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

By Dominic Joyce, Yinan Song

This publication 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 needs to be counted with complex rational weights. The $\bar{DT}{}^\alpha(\tau)$ are outlined for all periods $\alpha$, and are equivalent to $DT^\alpha(\tau)$ while it really is outlined. they're unchanged below deformations of $X$, and rework via a wall-crossing formulation less than swap 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 community 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 houses. additionally they expand the speculation 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

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

**Similar algebraic geometry books**

**Singularities, Representation of Algebras and Vector Bundles**

It's popular that there are shut kinfolk among periods of singularities and illustration thought through the McKay correspondence and among illustration idea and vector bundles on projective areas through the Bernstein-Gelfand-Gelfand development. those relatives in spite of the fact that can't be thought of to be both thoroughly understood or absolutely exploited.

**Understanding Geometric Algebra for Electromagnetic Theory**

This booklet goals to disseminate geometric algebra as a simple mathematical software set for operating with and figuring out classical electromagnetic concept. it is goal readership is someone who has a few wisdom of electromagnetic idea, predominantly traditional scientists and engineers who use it during their paintings, or postgraduate scholars and senior undergraduates who're trying to expand their wisdom and raise their knowing of the topic.

- Spectral Theory and Analytic Geometry over Non-Archimedean Fields (Mathematical Surveys and Monographs)
- Transseries in difference and differential equations
- Brauer Groups in Ring Theory and Algebraic Geometry: Proceedings, Univ. of Antwerp Wilrijk, Belgium, Aug 17-28, 1981 (Lecture notes in mathematics)
- Diophantus of Alexandria: A Study in the History of Greek Algebra
- Invitation to Noncommutative Geometry
- Introduction to Algebraic Geometry [Lecture notes]

**Extra info for A theory of generalized Donaldson-Thomas invariants**

**Example text**

2 to study the Behrend function νM . However, as Behrend says [3, p. 5]: ‘We do not know if every scheme admitting a symmetric obstruction theory can locally be written as the critical locus of a regular function on a smooth scheme. 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 ).

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. 7) α∈C(A):τ (α)=t where δ¯0 is the identity 1 in SFal (MA ). For α ∈ C(A) and t = τ (α), using the n−1 1 n xn and exp(x) = 1 + n 1 n! 7) to MA . 5) are inverse, since log and exp are inverse. Thus, knowing the ¯α (τ ) α (τ ). is equivalent to knowing the δ¯ss α α α (τ ).

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. 5. 3 also holds for Artin K-stacks X, Y locally of ﬁnite type. 2. Milnor ﬁbres and vanishing cycles We deﬁne Milnor ﬁbres for holomorphic functions on complex analytic spaces. 6. Let U be a complex analytic space, f : U → C a holomorphic function, and x ∈ U . Let d( , ) be a metric on U near x induced by a local embedding of U in some CN .