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Donaldson-Thomas invariants DT α (τ ) are integers which 'count' τstable coherent sheaves with Chern character α on a Calabi-Yau 3-fold X, where τ denotes Gieseker stability for some ample line bundle on X. They are unchanged under deformations of X. The conventional definition works only for classes α containing no strictly τ -semistable sheaves. Behrend showed that DT α (τ ) can be written as a weighted Euler characteristic χ M α st (τ ), ν M α st (τ ) of the stable moduli scheme M α st (τ ) by a constructible function ν M α st (τ ) we call the 'Behrend function'. This book studies generalized Donaldson-Thomas invariants DT α (τ ). They are rational numbers which 'count' both τ -stable and τ -semistable coherent sheaves with Chern character α on X; strictly τ -semistable sheaves must be counted with complicated rational weights. The DT α (τ ) are defined for all classes α, and are equal to DT α (τ ) when it is defined. They are unchanged under deformations of X, and transform by a wallcrossing formula under change of stability condition τ .To prove all this we study the local structure of the moduli stack M of coherent sheaves on X. We show that an atlas for M may be written locally as Crit(f ) for f : U → C holomorphic and U smooth, and use this to deduce identities on the Behrend function ν M . We compute our invariants DT α (τ ) in examples, and make a conjecture about their integrality properties. We also extend the theory to abelian categories mod-CQ/I of representations of a quiver Q with relations I coming from a superpotential W on Q, and connect our ideas with Szendrői's noncommutative Donaldson-Thomas invariants, and work by Reineke and others on invariants counting quiver representations. Our book is closely related to Kontsevich and Soibelman's independent paper [63].

This is the last in a series on configurations in an abelian category A. Given a finite poset (I, ), an (I, )-configuration (σ, ι, π) is a finite collection of objects σ(J) and morphisms ι(J, K) or π(J, K) : σ(J) → σ(K) in A satisfying some axioms, where J, K are subsets of I. Configurations describe how an object X in A decomposes into subobjects.The first paper defined configurations and studied moduli spaces of configurations in A, using Artin stacks. It showed well-behaved moduli stacks ObjA, M(I, )A of objects and configurations in A exist when A is the abelian category coh(P ) of coherent sheaves on a projective scheme P , or mod-KQ of representations of a quiver Q. The second studied algebras of constructible functions and stack functions on ObjA.The third introduced stability conditions (τ, T, ) on A, and showed the moduli space Obj α ss (τ ) of τ -semistable objects in class α is a constructible subset in ObjA, so its characteristic function δ α ss (τ ) is a constructible function. It formed algebras H pa τ , H to τ ,H pa τ ,H to τ of constructible and stack functions on ObjA, and proved many identities in them.In this paper, if (τ, T, ) and (τ ,T , ) are stability conditions on A we write δ α ss (τ ) in terms of the δ β ss (τ ), and deduce the algebras H pa τ , . . . ,H to τ are independent of (τ, T, ). We study invariants I α ss (τ ) or Iss(I, , κ, τ ) 'counting' τ -semistable objects or configurations in A, which satisfy additive and multiplicative identities. We compute them completely when A = mod-KQ or A = coh(P ) for P a smooth curve. We also find invariants with special properties when A = coh(P ) for P a smooth surface with K −1 P nef, or a Calabi-Yau 3-fold.

This is the second in a series on configurations in an abelian category A. Given a finite poset (I, ), an (I, )-configuration (σ, ι, π) is a finite collection of objects σ(J) and morphisms ι(J, K) or π(J, K) : σ(J) → σ(K) in A satisfying some axioms, where J, K ⊆ I. Configurations describe how an object X in A decomposes into subobjects.The first paper defined configurations and studied moduli spaces of (I, )-configurations in A, using the theory of Artin stacks. It showed well-behaved moduli stacks ObjA, M(I, )A of objects and configurations in A exist when A is the abelian category coh(P ) of coherent sheaves on a projective scheme P , or mod-KQ of representations of a quiver Q.Write CF(ObjA) for the vector space of Q-valued constructible functions on the stack ObjA. Motivated by the idea of Ringel-Hall algebras, we define an associative multiplication * on CF(ObjA) using pushforwards and pullbacks along 1-morphisms between configuration moduli stacks, so that CF(ObjA) is a Q-algebra. We also study representations of CF(ObjA), the Lie subalgebra CF ind (ObjA) of functions supported on indecomposables, and other algebraic structures on CF(ObjA).Then we generalize all these ideas to stack functions SF(ObjA), a universal generalization of constructible functions, containing more information. When Ext i (X, Y ) = 0 for all X, Y ∈ A and i > 1, or when A = coh(P ) for P a Calabi-Yau 3-fold, we construct (Lie) algebra morphisms from stack algebras to explicit algebras, which will be important in the sequels on invariants counting τ -semistable objects in A.

In the theory of Riemannian holonomy groups, perhaps the most mysterious are the two exceptional cases, the holonomy group G 2 in 7 dimensions and the holonomy group Spin(7) in 8 dimensions. This is a survey paper on the exceptional holonomy groups, in two parts. Part I collects together useful facts about G 2 and Spin(7) in §2, and explains constructions of compact 7-manifolds with holonomy G 2 in §3, and of compact 8-manifolds with holonomy Spin (7) in §4.Part II discusses the calibrated submanifolds of manifolds of exceptional holonomy, namely associative 3-folds and coassociative 4-folds in G 2 -manifolds, and Cayley 4-folds in Spin(7)-manifolds. We introduce calibrations in §5, defining the three geometries and giving examples. Finally, §6 explains their deformation theory.Sections 3 and 4 describe my own work, for which the main reference is my book [18]. Part II describes work by other people, principally the very important papers by Harvey and Lawson [12] and McLean [28], but also more recent developments.This paper was written to accompany lectures at the 11 th Gökova Geometry and Topology Conference in May 2004, sponsored by TUBITAK. In keeping with the theme of the conference, I have focussed mostly on G 2 , at the expense of Spin(7). The paper is based in part on the books [18] and [11, Part I], and the survey paper [21].

This is the third in a series on configurations in an abelian category A. Given a finite poset (I, ), an (I, )-configuration (σ, ι, π) is a finite collection of objects σ (J ) and morphisms ι(J, K) or π(J, K) : σ (J ) → σ (K) in A satisfying some axioms, where J, K are subsets of I . Configurations describe how an object X in A decomposes into subobjects.The first paper defined configurations and studied moduli spaces of configurations in A, using the theory of Artin stacks. It showed well-behaved moduli stacks Obj A , M(I, ) A of objects and configurations in A exist when A is the abelian category coh(P ) of coherent sheaves on a projective scheme P , or mod-KQ of representations of a quiver Q. The second studied algebras of constructible functions and stack functions on Obj A . This paper introduces (weak) stability conditions (τ, T , ) on A. We show the moduli spaces Obj α ss , Obj α si , Obj α st (τ ) of τ -semistable, indecomposable τ -semistable and τ -stable objects in class α are constructible sets in Obj A , and some associated configuration moduli spaces M ss ,so their characteristic functions δ α ss , δ α si , δ α st (τ ) and δ ss , . . . , δ b st (I, , κ, τ ) are constructible. We prove many identities relating these constructible functions, and their stack function analogues, under pushforwards. We introduce interesting algebras H pa τ , H to τ , H pa τ , H to τ of constructible and stack functions, and study their structure. In the fourth paper we show H pa τ , . . . , H to τ are independent of (τ, T , ), and construct invariants of A, (τ, T , ).