Filtrations on the homology of algebraic varieties pdf

The most useful fact about singular complex algebraic varieties is. Filtrations on the homology of algebraic varieties about this title. Such a decomposition might be folklore for nonregular schemes as well but does not seem to exist in the literature. In this paper, we study the persistent homology of the offset filtration of. In this paper, a real algebraic variety will be a reduced scheme of. Homology theory on algebraic varieties dover books on. Quillens work in algebraic ktheory volume 11 issue 3 daniel r. Operations on lawson homology are introduced and analyzed.

Supported by grant mog 000 from the international science foundation, supported by nsf grant dms9100383. If the group is of even order, we can not extract additive invariants directly from the induced spectral sequence. I will discuss recent progress by many people in the program of extending natural topological invariants from manifolds to singular spaces. In algebra, filtrations are ordinarily indexed by, the set of natural numbers.

The combinatorial data in the boundary complex captures one piece of the weight ltration. Filtrations on the homology of algebraic varieties. I know that ravenel spent a long time looking for a good abstract framework to encapsulate these issues, and never came up with something that he found satisfactory. Topological ktheory of complex noncommutative spaces volume 152 issue 3 anthony blanc. Filtrations on algebraic cycles and homology eric m. Topological ktheory of complex noncommutative spaces.

Deland in this thesis we study the geometry of the space of rational curves on various projective varieties. A very influential paper, along the likes that mark grant mentioned, and one which weve been trying to unravel the consequences of ever since. These varieties include projective spaces, and smooth hypersurfaces contained within them. Therefore pure homology is topologically invariant. Geometric interpretations of zeemans filtrations of the homology and cohomology of a triangulable space are given, using an. Offset hypersurfaces and persistent homology of algebraic.

This began with the fundamental paper of blaine lawson l which introduced in the context of. Supports and filtrations 3 in algebraic ktheory and in witt theory. The treatment begins with a brief introduction and considerations of linear sections of an algebraic variety as well as singular and hyperplane sections. Schcr the category of real algebraic varieties and proper regular morphisms. For smooth manifolds, the singularity filtration is trivial, while for. Eric friedlander, barry mazur, filtrations on the homology of algebraic varieties, memoir of the a. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition. Weight filtration on the cohomology of algebraic varieties authors. We show that for a complete complex algebraic variety the pure term of the weight filtration in homology coincides with the image of intersection homology. We will also use various sources for commutative algebra. Filtrations on algebraic cycles and homology numdam. The resultant lawson homology has numerous good properties, most notably that reflected in the lawson suspension theorem. Pdf topology of singular algebraic varieties semantic.

These operations lead to a filtration on the singular homology of algebraic varieties, which is identified in terms of correspondences and related to. Prerequisites homology theory of algebraic varieties by wallace. Professor voisin will led a special program on the topology of algebraic varieties. Filtrations on the homology of algebraic varieties book. Topology of algebraic varieties notes for a course taught at the ymsc, spring 2016 eduard looijenga. All homology and cohomology groups of x are taken with. Use features like bookmarks, note taking and highlighting while reading homology theory on algebraic varieties dover books on mathematics. The treatment is linear, and many simple statements are left for the reader to prove as exercises. Masaki hanamura, morihiko saito submitted on 23 may 2006 v1, last revised 16 jun 2006 this version, v2. Homology 5 union of the spheres, with the equatorial identi. Cohen, concerning the homology of the second johnson subgroup of torelli groups. On the formal group laws of unoriented and complex cobordism theory.

The homology of the variety can be inferred from the persistent homology of its approximation if the approximation has high accuracy. Barry mazur this work provides a detailed exposition of a classical topic from a very recent viewpoint. To rst approximation, a projective variety is the locus of zeroes of. Homology theory on algebraic varieties dover books on mathematics kindle edition by wallace, andrew h download it once and read it on your kindle device, pc, phones or tablets. Weight filtrations in algebraic ktheory 5 we can now describe quillens rst construction of algebraic ktheory for a ring r. The parameter space we will use is the kontsevich moduli space m 0. This homology theory does admit a natural grading whenever there is a small resolution. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. At this stage it is not clear whether it is possible to obtain a natural grading in general.

The picardlefschetz formula and vanishing lattices18. Diophantine geometry, representation theory and homology of the johnson. Review of the birational geometry of curves and surfaces the minimal model program for 3folds towards the minimal model program in higher dimensions the strategy the conjectures of the mmp mild singularities. On the group completion of a simplicial monoid, appendix q in. Prerequisites homology theory of algebraic varieties by. The authors study properties of chow varieties, especially in connection with algebraic correspondences relating algebraic varieties. Action of correspondences on filtrations on cohomology and. The past 20 years have seen a series of new invariants, partly inspired by string theory, such as motivic integration and the elliptic genus of. It is made up mainly from the material in referativnyi zhurnal matematika during 19651973 and is devoted to the geometric aspects of the theory of algebraic varieties.

Thanks for contributing an answer to mathematics stack exchange. Geometry of rational curves on algebraic varieties matthew f. Intersection homology theory and mixed hodge theory are model examples of such invariants. There is an analogy between the symmetric group n on nletters and the general linear group gln. Topology of singular algebraic varieties ucla math. Let x be a smooth complex projective algebraic variety, y a normal crossing divisor in x, and u a tubular neighbourhood of y in x. Weight filtration on the cohomology of algebraic varieties. In particular, if x is a complex algebraic variety, the weight filtration is well defined on hnx. Quillens work in algebraic ktheory journal of ktheory. Algebraic varieties are the central objects of study in algebraic geometry. On the weight filtration of the homology of algebraic varieties. Topology of algebraic varieties universiteit utrecht.

We show the existence of a weight filtration on the equivariant homology of real algebraic varieties equipped with a finite group action, by applying group homology to the weight complex of mccrory and parusinski. Pdf remarks on filtrations of the homology of real varieties. Friedlander in recent years, there has been a renewed interest in obtaining invariants for an algebraic variety x using the chow monoid crx of e ective rcycles on x. Note that this use of the word filtration corresponds to our descending filtration.

Pure homology of algebraic varieties sciencedirect. He used it to get a result in representation theory. Appendix q in friedlandermazurs filtrations on the homology of algebraic varieties. To obtain slightly more general results we introduce image homology for noncomplete varieties. We describe some foundational aspects of lawson homology for complex projective algebraic varieties, a. C, and their subspaces known as algebraic varieties. Towards an intersection homology theory for real algebraic. Intersection homology, weight filtration, elliptic genus. Throughout the paper, x is an irreducible algebraic variety of dimension n over the complex numbers.

On thecohomology of algebraic varieties clairevoisin. Given a group and a filtration, there is a natural way to define a topology on, said to be associated to the filtration. The topology of complex projective varieties afler s. We prove that, given a symmetrically distinguished correspondence of a suitable complex abelian variety which includes any abelian variety of dimension at most 5, powers of complex elliptic curves, etc. An algebraic version of lawsons analytic approach was developed by the author in f, permitting a study of projective varieties over.