Frame bundle

Abstract: We prove that a holomorphic line bundle on a projective manifold is pseudo-effective if and only if its degree on any member of a covering family of curves is non-negative. This is a consequence of a duality statement between the cone of pseudo-effective divisors and the cone of " movable curves " , which is obtained from a general theory of movable intersections and approximate Zariski decomposition for closed positive (1, 1)-currents. As a corollary, a projective manifold has a pseudo-effective canonical bundle if and only if it is not uniruled. We also prove that a 4-fold with a canonical bundle which is pseudo-effective and of numerical class zero in restriction to curves of a good covering family, has non-negative Kodaira dimension.

Abstract: A radiator device adapted for attachment to a radiator pipe, comprising a mounting base wrapped about the radiator pipe and secured thereto by wire fastenings; a plurality of outwardly disposed radiating fins circularly positioned about said mounting base and extending outwardly therefrom. This radiating device is adapted to be installed upon any size flue pipe extending from a furnace to a chimney to transmit flue pipe heat to the surrounding atmosphere.

Abstract: Using the space of holomorphic symmetric tensors on the moduli space of stable bundles over a Riemann surface we construct a projectively flat connection on a vector bundle over Teichmuller space. The fibre of the vector bundle consists of the global sections of a power of the determinant bundle on the moduli space. Both Dolbeault and Cech techniques are used.

Abstract: If M is a differentiable ra-dimensional manifold and V a linear connection for M, then the 2 rc-dimensional manifold TM, which is the total space of the tangent bündle of M, admits an almost complex structure /, naturally determined by V *). (I learned of this almost complex structure, which occurs e. g. in the theory of partial differential equations on Riemannian manifolds, frorn Professor W. Ambrose. I wish to thank him very much for the stimulating conversations which I have had with him on that topic.) We shall give here a computation of the Eckmann-Frölicher torsion tensor for this almost complex structure /, which implies the following result: / is integrable if and only if the linear connection has vanishing torsion and curvature). An appendix is devoted to some questions on the geometry of the tangent bündle TM which arise in connection with the construction of / and which can be answered easily by methods similar to those which we have used in order to compute the EckmannFrölicher torsion tensor of /. We list here only two of these results: If g is a Riemann metric for M and V its Levi-Civita connection, then TM admits a canonical hermitian metric hg with respect to the almost complex structure / on TM, which is determined (see above) by VWe prove, (confer Appendix (iii)) that hg is kählerian. If V is any linear connection for M, then the distribution of the \"horizontal subspaces\" on TM is invariant under the action of the multiplicative group R* of non vanishing real numbers on TM. We prove (confer Appendix (iv)) that if oppositely an n-dimensional distribution on TM is given, which is invariant under the action of the group JR* on TM and which contains no nonzero vertical\" vector, then this distribution

Abstract: The notion of a singular hermitian metric on a holomorphic line bundle is introduced as a tool for the study of various algebraic questions. One of the main interests of such metrics is the corresponding L2 vanishing theorem for Open image in new window cohomology, which gives a useful criterion for the existence of sections. In this context, numerically effective line bundles and line bundles with maximum Kodaira dimension are characterized by means of positivity properties of the curvature in the sense of currents. The coefficients of isolated logarithmic poles of a plurisubharmonic singular metric are shown to have a simple interpretation in terms of the constant e of Seshadri’s ampleness criterion. Finally, we use singular metrics and approximations of the curvature current to prove a new asymptotic estimate for the dimension of cohomology groups with values in high multiples O(kL) of a line bundle L with maximum Kodaira dimension.