential $W$ on \$Q

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Extra info for A theory of generalized Donaldson-Thomas invariants

Sample text

The same proof shows that DT α (τ ) for α ∈ K num (coh(X)) is deformationinvariant, provided we know that the group K num (coh(X)) is deformation-invariant, so that this statement makes sense. 5. We show that when K = C we can describe K num (coh(X)) in terms of cohomology groups H ∗ (X; Z), H ∗ (X; Q), so that K num (coh(X)) is manifestly deformationinvariant, and therefore DT α (τ ) is also deformation-invariant. Here is a property of Behrend functions which is crucial for Donaldson–Thomas theory.

But here we sketch an alternative approach due to Behrend [3], which could perhaps be used to give a strictly algebraic proof of the same identities. 15. Let K be an algebraically closed ﬁeld, and M a smooth Kscheme. Let ω be a 1-form on M , that is, ω ∈ H 0 (T ∗ M ). We call ω almost closed if dω is a section of Iω · Λ2 T ∗ M , where Iω is the ideal sheaf of the zero locus ω −1 (0) of ω. Equivalently, dω|ω−1 (0) is zero as a section of Λ2 T ∗ M |ω−1 (0) . In (´etale) local coordinates (z1 , . .

BEHREND FUNCTIONS AND DONALDSON–THOMAS THEORY The functors ψf , φf above have analogues Ψf , Φf on constructible functions deﬁned by Verdier [102, Props. 1]. 5) whenever Z is a closed complex analytic subspace of X, and 1Z ∈ CFan Z (X) is given / Z. We set Φf = Ψf − i∗ , where by 1Z (x) = 1 if x ∈ Z and 1Z (x) = 0 if x ∈ i : X0 → X is the inclusion. Then we have commutative diagrams b (X) DCon  χX CFan Z (X) ψf Ψf b G DCon (X0 ) b DCon (X) χX0  G CFan Z (X0 ),  χX CFan Z (X) φf Φf b G DCon (X0 ) χX0  G CFan Z (X0 ).