Introduction Polytopes Topology and combinatorics of simplicial complexes Commutative and homological algebra of simplicial complexes Cubical complexes Toric and quasitoric manifolds Moment-angle complexes Cohomology of moment-angle complexes and combinatorics of triangulated manifolds Cohomology rings of subspace arrangement complements Bibliography Index.

/pdf/torus-actions-and-their-applications-in-topology-and-36j92o51jp.pdf

Torus Actions and their Applications in Topology and Combinatorics

ALGEBRAIC VARIETIES By ALEXANDER GROTHENDIEGK It is less than four years since eohomologieal methods (i.e. methods of Homologieal Algebra) were introduced into Algebraic Geometry in Serre's fundamental paper, and it seems already certain that they are to overflow this part of mathematics in the coming years, from the foundations up to the most advanced parts. All we can do here is to sketch briefly some of the ideas and results. None of these have been published in their final form, but most of them originated in or were suggested by Serre's paper. Let us first give an outline of the main topics of eohomologieal investigation in Algebraic Geometry, as they appear at present. The need of a theory of cohomology for 'abstract' algebraic varieties was first emphasized by Weil, in order to be able to give a precise meaning to his celebrated conjectures in Diophantine Geometry. Therefore the initial aim was to find the 'Weil cohomology' of an algebraic variety, which should have as coefficients something 'at least as good as a field of characteristic 0, and have such formal properties (e.g. duality, Runneth formula) as to yield the analogue of Lefschetz's 'fixed-point formula '. Serre's general idea has been that the usual ' Zariski topology ' of a variety (in which the closed sets are the algebraic subsets) is a suitable one for applying methods of Algebraic Topology. His first approach was hoped to yield at least the right Betti numbers of a variety, it being evident from the start that it could not be considered as the Weil cohomology itself, as the coefficient field for cohomology was the ground field of the variety, and therefore not in general of characteristic 0. In fact, even the hope of getting the 'true ' Betti numbers has failed, and so have other attempts of Serre's to get Weil's cohomology by taking the cohomology of the variety with values, not in the sheaf of local rings themselves, but in the sheaves of Witt-vectors constructed on the latter. He gets in this way modules over the ring W{h) of infinite Witt vectors on the ground field h, and W{h) is a ring of characteristic 0 even if h is of characteristic p =t= 0. Unfortunately, modules thus obtained over W{h) may be infinitely generated, even when the variety V is an abelian variety. Although interesting relations must certainly exist between these cohomology groups and the 'true ones', it seems certain 104 ALEXANDER GROTHENDIECK now that the Weil cohomology has to be defined by a completely different approach. Such an approach was recently suggested to me by the connections between sheaf-theoretic cohomology and cohomology of Galois groups on the one hand, and the classification of unramified coverings of a variety on the other (as explained quite unsystematically in Serre's tentative Mexico paper), and by Serre's idea that a 'reasonable' algebraic principal fiber space with structure group G, defined on a variety V, if it is not locally trivial, should become locally trivial on some covering of V unramified over a given point of V. This has been the startingpoint of a definition of the Weil cohomology (involving both 'spatial' and Galois cohomology), which seems to be the right one, and which gives clear suggestions how Weil's conjectures may be attacked by the machinery of Homological Algebra. As I have not begun these investigations seriously as yet, and as moreover this theory has a quite distinct flavor from the one of the theory of algebraic coherent sheaves which we shall now be concerned with, we shall not dwell any longer on Weil's cohomology. Let us merely remark that the definition alluded to has already been the starting-point of a theory of eohomologieal dimension of fields, developed recently by Tate. The second main topic for eohomologieal methods is the cohomology theory of algebraic coherent sheaves, as initiated by Serre. Although inadequate for Weil's purposes, it is at present yielding a wealth of new methods and new notions, and gives the key even for results which were not commonly thought to be concerned with sheaves, still less with cohomology, such as Zariski's theorem on 'holomorphic functions' and his 'main theorem'—which can be stated now in a more satisfactory way, as we shall see, and proved by the same uniform elementary methods. The main parts of the theory, at present, can be fisted as follows: (a) General finiteness and asymptotic behaviour theorems. (6) Duality theorems, including (respectively identical with) a eohomologieal theory of residues. (c) Riemann-Roch theorem, including the theory of Chern classes for algebraic coherent sheaves. (cu) Some special results, concerning mainly abelian varieties. The third main topic consists in the application of the eohomologieal methods to local algebra. Initiated by Koszul and Cartan-Eilenberg in connection with Hubert's 'theorem of syzygies', the systematic use of COHOMOLOGY THEORY 105 these methods is mainly due again to Serre. The results are the characterization of regular local rings as those whose global eohomologieal dimension is finite, the clarification of Cohen-Macaulay's equidimensionality theorem by means of the notion of eohomologieal codimension\ and especially the possibility of giving (for the first time as it seems) a theory of intersections, really satisfactory by its algebraic simplicity and its generality. Serre's result just quoted, that regular local rings are the only ones of finite global eohomologieal dimension, accounts for the fact that only for such local rings does a satisfactory theory of intersections exist. I cannot give any details here on these subjects, nor on various results I have obtained by means of a local duality theory, which seems to be the tool which is to replace differential forms in the case of unequal characteristics, and gives, in the general context of commutative algebra, a clarification of the notion of residue, which as yet was not at all well understood. The motivation of this latter work has been the attempt to get a global theory of duality in cohomology for algebraic varieties admitting arbitrary singularities, in order to be able to develop intersection formulae for cycles with arbitrary singularities, in a nonsingular algebraic variety, formulas which contain also a 'Lefschetz formula mod.p'. In fact, once a proper local formalism is obtained, the global statements become almost trivial. As a general fact, it appears that, to a great extent, the 'local' results already contain a global one; more precisely, global results on varieties of dimension n can frequently be deduced from corresponding local ones for rings of Krull dimension n + l. We will therefore turn now to giving some main ideas in the second topic, that is, the cohomology theory of algebraic coherent sheaves. First, I would like, however, to emphasize one point common to all of the topics considered (except perhaps for {d)), and in fact to all of the standard techniques in Algebraic Geometry. Namely, that the natural range of the notions dealt with, and the methods used, are not really algebraic varieties. Thus, we know that an affine algebraic variety with ground field h is determined by its co-ordinate ring, which is an arbitrary finitely generated ^-algebra without nilpotent elements; therefore, any statement concerning affine algebraic varieties can be viewed also as a statement concerning rings A of the previous type. Now it appears that most of such statements make sense, and are true, if we assume only J. to be a commutative ring with unit, provided we sometimes submit it to some mild restriction, as being noetherian, for instance. In 106 ALEXANDER GROTHENDIECK the same way, most of the results proved for the local rings of algebraic geometry, make sense and are true for arbitrary noetherian local rings. Besides, frequently when it seemed at first sight that the statement only made sense when a ground field k was involved, as in questions in which differential forms are considered, further consideration of the matter showed that this impression was erroneous, and that a better understanding is obtained by replacing k by a ring B such that A is a finitely generated J5-algebra. Geometrically, this means that instead of a single affine algebraic variety V (as defined by A) we are considering a 'regular map' or 'morphism' of V into another affine variety W, and properties of the variety V then are generalized to properties of a morphism V -> W (the 'absolute' notion for V being obtained from the more general 'relative' notion by taking W reduced to a point). On the other hand, one should not prevent the rings having nilpotent elements, and by no means exclude them without serious reasons. Now just as arbitrary commutative rings can be thought of as a proper generalization of affine algebraic varieties, one can find a corresponding suitable generalization of arbitrary algebraic varieties (defined over an arbitrary field). This was done by Nagata in a particular case, yet following the definition of schemata given by Chevalley he had to stick to the irreducible case, and with no nilpotent elements involved. The principle of the right definition is again to be found in Serre's fundamental paper, and is as follows. If A is any commutative ring, then the set Spec {A) of all prime ideals of A can be turned into a topological space in a classical way, the closed subsets consisting of those prime ideals which contain a given subset of A. On the other hand, there is a sheaf of rings defined in a natural way on Spec {A), the fiber of this sheaf at the point p being the local ring Ap. More generally, every module M over A defines a sheaf of modules on Spec {A), the fiber of which at the point p is the localized moduleMp over Ap. Now we call pre-schema a topological space X with a sheaf of rings Gx on X, called its structure sheaf, such that every point of X has an open neighborhood isomorphic to some Spec {A). If X and Y are two pre-schemas, a morphism f fro

Proceedings of the International Congress of Mathematicians

1. Lie Groups and Homogeneous Spaces 2. Minimal Models 3. Manifolds 4. Complex and Symplectic Manifolds 5. Geodesics 6. Curvature 7. G-Spaces 8. Blow-ups and Intersection Products 9. A Florilege of Geometric Applications APPENDICES A. De Rham Forms B. Spectral Sequences C. Basic Homotopy Recollections

/pdf/algebraic-models-in-geometry-4m4p1db1xw.pdf

Algebraic models in geometry

Toric topology emerged in the end of the 1990s on the borders of equivariant topology, algebraic and symplectic geometry, combinatorics and commutative algebra It has quickly grown up into a very active area with many interdisciplinary links and applications, and continues to attract experts from different fields The key players in toric topology are moment-angle manifolds, a family of manifolds with torus actions defined in combinatorial terms Their construction links to combinatorial geometry and algebraic geometry of toric varieties via the related notion of a quasitoric manifold Discovery of remarkable geometric structures on moment-angle manifolds led to seminal connections with the classical and modern areas of symplectic, Lagrangian and non-Kaehler complex geometry A related categorical construction of moment-angle complexes and their generalisations, polyhedral products, provides a universal framework for many fundamental constructions of homotopical topology The study of polyhedral products is now evolving into a separate area of homotopy theory, with strong links to other areas of toric topology A new perspective on torus action has also contributed to the development of classical areas of algebraic topology, such as complex cobordism The book contains lots of open problems and is addressed to experts interested in new ideas linking all the subjects involved, as well as to graduate students and young researchers ready to enter into a beautiful new area

Toric Topology

The polyhedral product functor: A method of decomposition for moment-angle complexes, arrangements and related spaces

A torus manifold is an even-dimensional manifold acted on by a half-dimensional torus with non-empty fixed point set and some additional orie ntation data. It may be considered as a far-reaching generalisation of toric manifolds from algebraic geometry. The orbit space of a torus manifold has a rich combinatorial structure, e.g., it is a manifold with corners provided that the action is locally standard. Here we investigate relationships between the cohomological properties of torus manifolds and the combinatorics of their orbit quotients. We show that the cohomology ring of a torus manifold is generated by two-dimensional classes if and only if the quotient is a homology polytope. In this case we retrieve the familiar picture from toric geo metry: the equivariant cohomology is the face ring of the nerve simplicial complex and the ordinary cohomology is obtained by factoring out certain linear forms. In a more general situation, we show that the odd-degree cohomology of a torus manifold vanishes if and only if the orbit space is face-acyclic. Although the cohomology is no longer generated in degree two under these circumstances, the equivariant cohomology is still isomorphic to the face ring of an appropriate simplicial poset.

/pdf/on-the-cohomology-of-torus-manifolds-4dno6jwtkx.pdf

On the cohomology of torus manifolds

Our aim is to bring the theory of analogous polytopes to bear on the study of quasitoric manifolds, in the context of stably complex manifolds with compatible torus action By way of application, we give an explicit construction of a quasitoric representative for every complex cobordism class as the quotient of a free torus action on a real quadratic complete intersection We suggest a systematic description for omnioriented quasitoric manifolds in terms of combinatorial data, and explain the relationship with non-singular projective toric varieties (otherwise known as toric manifolds) By expressing the first and third authors' approach to the representability of cobordism classes in these terms, we simplify and correct two of their original proofs concerning quotient polytopes; the first relates to framed embeddings in the positive cone, and the second involves modifying the operation of connected sum to take account of orientations Analogous polytopes provide an informative setting for several of the details

/pdf/spaces-of-polytopes-and-cobordism-of-quasitoric-manifolds-52hnualj3h.pdf

Spaces of polytopes and cobordism of quasitoric manifolds

This paper is a survey of new results and open problems connected with fundamental combinatorial concepts, including polytopes, simplicial complexes, cubical complexes, and arrangements of subspaces. Attention is concentrated on simplicial and cubical subdivisions of manifolds, and especially on spheres. Important constructions are described that enable one to study these combinatorial objects by using commutative and homological algebra. The proposed approach to combinatorial problems is based on the theory of moment-angle complexes recently developed by the authors. The crucial construction assigns to each simplicial complex with vertices a -space with special bigraded cellular decomposition. In the framework of this theory, well-known non-singular toric varieties arise as orbit spaces of maximally free actions of subtori on moment-angle complexes corresponding to simplicial spheres. It is shown that diverse invariants of simplicial complexes and related combinatorial-geometric objects can be expressed in terms of bigraded cohomology rings of the corresponding moment-angle complexes. Finally, it is shown that the new relationships between combinatorics, geometry, and topology lead to solutions of some well-known topological problems.

https://iopscience.iop.org/article/10.1070/RM2000v055n05ABEH000320/pdf

Taras Panov

Papers

Torus Actions and their Applications in Topology and Combinatorics

Toric Topology

On the cohomology of torus manifolds

Spaces of polytopes and cobordism of quasitoric manifolds

Torus actions, combinatorial topology, and homological algebra