Vector fields and the cohomology ring of toric varieties. Section 4 of discriminant complements and kernels of monodromy representations. Using this we can compute the ring structure on h dr pn. In par ticular, we make explicit the algebra hxs when x is a suspended space, a complex projective space or a finite cw complex of dimension p such that t. Introduction let c be a complete smooth curve of genus g. For example, in 16 the equivariant cohomology ring h. The reduced integral homology h cp is wellknown to be the polynomial ring. I n 0 in this note we compute the rational cohomology ring of sp 3z,or equivalently, of a 3,the moduli space of. These generators are subject to a set of relations, which defines the ring. Cohomology of twisted projective spaces and lens complexes. The important role of the steenrod operations sqi in the description of the cohomology of.
The equivariant cohomology ring of weighted projective space. Denote by sp nr the group of automorphisms of r2n that preserve the unimodular alternating form given by the matrix 0 i n. In the second part, we give a natural cbasis of the orbifold cohomology. Examples of such invariants include homology, cohomology, and the euler characteristic. For a general reduced weighted projective space, we give a formula to compute the 3point function which is the key in the definition of chenruan cohomology ring. The cohomology of the total space of a fiber bundle is a module over the cohomology of the base space by pulling back an element and cupping. In such cases, we can say that the chow ring of bg is. Recall an explicit basis for the cohomology of p2n due to mallavibarrena and sols.
The latter statement means that if xis a complex vector bundle of dimension nthen we are given a class u u. The chenruan cohomology of weighted projective spaces. To understand this we need to know what a representation of gis. We describe the integral equivariant cohomology ring of a weighted projective space in terms of piecewise polynomials, and thence by generators and relations. Usingexcessive intersection theory, we compute the leading coe cients in the relations among the generators of the quantum cohomology ring structure. A constructive new proof of the bott formula is given by. Xcommutes with the involution and the map of bers e x. The same goes for any other coe cient ring considered as a local free sheaf. The lerayhirsch theorem is a theorem about what conditions are necessary on a fiber bundle to have this module be free i. The roggraded equivariant ordinary cohomology of complex projective spaces with linear 2p actions l. In this paper we determine the integral cohomology ring structures of the twisted projective space and the lens complex.
Noetherian ring and let ox1 be a very ample line bundle on x. Cohomology of projective space seen by residual complex. Transformation groups on cohomology protective spacesi. Borel construction, configuration space, integral cohomology ring. In particular, batyrevs conjectural formula for quantum cohomology of projective bundles associated. In particular, all of the integral cohomology is at even degree as in the case of a projective space. The ring structure of the complex projective space is brie y described in the following section,2. However the equivariant cohomology is often easier to understand as a consequence of the localization theorem 3. The equivariant cohomology of weighted projective space. The multiplicative structure of the cohomology ring of the moduli space of stable rank 2 bundles on a smooth projective curve is computed. The stabilization of the cohomology of moduli spaces of sheaves.
The rearrangement includes a renumbering according to the following scheme. The only ring automorphisms of arising from selfhomeomorphisms of the complex projective space are the identity map and the automorphism that acts as the negation map on and induces corresponding. A projective space rp1 is homeomorphic to the circle s1. If no coefficient groups are specified for homology or cohomology groups, the coefficients are assumed to be some commutative ring with unit. Then the only job is computing the sheaf cohomology of lf o xm for any integer m. We deduce that the ring is a perfect invariant, and prove a chern class formula for weighted projective bundles. Let x be a projective variety over a noetherian ring and let o x1 be a very ample line bundle on x. Jul 28, 2011 in particular, is identified with a generator for the top cohomology, or a fundamental class in cohomology.
If xis a simplicial complex or a cwcomplex then hix. The cohomology ring of the moduli space of stable vector. In mathematics, the grassmannian grk, v is a space that parameterizes all kdimensional linear subspaces of the ndimensional vector space v. From the results of carrell and lieberman there exists a. On the symmetric squares of complex and quaternionic projective. Over a regular scheme s such as a smooth variety, every projective bundle is of the form for some vector bundle locally free sheaf e. Sep 30, 2011 odddimensional projective space with coefficients in integers. In this note, for a smooth projective toric variety and a vector. A special class of z2 equivariant cohomology theories are called realoriented theories. Orbifold cohomology ring of weighted projective spaces in this section, we will describe explicitly the orbifold cohomology ring of weighted projective spaces. If, then is the collection of polynomials of degree if, and otherwise. Ellingsrud and stromme computed betti numbers of p2n.
If x is a cw complex with cells only in even dimensions and r is a ring, then, by an elementary result in cellular cohomology theory, the ordinary. In particular, batyrevs conjectural formula for quantum cohomology of projective. In general, g ottsche computed the betti numbers of xn. Each basis element depends on 3 partitions, whose total sum is n.
The cohomology of projective space climbing mount bourbaki. Line bundles on projective space daniel litt we wish to show that any line bundle over pn k is isomorphic to om for some m. The moduli space is isomorphic to picx xn for some n. It is clear from the computations in the proof of lemma 30. Then the cohomology ring h pe is an algebra over h x through the pullback p. C of an equivariantly formal space x was described in terms of the. Pdf the quantum orbifold cohomology of weighted projective. Odddimensional projective space with coefficients in an abelian group. Pdf equivariant structure constants for ordinary and. The original reference for this material is ega iii, but most graduate students would prob ably encounter it in hartshornes book har77 where many proofs are given only for noetherian schemes, probably because the only known proofs in the general. In mathematics, a projective bundle is a fiber bundle whose fibers are projective spaces by definition, a scheme x over a noetherian scheme s is a p nbundle if it is locally a projective n space. In many cases where we compute the chow ring of bg, it maps injectively to the integral cohomology of bg. Our main result is an explicit connection between this. The following 25 pages are a revision of the published version of 3.
Weighted projective space is a toric delignemumford stack this is spelled out in so one can compute the orbifold cohomology ring using results of borisovchensmith 14. Later in this course we will see a shorter proof of this theorem using poincar e duality. For the primitive part, there is an extension by carlson and toledo of the griffiths residue calculus to complete intersections in weighted projective space, cf. The chow ring of a classifying space ucla mathematics. Cech cohomology and use it to calculate cohomology of projective space. The stabilization of the cohomology of moduli spaces of. The cohomology of the line bundle on projective space is as follows. Let x be a complex smooth projective variety and e a complex vector bundle of rank r on it. Az, the ring of cvalued functions on z, such that gr az. A subcomplex of a residual complex on projective space is con structed for computing the cohomology modules of locally free sheaves.
We start with the real projective spaces rpn, which we think of as ob. Chenruan orbifold cohomology ring of weighted projective space, which is obtained from the quantum cohomology ring by setting q 0, has been studied by a number of authors. R r0 is a ring homomorphism then there is a natural transformation of graded ring valued functors extending which agrees with when evaluated on a point. It is the same thing as zgmodule, but for this we need to know what the group ring zgis, so some preparation is required. We do not assume kalgebraically closed since the most interesting case of this theorem is the case k q. Weighted projective space is a toric delignemumford stack this is spelled out in so one can compute the orbifold cohomology ring using. Complex oriented cohomology theories a complex oriented cohomology theory is a generalized cohomology theory ewhich is multiplicative and has a choice of thom class for every complex vector bundle. The main change is a rearrangement of the material into an order that may be slightly more e.
For example, the grassmannian gr1, v is the space of lines through the origin in v, so it is the same as the projective space of one dimension lower than v when v is a real or complex vector space, grassmannians are compact smooth manifolds. Pn has a natural basis of scalar mul tiples of the ordinary schubert classes in ht. The defining relations are easy to express for a larger set of generators, which consists of the chern classes of e and f. Let x be a smooth complex projective variety with a holomorphic vector. We describe a few below, starting with those that require the least additional structure on a. If, then is zero for, but for other is free on the set of negative monomials for. By general facts in representation theory, we have lf s xk o x1 where s is schur functor. X the graded abelian group of singular chains on x, so snx is the free abelian group generated by the singular nsimplices. Cohomology of projective space let us calculate the cohomology of projective space. In many cases where we compute the chow ring of bg, it maps injectively to. One can also apply the methods of chenhu 15, goldinholmknutson 28, or jiang 38. Consider the moduli space n of stable rank 2 vector bundles on c with.
I n 0 in this note we compute the rational cohomology ring of sp 3z,or equivalently, of a 3,the moduli space of principally polarized abelian 3folds. Rpn and all coe ecients for the cohomology groups are z2z coe cients. From the above theorem, one way to compute local cohomology of l is considering its shea ed version, lf on projective space pn k. As an application of general techniques of cohomology we prove the grothendieck and serre vanishing theorems. When xis projective, in addition to being symplectic, we conjecture that there is furthermore an algebra isomorphism hh xg. The integral cohomology ring of the complement of an arrangement of linear subspaces of a. Cohomology of projective space let us calculate the. We then compute its homology groups in terms of those of the complement to the hypersurface, and make a remark on the homotopy groups. Evendimensional projective space with coefficients in integers.
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