By Dominic Joyce, Yinan Song

ISBN-10: 0821852795

ISBN-13: 9780821852798

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 periods $\alpha$, and are equivalent to $DT^\alpha(\tau)$ whilst it truly is outlined. they're unchanged less than deformations of $X$, and rework through a wall-crossing formulation below switch of balance $\tau$. To turn out all this, the authors research 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 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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Wn ) representing the 1-form w1 dz1 + · · · + wn dzn at (z1 , . . , zn ), so that we identify T ∗ M near x with C2n . 17) where S = (z1 , . . , wn ) ∈ C2n : |z1 |2 + · · · + |wn |2 = 2 is the sphere of radius in C2n , and Γη−1 ω the graph of η −1 ω regarded locally as a complex submanifold of C2n , and Δ = (z1 , . . , wn ) ∈ C2n : wj = z¯j , j = 1, . . , n , and LS ( , ) the linking number of two disjoint, closed, oriented (n−1)-submanifolds in S . Here are some questions which seem interesting.

17? 11 true with df replaced by an almost closed 1-form ω, and df˜ replaced by π ∗ (ω)? (b) Can one deﬁne a natural perverse sheaf P supported on X, with χX (P) = νX , such that P ∼ = φf (Q[n − 1]) when ω = df for f : M → C holomorphic? (c) If the answer to (a) or (b) is yes, are there generalizations to the algebraic setting, which work say over K algebraically closed of characteristic zero? 18(b) for Saito’s mixed Hodge modules [92]. 5. Characterizing K num (coh(X)) for Calabi–Yau 3-folds Let X be a Calabi–Yau 3-fold over C, with H 1 (OX ) = 0.

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. It is proved by Behrend [3, Th. 18] when K = C, but his proof is valid for general K. 14. Let K be an algebraically closed ﬁeld of characteristic zero, X a proper K-scheme with a symmetric obstruction theory, and [X]vir ∈ A0 (X) the corresponding virtual class from Behrend and Fantechi [5].

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

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