Products Of Projective Spaces Research Articles

Cobordism classes of fiberings over product of projective spaces

Let RP(k) denotethe k-dimensional real projective space. This article deter mines which cobordism classes are represented by the total space of a fibering with pre scribed base space RP (3) × RP (1), RP (2) × RP (1), RP (2) × RP (1) × RP (1) or RP (3) × RP (2).

The equivariant Chow rings of quot schemes

We give a presentation for the (integral) torus-equivariant Chow ring of the quot scheme, a smooth compactiflcation of the space of rational curves of degree d in the Grassmannian. For this presentation, we reflne Evain's extension of the method of Goresky, Kottwitz, and MacPherson to express the torus-equivariant Chow ring in terms of the torus-flxed points and explicit relations coming from the geometry of families of torus-invariant curves. As part of this calculation, we give a complete description of the torus-invariant curves on the quot scheme and show that each family is a product of projective spaces.

Geometric invariant theory via Cox rings

We consider actions of reductive groups on a variety with finitely generated Cox ring, e.g., the classical case of a diagonal action on a product of projective spaces. Given such an action, we construct via combinatorial data in the Cox ring all maximal open subsets such that the quotient is quasiprojective or embeddable into a toric variety. As applications, we obtain an explicit description of the chamber structure of the linearized ample cone and several Gelfand–MacPherson type correspondences relating quotients by reductive groups to quotients by torus actions. Moreover, our approach provides a general access to the geometry of many of the resulting quotient spaces.

Compact Manifolds Covered by a Torus

Let X be a compact complex manifold which is the image of a complex torus by a holomorphic surjective map A→X. We prove that X is Kähler and that up to a finite étale cover, X is a product of projective spaces by a torus.

Mirror symmetry formulae for the elliptic genus of complete intersections

In this paper we calculate the elliptic genus of certain complete intersections in products of projective spaces. We show that it is equal to the elliptic genus of the Landau-Ginzburg models that are, according to Hori and Vafa, mirror partners of these complete intersections. This provides additional evidence of the validity of their construction.

On the zeta function of a projective complete intersection

We compute a basis for the $p$-adic Dwork cohomology of a smooth complete intersection in projective space over a finite field and use it to give $p$-adic estimates for the action of Frobenius on this cohomology. In particular, we prove that the Newton polygon of the characteristic polynomial of Frobenius lies on or above the associated Hodge polygon. This result was first proved by B. Mazur using crystalline cohomology.

The Gale transform and multi-graded determinantal schemes

Eisenbud and Popescu showed that certain finite determinantal subschemes of projective spaces defined by maximal minors of adjoint matrices of homogeneous linear forms are related by Veronese embeddings and a Gale transform. We extend this result to adjoint matrices of multihomogeneous multilinear forms. The subschemes now lie in products of projective spaces and the Veronese embeddings are replaced with Segre embeddings.

Near-optimal parameterization of the intersection of quadrics: I. The generic algorithm

We present an exact and efficient algorithm for computing a proper parametric representation of the intersection of two quadrics in three-dimensional real space given by implicit equations with rational coefficients. The output functions parameterizing the intersection in projective space are polynomial, whenever it is possible, which is the case when the intersection is not a smooth quartic (for example, a singular quartic, a cubic and a line, and two conics). Furthermore, the parameterization is near-optimal in the sense that the number of distinct square roots appearing in the coefficients of these functions is minimal, except in a small number of well-identified cases where there may be an extra square root. In addition, the algorithm is practical: a complete and efficient C++ implementation is described in Lazard et al. [Lazard, S., Peñaranda, L.M., Petitjean, S., 2006. Intersecting quadrics: An efficient and exact implementation. In: 20th ACM Symposium on Computational Geometry, 2004. Computational Geometry: Theory and Applications 35 (1–2), 74–99 (special issue)]. In Part I, we present an algorithm for computing a parameterization of the intersection of two arbitrary quadrics which we prove to be near-optimal in the generic, smooth quartic, case. Parts II and III treat the singular cases. We present in Part II the first classification of pencils of quadrics according to the real type of the intersection and we show how this classification can be used to efficiently determine the type of the real part of the intersection of two arbitrary quadrics. This classification is at the core of the design of our algorithms for computing near-optimal parameterizations of the real part of the intersection in all singular cases. We present these algorithms in Part III and give examples covering all the possible situations in terms of both the real type of intersection and the number and depth of square roots appearing in the coefficients.

A Characterization of Products of Projective Spaces

AbstractWe give a characterization of products of projective spaces using unsplit covering families of rational curves.

Residual intersection theory with reducible schemes

We develop a formula (Theorem 5.2) which allows to compute top Chern classes of vector bundles on the vanishing locus V(s) of a section of this bundle. This formula particularly applies in the case when V(s) is the union of local complete intersections giving the individual contribution of each component and their mutual intersections. We conclude with applications to the enumeration of rational curves in complete intersections in projective space.

Projective Bundles on Infinite-Dimensional Complex Spaces

Abstract Let V be a complex localizing Banach space with countable unconditional basis and E a rank r holomorphic vector bundle on P(V). Here we study the holomorphic embeddings of P(E) into products of projective spaces and the holomorphic line bundles on P(E). In particular we prove that if r ≥ 3, then H 1(P(E), L) = 0 for every holomorphic line bundle L on P(E).

Stabilizers for Nondegenerate Matrices of Boundary Format and Steiner Bundles

In this paper nondegenerate multidimensional matrices of boundary format in V0• • •Vp are investigated by their link with Steiner vector bundles on product of projective spaces. For any nondegenerate matrix A the stabilizer for the SL(V0) × • • • × SL(Vp)-action, Stab(A), is completely described. In particular we prove that there exists an explicit action of SL(2) on V0 • • •Vp such that Stab(A)0 _ SL(2) and the equality holds if and only if A belongs to a unique SL(V0) × • • • × SL(Vp)-orbit containing the identity matrices, according to [1].

Automorphisms of Fermat-like varieties

We study the automorphisms of some nice hypersurfaces and complete intersections in projective space by reducing the problem to the determination of the linear automorphisms of the ambient space that leave the algebraic set invariant.

Semi-topological K-theory using function complexes

The semi-topological K-theory K ∗ semi (X) of a quasi-projective complex algebraic variety X is based on the notion of algebraic vector bundles modulo algebraic equivalence. This theory is given as the homotopy groups of an infinite loop space K semi (X) which is equipped with maps K alg (X)→ K semi (X) , K semi (X)→ K top (X an ) whose composition is the natural map from the algebraic K-theory of X to the topological K-theory of the underlying analytic space X an of X. The theory K semi (X) defined and studied here is equivalent (when X is projective and weakly normal) to the so-called “holomorphic K-theory”, K hol (X) , of projective varieties, which is studied by Cohen and Lima-Filho. We give an explicit description of K 0 semi( X) in terms of K 0( X), a description of K q semi(−) in terms of K 0 semi(−) for projective varieties, a Poincaré duality theorem for projective varieties, and a computation of K semi (X) whenever X is a product of projective spaces or a smooth complete curve. For X a smooth quasi-projective variety, there are natural Chern class maps from K ∗ semi (X) to morphic cohomology compatible with similarly defined Chern class maps from algebraic K-theory to motivic cohomology and compatible with the classical Chern class maps from topological K-theory to the singular cohomology of X an.

Toric Residues and Mirror Symmetry

We develop some ideas of Morrison and Plesser and formulate a precise mathematical conjecture which has close relations to toric mirror symmetry. Our conjecture, we call it Toric Residue Mirror Conjecture, claims that the generating functions of intersection numbers of divisors on a special sequence of simplicial toric varieties are power series expansions of some rational functions obtained as toric residues. We expect that this conjecture holds true for all Gorenstein toric Fano varieties associated with reflexive polytopes and give some evidences for that. The proposed conjecture suggests a simple method for computing Yukawa couplings for toric mirror Calabi-Yau hypersurfaces without solving systems of differential equations. We make several explicit computations for Calabi-Yau hypersurfaces in weighted projective spaces and in products of projective spaces.

Effective bounds for Hodge-theoretic connectivity

We prove an effective version of Nori’s connectivity theorem using Koszul cohomology computations. We apply this result to study the cycle class, Abel-Jacobi and regulator maps and the nonvanishing of certain Griffiths groups for complete intersections in projective spaces, abelian varieties and quadrics.

Riemann-Roch variations

We construct a mirror-type correspondence that assigns variations (that is, local systems, -modules or -adic sheaves) to pairs , where is a variety and is a complex of densely filtered vector bundles over . We consider Calabi-Yau complete intersections in projective spaces. In the particular case when the complex is quasi-isomorphic to the tangent bundle on a generic Calabi-Yau complete intersection, this construction yields the variation that arises in the relative cohomology of the mirror-dual pencil. We call it the Riemann-Roch variation. The Riemann-Roch data of the divisorial sublattice in the -group can be read off the Riemann-Roch local system since it encodes the information about the Euler characteristics of all sheaves (in an essentially non-commutative way).

Einstein–Kähler metrics on certain complex bundles

We study a class of compact complex manifolds, with positive first Chern class, fibered over products of projective spaces. We prove that these bundles carry Einstein–Kähler metrics when the projective spaces of the basis have the same dimension. When this dimensional condition is not satisfied, we estimate their Ricci tensor in another paper.

On Ricci curvature of certain complex bundles

We study a class of compact complex manifolds, with positive first Chern class C 1 , fibered over products of projective spaces. When the spaces of the basis don't have the same dimension, we know that these bundles cannot carry Einstein–Kähler metrics. In this article, we construct a metric belonging to C 1 , with an optimal lower bound of the Ricci curvature.

D‐Affinity and Toric Varieties

1991 Mathematics Subject Classification 32C38.