By Dominic Joyce, Yinan Song

This e-book 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 complex rational weights. The $\bar{DT}{}^\alpha(\tau)$ are outlined for all sessions $\alpha$, and are equivalent to $DT^\alpha(\tau)$ while it truly is outlined. they're unchanged lower than deformations of $X$, and remodel by means of a wall-crossing formulation lower than switch of balance situation $\tau$. To turn out all this, the authors learn the neighborhood constitution of the moduli stack $\mathfrak M$ of coherent sheaves on $X$. They convey that an atlas for $\mathfrak M$ might be written in the neighborhood as $\mathrm{Crit}(f)$ for $f:U\to{\mathbb C}$ holomorphic and $U$ tender, 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. additionally 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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20) yields [δ¯E , δ¯F ] = dim Hom(E, F ) − dim Hom(F, E) [Spec K/Gm ], ρ5 + P(Ext1 (F, E))×[Spec K/Gm ], ρ6 − P(Ext1 (E, F ))×[Spec K/Gm ], ρ7 . 14) and χ P(Ext1 (E, F )) = dim Ext1 (E, F ). 5. 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 ).

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). Therefore χ ¯ is also given by χ ¯ [E], [F ] = dim Hom(E, F ) − dim Ext1 (E, F ) − dim Hom(F, E) − dim Ext1 (F, E) . 14) Thus the Euler form χ ¯ on K(coh(X)) is antisymmetric. 15) 28 3. BACKGROUND MATERIAL FROM [51, 52, 53, 54] for α, β ∈ K(coh(X)). 15) satisﬁes the Jacobi identity and makes L(X) into an inﬁnite-dimensional Lie algebra over Q.

Let X, Y be complex analytic spaces, ϕ : Y → X a proper morphism, and f : X → C a holomorphic function. Set g = f ◦ ϕ, and write X0 = f −1 (0) and Y0 = g −1 (0). Then the following diagrams commute: CFan Z (Y ) Ψg CFan Z (Y0 ) CF(ϕ) CF(ϕ) G CFan Z (X) CFan Z (Y ) Ψf G CFan Z (X0 ), Φg CFan Z (Y0 ) CF(ϕ) CF(ϕ) G CFan Z (X) Φf G CFan Z (X0 ). 9) We use this to prove a property of Milnor ﬁbres that we will need later. 10, which replaces a longer proof using Lagrangian cycles in an earlier version of this book.