Lectures on the Theory of Pure Motives (University Lecture) by Jacob P. Murre (PDF)

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The theory of motives was created by Grothendieck in the 1960s as he searched for a universal cohomology theory for algebraic varieties. The theory of pure motives is well established as far as the construction is concerned. Pure motives are expected to have a number of additional properties predicted by Grothendieck’s standard conjectures, but these conjectures remain wide open. The theory for mixed motives is still incomplete. This book deals primarily with the theory of pure motives. The exposition begins with the fundamentals: Grothendieck’s construction of the category of pure motives and examples. Next, the standard conjectures and the famous theorem of Jannsen on the category of the numerical motives are discussed. Following this, the important theory of finite dimensionality is covered. The concept of Chow-Künneth decomposition is introduced, with discussion of the known results and the related conjectures, in particular the conjectures of Bloch-Beilinson type. We finish with a chapter on relative motives and a chapter giving a short introduction to Voevodsky’s theory of mixed motives.

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⭐The theory of motives originated with the mathematician Alexander Grothendieck and can loosely be described as an attempt to find a ‘universal cohomology theory’ for algebraic varieties. In this book, attention is focused for the most part on the theory of pure motives, which are those motives that are related to smooth projective varieties. Every such variety X is viewed as a motive, which is viewed as a “piece” of X that accounts for the geometric and arithmetic properties of X. This view is inspired from algebraic geometry, wherein for an algebraic curve X, the “essential” part of X is determined by the Jacobian variety of X, and every abelian variety is an abelian subvariety of a Jacobian. This fact motivates the search for finding a Jacobian variety for any arbitrary variety. Loosely speaking, a motive could then be viewed as an analog in higher dimensions of the Jacobian of a curve.Crucial to the understanding of pure motives as outlined in this book is the concept of a correspondence. Given smooth projective varieties X and Y, correspondences are special types of maps between X and Y which form an abelian group. Compositions of correspondences are defined using the fiber product and involve a complex set of operations using projections on factors. Algebraic cycles are then taken to be formal linear combinations of correspondences. Correspondences reflect the idea that in general, there do not exist regular maps from one algebraic variety to a second one, and hence correspondences are the “many-valued maps” that reflect the absence of regularity.More specifically, a correspondence from one algebraic variety X to another algebraic variety Y is a cycle in the product X x Y. If Z is such a correspondence and if T is a cycle in X of codimension i with d = dim(X), then Z will “push forward” T to a cycle Y of codimension i + t – d where t = dim(X x Y). If it happens that t = d, then Z is said to preserve the codimension of the cycle T. The fact that Z can raise the dimension in this way is a sign that correspondences are not so simple as one might imagine at first glance. This complexity is responsible for some of the nagging issues that must be settled in order to get a reasonable notion of intersection of cycles and definitions of push forward and pullback maps. As in most areas of mathematics, the strategy for dealing with such complexity is define an equivalence relation on cycles, in order that the operations of intersection, pullback, etc can be defined.The goal therefore is to find an “adequate” equivalence relation, and a few proposals have been made, going by the names of rational, algebraic, numerical, and homological equivalence. iRational equivalence is the easiest to understand, being that it is a generalization of the classical notion of linear equivalence of divisors. The original definition of rational equivalence involved the notion of ‘specialization’, which in turn relied on the notion of an ‘associated form.’ A more specialized notion of rational equivalence is that of ‘smash-nilpotent’ equivalence, which means that some integer power of the cycle is rationally equivalent to the zero cycle. Algebraic equivalence is somewhat similar to rational equivalence, with the difference being that one can find a smooth irreducible curve that in a sense that can be rigorously defined serves as an interpolation between the cycle and the zero cycle. Readers will find a fairly detailed discussion of the conjectured equivalences between some of these notions of equivalence, such as that of homological and numerical equivalence etc.If readers look into the history of the notion of a correspondence, they will find that correspondences were widely used in “classical” algebraic geometry. This is readily apparent when considering them as examples of a graph of a morphism or the closure of such a graph. Correspondences have a product operation and there exists homomorphisms that generalize the notion of composition, push-forward, and pull-back for morphisms. In addition, one can view obtain a motivating example of the intersection theory of correspondences by remembering that the Lefschetz fixed point formula allows one to study intersections of an object with the “diagonal”. If X and Y are both n-dimensional and T is an n-dimensional correspondence then the degree of the intersection of T with the diagonal will give the number of virtual fixed points of T. The classical theory of correspondences is very rich and touches on many “modern” topics such as enumerative geometry. Correspondences have been shown to have a connection with the theory of Hecke operators, but this connection is not discussed in this book. An immediate question concerns the issue of whether a motive is finite-dimensional in some sense. This issue is discussed in this book using the theory of group representations. Finite-dimensionality of motives is defined in terms of what happens to a product motive under the action of the symmetric group. One says a motive is ‘evenly finite dimensional’ if there exists a positive integer n such that the nth exterior product is zero. A motive is ‘oddly finite dimensional’ if there exists a positive integer n such that the nth symmetric product of M is zero. A motive is then said to have finite dimension if it can be written as the direct sum of evenly finite dimensional and oddly finite dimensional motives. The dimension of the motive is then the sum of the dimensions of the summands. If a motive is both evenly and oddly finitely dimensional then the motive is identically zero. Dimension is preserved under surjective morphisms of motives, and so it is important to pinpoint the cases where a morphism between motives is surjective. It is an open question as to whether every Chow motive is finite dimensional. One can show that the dimension of objects of a full tensor pseudo abelian subcategory in the category of Chow motives generated by motives of smooth projective curves is finite.Also of importance in this discussion of finite-dimensionality of motives is the notion of a ‘phantom motive’ which becomes zero after passing to numerical equivalence. Phantom motives arise from the fact that the forget functor from motives under rational equivalence to motives under homological equivalence is not faithful. A Chow motive is a phantom motive if it is not zero but equal to zero under homological equivalence. Phantom motives do not arise for motives of finite dimension.The underlying need for discussing the finite-dimensionality of motives is that there exists a big difference between the theory of divisors and the theory of algebraic cycles of codimension greater than one. For example, for divisors, the Chow group is nothing other than the Picard group which is finitely-generated, as is the Neron-Severi group, which is the Chow group modulo algebraic equivalence.As the authors show, one can take the direct sum of pure motives, along with their tensor product, and one has a notion of ‘unit’ motive, which is the identity for the tensor product. There is also a notion of a ‘Lefschetz motive’, which should be viewed as one of the “elementary” motives, in the sense that any motive can be expressed as a direct factor of a power of the Lefschetz motive. Interestingly, this result alleviates the need to deal with motives based on varieties with components of different dimensions. The dimensionality of direct sums of motives is straightforward to define, whereas for tensor products it is somewhat more involved. It is an open question as to whether every Chow motive is finite-dimensional.

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