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

We 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.     

Certaines formes quadratiques de dimensin au plus 6 et corps des fonctions en caractéristique 2

LetF be a field of characteristic 2. The aim of this paper is to study the isotropy of someF-quadratic forms of dimension ≤6 over the function field of a projective quadric.      

Homogeneous almost complex structures in dimension 6 with semi-simple isotropy

We classify invariant almost complex structures on homogeneous manifolds of dimension 6 with semi-simple isotropy. Those with non-degenerate Nijenhuis tensor have the automorphism group of dimension either 14 or 9. An invariant almost complex structure with semi-simple isotropy is necessarily either of specified 6 homogeneous types or a left-invariant structure on a Lie group. For integrable invariant almost complex structures we classify all compatible invariant Hermitian structures on these homogeneous manifolds, indicate their integrability properties (Kähler, SNK, SKT) and mark the other interesting geometric properties (including the Gray-Hervella type).     

Ovoids and monomial ovals

This paper is a contribution to the classification of ovoids. We show, under some rather technical assumptions, that if an ovoid of PG(3, q) has a pencil of monomial ovals, then it is either an elliptic quadric or a Tits ovoid. Further, we show that if an ovoid of PG(3, q) has a bundle of translation ovals, again under some extra assumptions, then the ovoid is an elliptic quadric or a Tits ovoid.     

On Ovoids of Parabolic Quadrics

It is known that every ovoid of the parabolic quadric Q(4, q), q=p ^h , p prime, intersects every three-dimensional elliptic quadric in 1 mod p points. We present a new approach which gives us a second proof of this result and, in the case when p=2, allows us to prove that every ovoid of Q(4, q) either intersects all the three-dimensional elliptic quadrics in 1 mod 4 points or intersects all the three-dimensional elliptic quadrics in 3 mod 4 points. We also prove that every ovoid of Q(4, q), q prime, is an elliptic quadric. This theorem has several applications, one of which is the non-existence of ovoids of Q(6, q), q prime, q>3. We conclude with a 1 mod p result for ovoids of Q(6, q), q=p ^h , p prime.     

On the intersection of ovoids sharing a polarity

In this article, an ovoidal fibration is used to show that any two ovoids of PG(3, q), q even, sharing a polarity, must meet in an odd number of points. This result was previously known only when one of the ovoids was an elliptic quadric or a Tits ovoid. It is also shown that an ovoid and an elliptic quadric of PG(3, q), sharing all of their tangents, must meet in 1 (mod 4) points.      

Linearizable circle diffeomorphisms in one-parameter families

We consider smooth families of diffeomorphisms of the circle. We prove that the set of parameter values which correspond to non-linearizable maps with irrational rotation numbers is of Hausdorff dimension 0.Partially supported by Polish KBN grant No 2109001     

Embedding a quantum rank three quadric in a quantum P3

We continue the classification, begun in [11], [14] and [12], of quadratic Artin-Schelter regular algebras of global dimension 4 which map onto a twisted homogeneous coordinate ring of a quadric hypersurfcice in P³. In this paper, we consider those cases where the quadric has rank 3. We also give sufficient conditions for the point scheme of any quadratic regular algebra of global dimension 4 to be the graph of an automorphism.     

Elliptic points of the Picard modular group

We explicitly compute the elliptic points and isotropy groups for the action of the Picard modular group over the Gaussian integers on 2-dimensional complex hyperbolic space.      

Fundamental Solutions to □ b on Certain Quadrics

The purpose of this article is to expand the number of examples for which the complex Green operator, that is, the fundamental solution to the Kohn Laplacian, can be computed. We use the Lie group structure of quadric submanifolds of ? ⁿ ×? ^m and the group Fourier transform to reduce the □ _b equation to ones that can be solved using modified Hermite functions. We use Mehler’s formula and investigate (1) quadric hypersurfaces, where the eigenvalues of the Levi form are not identical (including possibly zero eigenvalues), and (2) the canonical quadrics in ?⁴ of codimension two.     

Real and Rational Forms of Certain O<Subscript>2</Subscript>(ℂ)-actions,and a Solution to the Weak Complexification Problem

The main result of this paper is that there is a non-linearizable real algebraic action of the circle S¹ on ⁴, an action which becomes linearizable over . This solves the Weak Complexification Problem. We also show that for any field k of characteristic zero, there are non-linearizable algebraic actions of the group O₂(k) on four-dimensional affine k-space, and if k contains a square root of 3, then this action restricts to a non-linearizable action of the symmetric group S₃ on four-dimensional affine k-space.     

Diffeomorphism classification of finite group actions on disks

In this paper we discuss the diffeomorphism classification of finite group actions on disks. We answer the question when an action on a space M can be extended to an action on a disk such that the action is free away from M. Let the singular set consist of the points with nontrivial isotropy group. We show (under some dimension assumptions) that disks with diffeomorphic neighborhoods of the singular set can be imbedded into each other. As a consequence we find a classification of group actions on disks in terms of the neighborhood of the singular set and an element in the Whitehead group of G.     

Pseudo-Riemannian almost quaternionic homogeneous spaces with irreducible isotropy

We show that pseudo-Riemannian almost quaternionic homogeneous spaces with index 4 and an \(\mathbb {H}\)-irreducible isotropy group are locally isometric to a pseudo-Riemannian quaternionic Kähler symmetric space if the dimension is at least 16. In dimension 12 we give a non-symmetric example.     

Ordinary abelian varieties having small embedding degree

Miyaji, Nakabayashi and Takano (MNT) gave families of group orders of ordinary elliptic curves with embedding degree suitable for pairing applications. In this paper we generalise their results by giving families corresponding to non-prime group orders. We also consider the case of ordinary abelian varieties of dimension 2. We give families of group orders with embedding degrees 5, 10 and 12.     

Einstein metrics via intrinsic or parallel torsion

The classification of Riemannian manifolds by the holonomy group of their Levi-Civita connection picks out many interesting classes of structures, several of which are solutions to the Einstein equations. The classification has two parts. The first consists of isolated examples: the Riemannian symmetric spaces. The second consists of geometries that can occur in continuous families: these include the Calabi-Yau structures and Joyce manifolds of string theory. One may ask how one can weaken the definitions and still obtain similar classifications. We present two closely related suggestions. The classifications for these give isolated examples that are isotropy irreducible spaces, and known families that are the nearly Kähler manifolds in dimension 6 and Grays weak holonomy G₂ structures in dimension 7.Mathematics Subject Classification (2000): 53C10, 17B10, 53C25, 53C29in final form: 11 June 2003     

Note: A remark on a result of B. Segre concerning pseudoregular points of an elliptic quadric ofAG(2,q), q odd

In [1] S. ILKKA conjectured that pqeudoregular points of an elliptic quadric ofAG(2,q),q odd, only exist for small values ofq. In [3] B. SEGRE proves that an elliptic quadric ofAG(2,q),q odd, has pseudoregular points iffq=3 or 5. In [2], however, F. KáRTESZI shows that an elliptic quadric ofAG(2,7) has eight pseudoregular points. In this note we prove that part of B. Segre's proof is not correct, and that an elliptic quadric ofAG(2,q),q odd, has pseudoregular points iffq=3, 5 or 7.     

Smoothness of holonomy covers for singular foliations and essential isotropy

Given a singular foliation, we attach an “essential isotropy” group to each of its leaves, and show that its discreteness is the integrability obstruction of a natural Lie algebroid over the leaf. We show that a condition ensuring discreteness is the local surjectivity of a transversal exponential map associated with the maximal ideal of vector fields prescribed to be tangent to the foliation. The essential isotropy group is also shown to control the smoothness of the holonomy cover of the leaf (the associated fiber of the holonomy groupoid), as well as the smoothness of the associated isotropy group. Namely, the (topological) closeness of the essential isotropy group is a necessary and sufficient condition for the holonomy cover to be a smooth (finite-dimensional) manifold and the isotropy group to be a Lie group. These results are useful towards understanding the normal form of a singular foliation around a compact leaf. At the end of this article we briefly outline work of ours on this normal form, to be presented in a subsequent paper.     

Linear and Steiner Bundles on Projective Varieties

We generalize the theory of Horrocks monads to ACM varieties, and use the generalization to establish a cohomological characterization of linear and Steiner bundles on projective space and on quadric hypersurfaces. We also characterize Steiner bundles on the Grassmannian G(1, 4) of lines in ?⁴. Finally, we study linear resolutions of bundles on ACM varieties, and characterize linear homological dimension on quadric hypersurfaces.     

A combinatorial characterization of quadrics

In this paper we prove that in a projective space of dimension three and orderq the two plane characterk-sets fork {q ²+ 1,(q+1)²} are of the same type as the elliptic or the hyperbolic quadric, respectively. As a corollary we obtain a characterization of the elliptic and the hyperbolic quadrics.     

The Euler-Poincaré-Hopf theorem for flat connections in some transitive Lie algebroids

This paper is a continuation of [19], [21], [22]. We study flat connections with isolated singularities in some transitive Lie algebroids for which either ℝ or sl(2, ℝ) or so(3) are isotropy Lie algebras. Under the assumption that the dimension of the isotropy Lie algebra is equal to n + 1, where n is the dimension of the base manifold, we assign to any such isolated singularity a real number called an index. For ℝ-Lie algebroids, this index cannot be an integer. We prove the index theorem (the Euler-Poincaré-Hopf theorem for flat connections) saying that the index sum is independent of the choice of a connection. Multiplying this index sum by the orientation class of M, we get the Euler class of this Lie algebroid. Some integral formulae for indices are given.