On normal surfaces with strictly nef anticanonical divisors

We study normal complete algebraic surfaces with strictly nef anticanonical divisors defined over an algebraically closed field of arbitrary characteristic. We prove that such surfaces have numerically ample anticanonical divisors.     

Ulrich bundles on rational surfaces with an anticanonical pencil

Ulrich bundles are the simplest sheaves from the viewpoint of cohomology tables. Eisenbud and Schreyer conjectured that every projective variety carries an Ulrich bundle, which implies it has the same cone of cohomology tables as the projective space of same dimension. In this paper we show the existence of stable rank 2 Ulrich bundles on rational surfaces with an anticanonical pencil, under a mild Brill–Noether assumption by using Lazarsfeld–Mukai bundles. As a consequence, the Chow form of a surface admitting a rank 2 Ulrich bundle of the form we construct is known to be the Pfaffian of a skew-symmetric matrix.     

Fano 5-folds with nef tangent bundles and Picard numbers greater than one

We prove that smooth Fano 5-folds with nef tangent bundles and Picard numbers greater than one are rational homogeneous manifolds.     

A criterion of openness of a family of nef line bundles

Letf:X→S be a projective morphism of noetherian schemes andL a line bundle onX. In this note we will give a condition for openness of the setS _L ^nef ={∈S‖L _s is nef onX _s }. This article was processed by the author using the Springer-Verlag TEX mamath macro package 1990     

Threefolds with vanishing Hodge cohomology

We consider algebraic manifolds of dimension 3 over with for all and 0$">. Let be a smooth completion of with , an effective divisor on with normal crossings. If the -dimension of is not zero, then is a fibre space over a smooth affine curve (i.e., we have a surjective morphism from to such that the general fibre is smooth and irreducible) such that every fibre satisfies the same vanishing condition. If an irreducible smooth fibre is not affine, then the Kodaira dimension of is and the -dimension of is 1. We also discuss sufficient conditions from the behavior of fibres or higher direct images to guarantee the global vanishing of Hodge cohomology and the affineness of .

     

Fano 3-folds with divisible anticanonical class

We show the nonvanishing of H ⁰(X,−K _X ) for any a Fano 3-fold X for which −K _X is a multiple of another Weil divisor in Cl(X). The main case we study is Fano 3-folds with Fano index 2: that is, 3-folds X with rank Pic(X)=1, -factorial terminal singularities and −K _X  = 2A for an ample Weil divisor A. We give a first classification of all possible Hilbert series of such polarised varieties (X,A) and deduce both the nonvanishing of H ⁰(X,−K _X ) and the sharp bound (−K _X )³≥ 8/165. We find the families that can be realised in codimension up to 4.     

Bordism rings with split normal bundles. II

Dedicated to Yuri Grigor'evich Reshetnyak on his sixtieth birthday.     

A Nadel vanishing theorem for metrics with minimal singularities on big line bundles

The purpose of this paper is to establish a Nadel vanishing theorem for big line bundles with multiplier ideal sheaves of singular metrics admitting an analytic Zariski decomposition (such as, metrics with minimal singularities and Siu's metrics). For this purpose, we apply the theory of harmonic integrals and generalize Enoki's proof of Kollár's injectivity theorem. Moreover we investigate the asymptotic behavior of harmonic forms with respect to a family of regularized metrics.     

Seshadri constants on rational surfaces with anticanonical pencils

We study a Seshadri constant at a general point on a rational surface whose anticanonical linear system contains a pencil. First, we describe a Seshadri constant of an ample line bundle on such a rational surface explicitly by the numerical data of the ample line bundle. Second, we classify log del Pezzo surfaces which are special in terms of the Seshadri constants of the anticanonical divisors when the anticanonical degree is between 4 and 9.     

Analysis in space-time bundles,II. The spinor and form bundles

The structures of the spin and form bundles over the universal cosmos M?, and their relations with corresponding bundles over the Minkowski space M₀ canonically imbedded in M?, are treated. Wave equations covariant with respect to the causal group G of M? are studied, their solution manifolds and other stable (essentially positive-energy) invariant subspaces of the section spaces of the bundles are determined, and the indecomposability of relevant invariant subspace chains is shown. Explicit parallelizations of the bundles are applied to the Dirac and Maxwell equations on M?. A basis for spinor fields that diagonalizes a complete set of K?-covariant quantum numbers (K? = maximal essentially compact subgroup of G?) is developed. Local multilinear invariants of bundles over M? are treated and specialized to convergent mathematical versions of the Fermi and Yukawa interaction Lagrangians that are G?-invariant for the appropriate conformal weights.     

Classifying Hilbert bundles. II

A Hilbert bundle (p, B, X) is a type of fibre space p: B → X such that each fibre p^?1(x) is a Hilbert space. However, p^?1(x) may vary in dimension as x varies in X, even when X is connected. We give two “homotopy” type classification theorems for Hilbert bundles having primarily finite dimensional fibres. An (m, n)-bundle over the pair (X, A) is a Hilbert bundle over (p, B, X) such that the dimension of p^?1(x) is m for x in A and n otherwise. As a special case, we show that if X is a compact metric space, C₊X the upper cone of the suspension SX, then the isomorphism classes of (m, n)-bundles over (SX, C₊X) are in one-to-one correspondence with the members of [X, V_m($\text{C}$ⁿ)] where V_m($\text{C}$ⁿ) is the Stiefel manifold. The results are all applicable to the classification of separable, continuous trace C¹-algebras, with specific results given to illustrate.