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

**Read or Download A theory of generalized Donaldson-Thomas invariants PDF**

**Best algebraic geometry books**

**A basic course in algebraic topology**

This e-book is meant to function a textbook for a path in algebraic topology first and foremost graduate point. the most issues lined are the type of compact 2-manifolds, the elemental staff, overlaying areas, singular homology concept, and singular cohomology concept. those subject matters are constructed systematically, warding off all unecessary definitions, terminology, and technical equipment.

This paintings offers a learn of the algebraic houses of compact correct topological semigroups often and the Stone-Cech compactification of a discrete semigroup particularly. a number of strong purposes to combinatorics, basically to the department of combinarotics referred to as Ramsey idea, are given, and connections with topological dynamics and ergodic concept are offered.

**Complex Analysis in One Variable**

This publication offers complicated research in a single variable within the context of contemporary arithmetic, with transparent connections to a number of complicated variables, de Rham idea, actual research, and different branches of arithmetic. hence, protecting areas are used explicitly in facing Cauchy's theorem, actual variable tools are illustrated within the Loman-Menchoff theorem and within the corona theorem, and the algebraic constitution of the hoop of holomorphic services is studied.

This e-book is an intensive monograph on Sasakian manifolds, targeting the complex courting among okay er and Sasakian geometries. the topic is brought via dialogue of a number of history themes, together with the idea of Riemannian foliations, compact complicated and okay er orbifolds, and the life and obstruction concept of ok er-Einstein metrics on complicated compact orbifolds.

- Generalized Polygons
- Weighted Expansions for Canonical Desingularization
- Nevanlinna theory and its relation to diophantine approximation
- Algebraic geometry 2. Sheaves and cohomology
- Fourier Analysis and Convexity
- Arithmetic of Quadratic Forms

**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 ).