Projective Space Bundle Research Articles

Free rank of symmetry of products of Dold manifolds

AbstractDold manifolds $P(m,n)$ are certain twisted complex projective space bundles over real projective spaces and serve as generators for the unoriented cobordism algebra of smooth manifolds. The paper investigates the structure of finite groups that act freely on products of Dold manifolds. It is proved that if a finite group G acts freely and $ \mathbb{Z}_2 $ cohomologically trivially on a finite CW-complex homotopy equivalent to ${\prod_{i=1}^{k} P(2m_i,n_i)}$, then $G\cong (\mathbb{Z}_2)^l$ for some $l\leq k$ (see Theorem A for the exact bound). We also determine some bounds in the case when for each i, ni is even and mi is arbitrary. As a consequence, the free rank of symmetry of these manifolds is determined for cohomologically trivial actions.

Simple fibrations in -surfaces

Abstract We introduce the notion of a simple fibration in $(1,2)$ -surfaces – that is, a hypersurface inside a certain weighted projective space bundle over a curve such that the general fibre is a minimal surface of general type with $p_g=2$ and $K^2=1$ . We prove that almost all Gorenstein simple fibrations over the projective line with at worst canonical singularities are canonical threefolds ‘on the Noether line’ with $K^3=\frac 43 p_g-\frac {10}3$ , and we classify them. Among them, we find all the canonical threefolds on the Noether line that have previously appeared in the literature. The Gorenstein simple fibrations over ${\mathbb {P}}^1$ are Cartier divisors in a toric $4$ -fold. This allows to us to show, among other things, that the previously known canonical threefolds on the Noether line form an open subset of the moduli space of canonical threefolds, that the general element of this component is a Mori Dream Space and that there is a second component when the geometric genus is congruent to $6$ modulo $8$ ; the threefolds in this component are new.

Toric Fano contractions associated to long extremal rays

We show that a toric Fano contraction associated to an extremal ray whose length is greater than the dimension of its fiber is a projective space bundle.

Kähler manifolds with geodesic holomorphic gradients

We prove a dichotomy theorem about compact Kähler manifolds admitting nontrivial real-holomorphic geodesic gradient vector fields, which has the following consequence: either such a manifold satisfies an additional integrability condition, or through every zero of the real-holomorphic geodesic gradient there passes an uncountable family of totally geodesic, holomorphically immersed complex projective spaces, each carrying a fixed multiple of the Fubini–Study metric. We also obtain a classification result for the case where the integrability condition holds, implying that the manifold must then be biholomorphically isometric to a bundle of complex projective spaces with a bundle-like metric.

Characterization of generic projective space bundles and algebraicity of foliations

In this paper we consider various notions of positivity for distributions on complex projective manifolds. We start by analyzing distributions having big slope with respect to curve classes, obtaining characterizations of generic projective space bundles in terms of movable curve classes. We then apply this result to investigate algebraicity of leaves of foliations, providing a lower bound for the algebraic rank of a foliation in terms of invariants measuring positivity. We classify foliations attaining this bound, and describe those whose algebraic rank slightly exceeds this bound.

Mean curvature flows in manifolds of special holonomy

We study the uniqueness of minimal submanifolds and the stability of the mean curvature flow in several well-known model spaces of manifolds of special holonomy. These include the Stenzel metric on the cotangent bundle of spheres, the Calabi metric on the cotangent bundle of complex projective spaces, and the Bryant--Salamon metrics on vector bundles over certain Einstein manifolds. In particular, we show that the zero sections, as calibrated submanifolds with respect to their respective ambient metrics, are unique among compact minimal submanifolds and are dynamically stable under the mean curvature flow. The proof relies on intricate interconnections of the Ricci flatness of the ambient space and the extrinsic geometry of the calibrated submanifolds.

Looijenga’s weighted projective space, Tate’s algorithm and Mordell-Weil lattice in F-theory and heterotic string theory

It is now well known that the moduli space of a vector bundle for heterotic string compactifications to four dimensions is parameterized by a set of sections of a weighted projective space bundle of a particular kind, known as Looijenga’s weighted projective space bundle. We show that the requisite weighted projective spaces and the Weierstrass equations describing the spectral covers for gauge groups E N (N = 4, · · · , 8) and SU(n + 1) (n = 1, 2, 3) can be obtained systematically by a series of blowing-up procedures according to Tate’s algorithm, thereby the sections of correct line bundles claimed to arise by Looijenga’s theorem can be automatically obtained. They are nothing but the four-dimensional analogue of the set of independent polynomials in the six-dimensional F-theory parameterizing the complex structure, which is further confirmed in the constructions of D 4, A 5, D 6, E 3 and SU(2) × SU(2) bundles. We also explain why we can obtain them in this way by using the structure theorem of the Mordell-Weil lattice, which is also useful for understanding the relation between the singularity and the occurrence of chiral matter in F-theory.

Splitting vector bundles and A1 -fundamental groups of higher-dimensional varieties

We study aspects of the A^1-homotopy classification problem in dimensions >= 3 and, to this end, we investigate the problem of computing A^1-homotopy groups of some A^1-connected smooth varieties of dimension >=. Using these computations, we construct pairs of A^1-connected smooth proper varieties all of whose A^1-homotopy groups are abstractly isomorphic, yet which are not A^1-weakly equivalent. The examples come from pairs of Zariski locally trivial projective space bundles over projective spaces and are of the smallest possible dimension. Projectivizations of vector bundles give rise to A^1-fiber sequences, and when the base of the fibration is an A^1-connected smooth variety, the associated long exact sequence of A^1-homotopy groups can be analyzed in detail. In the case of the projectivization of a rank 2 vector bundle, the structure of the A^1-fundamental group depends on the splitting behavior of the vector bundle via a certain obstruction class. For projective bundles of vector bundles of rank >=, the A^1-fundamental group is insensitive to the splitting behavior of the vector bundle, but the structure of higher A^1-homotopy groups is influenced by an appropriately defined higher obstruction class.

Certain algebraic surfaces with Eisenbud‐Harris general fibration of genus 4

AbstractWe classify the algebraic surfaces with Eisenbud‐Harris general fibration of genus 4 over a rational curve or an elliptic curve whose slope attains the lower bound. The classification of our surfaces is strongly related to the result of the classification for certain relative quadric hypersurfaces in 3‐dimensional projective space bundles over a rational curve and an elliptic curve. We further prove some results about the canonical maps, the quadric hulls of the canonical images and the deformation for these surfaces.

The Faddeev–Mickelsson–Shatashvili Anomaly and Lifting Bundle Gerbes

In gauge theory, the Faddeev–Mickelsson–Shatashvili anomaly arises as a prolongation problem for the action of the gauge group on a bundle of projective Fock spaces. In this paper, we study this anomaly from the point of view of bundle gerbes and give several equivalent descriptions of the obstruction. These include lifting bundle gerbes with non-trivial structure group bundle and bundle gerbes related to the caloron correspondence.

Certain relative quadric hypersurfaces in 3-dimensional projective space bundles over a projective curve

We classify the certain type of relative quadric hypersurfaces of 3-dimensional projective space bundles over a projective line or an elliptic curve whose fiber is the direct product of 2 projective lines.

Extension of twisted Hodge metrics for Kähler morphisms

Let f : X --> Y be a holomorphic map of complex manifolds, which is proper, Kahler, and surjective with connected fibers, and which is smooth over Y-Z the complement of an analytic subset Z. Let E be a Nakano semi-positive vector bundle on X, and consider direct image sheaves F = R^qf_*(K_{X/Y} \otimes E) for q \geq 0. In our previous paper, we discussed the Nakano semi-positivity of F with respect to the so-called Hodge metric, when the map f is smooth. In this paper, we discuss the extension of the induced metric on the tautological line bundle O(1) on the projective space bundle P(F) ``over Y-Z'' as a singular Hermitian metric with semi-positive curvature ``over Y''. As a particular consequence, if Y is projective, R^qf_*(K_{X/Y} \otimes E) is weakly positive over Y-Z in the sense of Viehweg.

The Grothendieck group of a quantum projective space bundle

We compute the Grothendieck group K_0 of non-commutative analogues of quantum projective space bundles. Our results specialize to give the Grothendieck groups of non-commutative analogues of projective spaces, and specialize to recover the Grothendieck group of a usual projective space bundle over a regular noetherian separated scheme. As an application we develop an intersection theory for the quantum ruled surfaces defined by Van den Bergh.

The index and co-index of the twisted tangent bundle over projective spaces

We determine the index and co-index of the twisted tangent bundle of projective spaces. We also discuss the stabilty of them, and determine the set of integers that can be realized as the stable co-index of a vector bundle over the projective space.

Characterization theorems for the projective space and vector bundle adjunction

We consider some conditions under which a smooth projective variety X is actually the projective space. We also extend to the case of positive characteristic some results in the theory of vector bundle adjunction. We use methods and techniques of the so called Mori theory, in particular the study of rational curves on projective manifolds.

The cobordism classification of projective space bundles

We show that the cobordism class of E determines the smooth fibering RP2n→E→RP2m.

Behavior of rational curves of the minimal degree in projective space bundle in any characteristic

Behavior of rational curves of the minimal degree in projective space bundle in any characteristic

Euler and Pontrjagin currents of a section of a compactified real bundle

The canonical family of Thom forms τ s for 0 < s < ∞ constructed by Harvey and Lawson on an oriented real vector bundle V → X with metric connection D v is shown to have an extension to the bundle of real projective spaces, P( R ⊕ V) → X , which compactifies V in the fibre directions. The current limit as r → ∞ and s → 0 of the smooth transgression formula τ r − τ s = dσ r, s is the current equation x ̃ (D v) + [P(V)] Res − [X] = dσ ∞,0 on P( R ⊕ V) . Here σ ∞,0 is an L loc 1 current on P( R ⊕ V) and [ X] is the current of integration over the zero section of V. If the rank of V is even, x ̃ (D v) is the twisted extension to P( R ⊕ V) of the Chern-Euler form of the connection D v on V, [ P( V)] integrates densities over the nonorientable submanifold P( V) of P( R ⊕ V) and Res is a smooth twisted form on P( V). If the rank of V is odd, x ̃ (D v) ≡ 0 , [ P( V)] is the current of integration over P( V) and Res is a smooth form on P( V). This construction can be pulled back to X by an atomic section ν:X → P( R ⊕ V) satisfying an appropriate orientability hypothesis to obtain a current equation on X relating the zero and pole sets of ν to the topology of the bundle. Applications to the differential topology of maps into spheres and real projective spaces are discussed.

A Kähler structure on the punctured cotangent bundle of complex and quaternion projective spaces and its application to a geometric quantization II

A Kähler structure on the punctured cotangent bundle of complex and quaternion projective spaces and its application to a geometric quantization II

A Kähler structure on the punctured cotangent bundle of complex and quaternion projective spaces and its application to a geometric quantization I

A Kähler structure on the punctured cotangent bundle of complex and quaternion projective spaces and its application to a geometric quantization I