Zerocycles on smooth complex projective varieties and global holomorphic forms 5 3. Pdf lawson homology for varieties with small chow groups. Chow groups murre june 28, 2010 1 murre 1 chow groups conventions. We investigate the chow groups of projective determinantal varieties and those of their strata of matrices of xed rank, using chern class computations. Projective invariants of projective structures 527 3 if g acts properly on v, and if a quasi projective orbit space vg exists, then for some projective embedding fcp every point of v is stable. The problem of computing the homotopy of real projective space therefore reduces to the problem of computing the homotopy of spheres. For rdimensional subvarieties of ndimensional projective space, the author proves that one can define the 2rdimensional volume of the variety, and this is accomplished via a riemannian metric on ndimensional projective space. The chow groups are defined by taking groups of cycles modulo rationally trivial cycles. Lawson homology for varieties with small chow groups and the induced filtration on the griffiths groups article pdf available in mathematische zeitschrift 2342. Curves of low degrees on projective varieties olivier debarre we work over the eld of complex numbers. Request pdf a filtration on the chow groups of a complex projective variety let x c be a projective algebraic manifold, and further let ch k x q be the chow group of codimension k. Cycles of codimension 3 on a projective quadric springerlink.
We observe that linear relations among chernmather classes of projective varieties are preserved by projective duality. Projective invariants of projective structures 527 3 if g acts properly on v, and if a quasiprojective orbit space vg exists, then for some projective embedding fcp every point of v is stable. Chow groups modulo 2 by burt totaro abstract for a very general principally polarized complex abelian 3fold, the chow group of algebraic cycles is in nite modulo every prime number. One then has a cycle class map to etale cohomology over the base field, and for a. Chow groups of quadrics and index reduction formula. We deduce the existence of an explicit involution on a part of the chow group of projective space, encoding the effect of duality on chernmather classes. Voisins conjecture for zerocycles on calabiyau varieties and. The space is a onepoint space and all its homotopy groups are trivial groups, and the set of path components is a onepoint space the case. This is a result that was originally proved by bertin and elencwajg. Both methods have their importance, but thesecond is more natural. Request pdf a filtration on the chow groups of a complex projective variety let x c be a projective algebraic manifold, and further let ch k x q be the chow group of codimension k algebraic. It is a compacti cation of the con guration space bx.
This article discusses a common choice of cw structure for real projective space, i. This article describes the homotopy groups of the real projective space. In particular, we nd an explicit formula for the brauer group of fourfolds bered in quadrics of dimension 2 over a rational surface. Projective duality and a chernmather involution core. In this paper, the chow groups of projective hypersurfaces are studied. All chow groups will be with rational coefficients. The chow associated forms give a description of the moduli space of the algebraic varieties in projective space. The chow group of x is similar to the total singular homology group of a.
By appointment, in 380383m third floor of the math building. It is proved that torsion in ch 3 x is either trivial or is a second order group. A note on the chow groups of projective determinantal varieties appendix to \a cascade of determinantal calabiyau threefolds by g. Chow groups of some generically twisted flag varieties. Prelog chow groups of selfproducts of degenerations of. This approach is introduced and developed by edidingraham and totaro. A filtration on the chow groups of a complex projective. The elements of the chow group are formed out of subvarieties socalled algebraic cycles in a similar way to how simplicial or cellular homology groups are formed out of subcomplexes. It gives a beautiful solution of an important problem. A filtration on the chow groups of a complex projective variety. These papers are based on the observation that if a smooth projective variety is stably rational, then its chow group of 0cycles is universally trivial, meaning that ch 0 does not increase when. Finally, if g is any reductive algebraic group in characteristic 0, then i can analyze the manner in which stability breaks down in the following way.
The chow ring has many advantages and is widely used. The projective space pn thus contains more points than the a. The complex projective line cp1 for purposes of complex analysis, a better description of a onepoint compacti cation of c is an instance of the complex projective space cpn, a compact space containing cn, described as follows. We show that the cycle map of the higher chow group to deligne cohomology is injective and the higher hodge. By analogy, whereas the points of a real projective space label the lines through the origin of a real euclidean space, the points of a complex projective space label the complex lines through the origin of a complex euclidean space see below for an intuitive account. Riemann sphere, projective space november 22, 2014 2. Schoen gave the rst examples of smooth complex projective. Friedlander in the monograph fm1, the author and barry mazur introduce a ltration on.
The arakelov chow group of xw denoted chx is the group of equivalence classes generated by pairs y, g. We find an explicit formula for the map p that assigns to a generic point x. The properties hausdorff quotient topology and proper action are equivalently characterized by the closure of the. Then z qx zix fzjz p n w irreducible codimension ivarietiesgbe. It is proved that torsion in ch3x is either trivial or is a second order group. The chow group chlx is the group of algebraic cycles of dimension lon xwith rational coe. We want to explain how theses spaces can be used, in very speci c cases, to study the geometry of x. More generally, the grassmannian gk, v of a vector space v over a field f is the moduli space. His theorem that a compact analytic variety in a projective space is algebraic is justly famous. We say a fibration of smooth projective varieties is chow constant if pushforward induces an. Then, we will give an example in which the formal tangent space is in nite dimensional. X the chow point of the closure of the orbit through x. X in the chow group of a smooth, projective variety x, we can find. On the arakelov chow group of arithmetic abelian schemes and other spaces with symmetries by eitan bachmat submitted to the department of mathematics on march 10, 1994 in partial fullfilment of the requirements for the degree of doctor of science in mathematics abstract we construct a fourier transform for arakelov chow groups of arithmetic.
Suppose x is a nonsingular projective variety over c of dimension d. Several notions, describing how close xis to projective space, have been developed. In particular, this gives the rst examples of complex varieties with in nite chow groups modulo 2. Chow groups of projective varieties of very small degree unidue.
Examples lines are hyperplanes of p2 and they form a projective space of dimension 2. Algebra and geometry through projective spaces sf2724 topics in mathematics iv spring 2015, 7. The chow group of zerocycles on x maps onto the integers by the degree homomorphism. Let g be a reductive complex linear algebraic group. Preliminaries schemes are of nite type over a eld k. If a stack x can be written as the quotient stack for some quasi projective variety y with a linearized action of a linear algebraic group g, then the chow group of x is defined as the gequivariant chow group of y. We will prove that if the degree of the hypersurface is su ciently high, its chow group is \small in the sense that its formal tangent space vanishes. For example, let x be a smooth complex projective surface. Hilbx be the quasiprojective moduli space of smooth, genusg, degreedcurves on x. For which varieties is the natural map from the chow ring. On the arakelov chow group of arithmetic abelian schemes and other spaces with symmetries by.
This metric only depends on r and the degree of the variety, and is in fact the famous kahler metric. Let x be a nonsingular quadratic hypersurface in a projective space over an arbitrary field of characteristic not two and let ch p x be a chow group of codimension p, that is, a group of classes of codimension p cycles on x with respect to rational equivalency. There is also related work by kaneyama, klyachko, and iltensuss. Rational, unirational and stably rational varieties.
The latter can be described as a quotient gp, where gis a semisimple algebraic group and pa parabolic subgroup. The universal regular quotient of the chow group of points on. The integral cohomology of the hilbert scheme of two points burt totaro for a complex manifold xand a natural number a, the hilbert scheme xa also called the douady space is the space of 0dimensional subschemes of degree ain x. W has a natural structure of projective space and its dimension is given by the. On the arakelov chow group of arithmetic abelian schemes and. Relative chow correspondences and the griffiths group eric m.
Introduction to intersection theory in algebraic geometry lectures. On the geometry of algebraic homogeneous spaces the. This includes the set of path components, the fundamental group, and all the higher homotopy groups the case. Let x be a smooth projective complex variety of dimension n. We will now investigate these additional points in detail. In mathematics, complex projective space is the projective space with respect to the field of complex numbers. The integral cohomology of the hilbert scheme of two points.
A category m of chow motives is constructed in mu2. In algebraic geometry, the chow groups named after weiliang chow by claude chevalley of an algebraic variety over any field are algebrogeometric analogs of the homology of a topological space. Hodgetype conjecture for higher chow groups morihiko saito dedicated to professor friedrich hirzebruch abstract. Let x be a nonsingular quadratic hypersurface in a projective space over an arbitrary field of characteristic not two and let chpx be a chow group of codimension p, that is, a group of classes of codimension p cycles on x with respect to rational equivalency.
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