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

By Dominic Joyce, Yinan Song

This e-book experiences 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 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 below deformations of $X$, and rework by means 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 $\mathfrak M$ will be written in the community 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 homes. in addition they expand the speculation 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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3). 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 , .

3 ) ∈ ΛX , as we want. = 1 2 0 12 As X is a Calabi–Yau 3-fold over C with H 1 (OX ) = 0 we have H 2,0 (X) = H 0,2 (X) = 0, so H 1,1 (X) = H 2 (X; C). Therefore every β ∈ H 2 (X; Z) is c1 (Lβ ) for some holomorphic line bundle Lβ , with ch(Lβ ) = 1, β, 12 β 2 , 16 β 3 , for any β ∈ H 2 (X; Z). 21) Pick x ∈ X, and let Ox be the skyscraper sheaf at x. 22) identifying H 6 (X; Q) ∼ = Q and H 6 (X; Z) ∼ = Z in the natural way. Suppose Σ is an reduced algebraic curve in X, with homology class [Σ] ∈ H2 (X; Z) ∼ = H 4 (X; Z).

6]. α In [53, §8] we define interesting stack functions δ¯ss (τ ), ¯α (τ ) in SFal (MA ). 10. 2, and (τ, T, ) be a α permissible weak stability condition on A. Define stack functions δ¯ss (τ ) = δ¯Mαss (τ ) α ¯ in SFal (MA ) for α ∈ C(A). 6, of the moduli substack Mα ss (τ ) of τ -semistable sheaves in MA . In [53, Def. ,αn ∈C(A): α1 +···+αn =α, τ (αi )=τ (α), all i where ∗ is the Ringel–Hall multiplication in SFal (MA ). Then [53, Th. 2] proves 1 α1 ¯ (τ ) ∗ ¯α2 (τ ) ∗ · · · ∗ ¯αn (τ ). n! 5), because as the family of τ -semistable sheaves in class α is bounded, there are only finitely ways to write α = α1 + · · · + αn with τ -semistable sheaves in class αi for all i.

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