Extension theorems, non-vanishing and the existence of good minimal models

We prove an extension theorem for effective purely log-terminal pairs (X, S + B) of non-negative Kodaira dimension ${\kappa (K_X+S+B)\ge 0}$ . The main new ingredient is a refinement of the Ohsawa–Takegoshi L ² extension theorem involving singular Hermitian metrics.     

Weak boundedness theorems for canonically fibered Gorenstein minimal 3-folds

Let be a Gorenstein minimal projective 3-fold with at worst locally factorial terminal singularities. Suppose the canonical map is of fiber type. Denote by a smooth model of a generic irreducible element in fibers of , and so is a curve or a smooth surface. The main result is that there is a computable constant independent of such that or whenever .

     

Varieties without minimal generators

While many familiar varieties have a minimal varietal generator, i.e., a regular projective finitely presentable regular generator such that none of its retracts is a regular generator, and even a unique one, we present (a) a variety having no minimal varietal generator at all and (b) a variety having two non-isomorphic minimal varietal generators. Moreover we demonstrate that the same effects can happen with respect to a weaker notion of minimality and are common even in module categories. Received April 7, 1999; accepted in final form July 10, 2000.     

Codimension one minimal cycles with coefficients inZ orZ p , and variational functionals on fibered spaces

Given a compact, oriented Riemannian manifold M, without boundary, and a codimension-one homology class in H_* (M, Z) (or, respectively, in H_* (M, Z_p) with p an odd prime), we consider the problem of finding a cycle of least area in the given class: this is known as the homological Plateau’s problem. We propose an elliptic regularization of this problem, by constructing suitable fiber bundles ξ (resp. ζ) on M, and one-parameter families of functionals defined on the regular sections of ξ, (resp. ζ), depending on a small parameter ε. As ε → 0, the minimizers of these functionals are shown to converge to some limiting section, whose discontinuity set is exactly the minimal cycle desired.     

Nef dimension of minimal models

We reduce the Abundance Conjecture in dimension 4 to the following numerical statement: if the canonical divisor K is nef and has maximal nef dimension, then K is big. From this point of view, we classify in dimension 2 nef divisors which have maximal nef dimension, but which are not big.Mathematics Subject Classification (2000): 14E30, 14J10, 14N30Revised version: 2 March 2004     

Noncritical W-strings and minimal models

We perform a BRST analysis of the physical states described by a general noncriticalW-string. A crucial feature of our analysis is that we introduce a special basis in the Hilbert space of physical states in which the BRST operator splits into a nested sum of nilpotent BRST operators. We argue that the cohomology of each nilpotent BRST operator occurring in the nested sum is closely related to a specificW mimimal model. We discuss in detail the special case of the noncriticalW ₃-string.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 98, No. 3, pp. 343–357, March, 1994.     

Varieties minimal over representable varieties of lattice-ordered groups

Weinberg showed that the variety of abelian lattice-ordered groups is the minimal nontrivial variety in the lattice of varieties of lattice-ordered groups. Scrimger showed that the abelian variety of lattice-ordered groups has countably infinitely many nonrepresentable covering varieties, and it is now known that his varieties are the only nonrepresentable covers of the abelian variety.

In this paper, a variation of the method used to construct the Scrimger varieties is developed that is shown to produce every nonrepresentable cover of any representable variety. Using this variation, all nonrepresentable covers of any weakly abelian l-variety are specifically identified, as are the nonrepresentable covers of any l-metabelian representable l-variety. In both instances, such il-varieties have only countably infinitely many such covers.

Any nonrepresentable cover of a representable il-variety is shown to be a subvariety of a quasi-representable il-variety as defined by Reilly. The class of these quasi-representable l-varieties is shown to contain the well-known L_n l-varieties and to generalize many of their properties.     

Operators of fibered groups

Let (G, ) denote a finite groupG with fibration . The group Aut (G, ) of operators of (G, ) is closely related to the group of collineations of (G, ). In this paper we investigate the geometric properties imposed on (G, ) by requiring that Aut (G, )=AutG. We find that in many instances this algebraic property restricts the geometry to a very special form.Dedicated to Professor Helmut Karzel on the occasion of his 60th birthday.     

Metrizably fibered generalized ordered spaces

In this paper we characterize generalized ordered spaces that are metrizably fibered in terms of certain quotient spaces and in terms of the existence of special open covers. We apply our results to give a new characterization of perfect generalized ordered spaces that have a σ-closed-discrete dense subset and to give examples of GO-spaces that are, or are not, metrizably fibered.     

Inventory models with minimal service level constraints

This paper investigates inventory models in which the stockout cost is replaced by a minimal service level constraint (SLC) that requires a certain level of service to be met in every period. The minimal service level approach has the virtue of simplifying the computation of an optimal ordering policy, because the optimal reorder level is solely determined by the minimal SLC and demand distributions. It is found that above a certain “critical” service level, the optimal (s,S) policy “collapses” to a simple base-stock or order-up-to level policy, which is independent on the cost parameters. This shows the minimal SLC models to be qualitatively different from their shortage cost counterparts. We also demonstrate that the “imputed shortage cost” transforming a minimal SLC model to a shortage cost model does not generally exist. The minimal SLC approach is extended to models with negligible set-up costs. The optimality of myopic base-stock policies is established under mild conditions.     

The geometric minimal models of analytic spaces

This paper shows that an analytic space X has a unique maximal model through which every proper surjective morphism from a non-singular analytic space to X factors. This is called the geometric minimal model of X and characterized by the contraction property of rational curves. Some other properties such as functoriality, the direct product property and the quotient property of the geometric minimal model are also studied here. The relation of the geometric minimal model with Mori's minimal model is discussed. Received: 25 June 2001 / Published online: 4 April 2002     

Asymptotic velocity for fibered Hamiltonians

We show the existence of the asymptotic velocity for a large class of Hamiltonians with a fibered structure.