On partially ample adjoint divisors

We study the MMP (minimal model program) invariant properties of partially ample adjoint divisors. Using these invariant properties, we prove that the three notions of (numerically, cohomologically, asymptotically) partial ampleness coincide for big divisors on Q$\mathbb{Q}$-factorial Fano type varieties and more generally Mori dream spaces. This gives a partial answer to a question raised by Totaro.     

Singularities of 3-dimensional varieties admitting an ample effective divisor of Kodaira dimension zero

For a normal threefoldX with an effective Cartier divisorH, which is a minimal model of Kodaira dimension zero, we prove that eitherX is a generalized cone overH, orX has quadruple singularities andH is either a K3 surface, or an Enriques surface. Translated fromMatematicheskie Zametki, Vol. 59, No. 4, pp. 617–625, April, 1996. The author is extremely grateful to V. A. Iskovskikh and Yu. G. Prokhorov for valuable advice and fruitful discussions. This research was partially supported by the International Science Foundation under grant No. M90 000.     

On an adjoint functor to the Thom functor

We construct a right adjoint functor to the Thom functor, i.e., to the functor which assigns the Thom space to a vector bundle .

     

Nefness of adjoint bundles for ample vector bundles of corank 3

Let be an ample vector bundle of rank on a smooth complex projective variety X of dimension n. The aim of this paper is to describe the structure of pairs as above whose adjoint bundles are not nef for . Furthermore, we give some immediate consequences of this result in adjunction theory.     

On an adjoint initial-boundary value problem

We study an adjoint initial-boundary value problem for linear parabolic equations; which arises when modeling the unidirectional motion of two viscous fluids with a common interface under the action of a pressure gradient. Under some conditions on the pressure gradient, we obtain a priori estimates and show that the solution enters a stationary mode. For semibounded layers, we find the solution in closed form and indicate the case of a self-similar solution. We determine the volume flow rates in the layers.     

On digit sums of multiples of an integer

Let g>1 be an integer and sg(m) be the sum of digits in base g of the positive integer m. In this paper, we study the positive integers n such that sg(n) and sg(kn) satisfy certain relations for a fixed, or arbitrary positive integer k. In the first part of the paper, we prove that if n is not a power of g, then there exists a nontrivial multiple of n say kn such that sg(n)=sg(kn). In the second part of the paper, we show that for any K>0 the set of the integers n satisfying sg(n)?Ksg(kn) for all k∈N is of asymptotic density 0. This gives an affirmative answer to a question of W.M. Schmidt.     

On the largest prime divisor of an odd harmonic number

A positive integer is called a (Ore's) harmonic number if its positive divisors have integral harmonic mean. Ore conjectured that every harmonic number greater than is even. If Ore's conjecture is true, there exist no odd perfect numbers. In this paper, we prove that every odd harmonic number greater than must be divisible by a prime greater than .

     

On an asymmetric divisor problem with congruence conditions

For fixed positive integersab, natural numbersl ₁k ₁,l ₂k ₂ andn, denote withd _a,b (l ₁,k ₁;l ₂,k ₂;n) the number of all (,)N² with ^{a b} =n,l ₁(modk ₁),l ₂(modk ₂). In the present paper we establish asymptotic formulas for the Dirichlet summatory function ofd _a,b (l ₁,k ₁;l ₂,k ₂;n) with both upper and lower estimates of the error term, all of them uniform in the moduli.     

On the extremizers of an adjoint Fourier restriction inequality

The adjoint Fourier restriction inequality for the sphere $S^2$ states that if $f\in L^2(S^2,\sigma )$ then $\stackrel{?}{f\sigma }\in L^4\left({\mathbb{R}}^3\right)$. We prove that all critical points $f$ of the functional ${6\stackrel{?}{f\sigma }6}_{L^4}/{6f6}_{L^2}$ are smooth, any complex-valued extremizer for the inequality is a nonnegative extremizer multiplied by the character $e^{ix?\xi }$ for some $\xi$, and complex-valued extremizing sequences for the inequality are precompact modulo multiplication by characters.     

On the positivity of high-degree Schur classes of an ample vector bundle

Let X be a smooth projective variety of dimension n,and let E be an ample vector bundle over X.We show that any Schur class of E,lying in the cohomology group of bidegree(n-1,n-1),has a representative which is strictly positive in the sense of smooth forms.This conforms the prediction of Griffiths conjecture on the positive polynomials of Chern classes/forms of an ample vector bundle on the form level,and thus strengthens the celebrated positivity results of Fulton and Lazarsfeld(1983)for certain degrees.