Volume 5, Issue 3

In this article we consider the connection between semifield flocks of a quadratic cone in PG(3,  q  n  ), eggs in PG(4 n  - 1,  q ) and ovoids of Q  (4,  q  n  ), when q is odd. Starting from a semifield flock ℱ of a quadratic cone in PG(3,  q  n  ), q odd, one can obtain an ovoid ## add figure 'advg.5.3.333_01.gif' ##(ℱ ) of Q  (4,  q  n  ) using the construction of Thas [J. A. Thas, Symplectic spreads in PG(3,  q  )inversive planes and projective planes. Discrete Math. 174 (1997), 329–336]. With a semifield flock there also corresponds a good egg ℰ of PG(4 n  - 1,  q ) (see, e.g., [M. Lavrauw, T. Penttila, On eggs and translation generalised quadrangles. J. Combin. Theory Ser. A 96 (2001), 303–315]) and the TGQ T  (ℰ ) contains at least q   3 n  +  q   2 n subquadrangles all isomorphic to Q  (4,  q  n  ) (Thas [J. A. Thas, Generalized quadrangles of order ( s ,  s  2 ). I. J. Combin. Theory Ser. A 67 (1994), 140–160]). Hence by subtending one can obtain ovoids of Q  (4,  q  n  ) (consider the set of points in the subquadrangle collinear with a point not in the subquadrangle). Here we prove that all the ovoids subtended from points of type (ii) are isomorphic to ## add figure 'advg.5.3.333_01.gif' ##(ℱ ), and that in at least 2 q  n subGQ’s the ovoids subtended from points of type (i) are isomorphic to the ovoids subtended from points of type (ii).

Let P be a polar space induced by a quasipolarity on a projective space G . In this article, we give sufficient conditions for an embedding (i.e. an incidence-preserving map which is injective on points and lines) of P into a projective space H to be extendable to an incidencepreserving map from G to H . As a consequence, we obtain conditions for such an embedding to be induced by a semilinear map between the overlying vector spaces.

For a manifold M , the structure set S  ( M , rel ∂ ) is the collection of manifolds homotopy equivalent to M relative to the boundary. Siebenmann [R. C. Kirby, L. C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations . Princeton Univ. Press 1977] showed that in the topological category, the structure set is 4-periodic: S  ( M , rel ∂ ) ≅  S  ( M ｘ D  4 , rel ∂ ) up to a copy of ℤ. The periodicity has been extended to topological manifolds with homotopically stratified group actions for various representations in place of D   4 , including twice any complex representation of a compact abelian group. In this paper, we extend the result to twice any complex representation of a compact Lie group. We also prove the bundle version of the periodicity.

We prove the finiteness property for the class of global semianalytic sets: we get it as a consequence of a global weak Łojasiewicz inequality for this class. This weaker result still implies the usual Hörmander form. Some other consequences are deduced.

We treat the almost differentiable left A-loops as images of global differentiable sharply transitive sections  σ  :  G   |   H  →  G for a Lie group G such that G | H is a reductive homogeneous manifold. In this paper we classify all 3-dimensional connected strongly left alternative almost differentiable left A-loops L such that for the corresponding section σ  :  G   |   H  →  G the Lie group G is non-solvable.

Let ( X ,  L ) be a polarized manifold defined over ℂ with dim X  = n ≥ 3 and let ( M ,  A  ) be a reduction of ( X ,  L ). In this paper, if X is not uniruled and K M  + ( n  - 2) A is nef and ( n  - 2)-big, then we give a lower bound for c 2 ( M  ) A n  - 2 , which improves a result in a previous paper. By using this bound, we prove that g 2  ( X ,  L ) ≥  h 1 (## add figure 'advg.5.3.431_01.gif' ## x   ) if n  ≥ 3, X is not uniruled, and K M  + ( n  - 2) A is nef and ( n  - 2)-big, where g 2  ( X ,  L ) is the second sectional geometric genus of ( X ,  L ). In particular, if n ≥ 3 and κ  ( X  ) ≥ 0, then g 2  ( X ,  L ) ≥  h 1 (## add figure 'advg.5.3.431_01.gif' ## x   ). Furthermore we prove that g 2  ( X ,  L ) ≥ 0 for any polarized 3-folds. For n  ≥ 3 we also study ( X ,  L ) with g 2  ( X ,  L ) ≥  h 1 (## add figure 'advg.5.3.431_01.gif' ## x  ). Finally we consider a lower bound of h 0 ( K X  +  L ) for polarized 3-folds ( X ,  L ).

Let V   be an n -dimensional left vector space over a division ring R and n  ≥ 3. Denote by ## add figure 'advg.5.3.455_01.gif'## k the Grassmann space of k -dimensional subspaces of V and write ## add figure 'advg.5.3.455_02.gif'## k for the set of all pairs ( S ,  U  ) ∈ ## add figure 'advg.5.3.455_01.gif'## k  ｘ ## add figure 'advg.5.3.455_01.gif'## n  -  k   such that S  +  U  =  V . We study bijective transformations of ## add figure 'advg.5.3.455_02.gif'## k preserving the class of base subsets and show that these mappings are induced by semilinear isomorphisms of V   to itself or to the dual space V*   if n ≠ 2 k  ; for n  = 2 k this fails. This result can be formulated as the following: if n  ≠ 2 k and the characteristic of R is not equal to 2 then any commutativity preserving transformation of the set of ( k ,  n  -  k  )-involutions can be extended to an automorphism of the group GL( V   ).

We prove a removable singularity theorem for p  -harmonic maps in the subquadratic case. The theorem states that an isolated singularity of a weakly p  -harmonic map is removable if the energy is sufficiently small in a neighbourhood of the singularity.