when 0 < k ≤ p. We will say that a smooth projective variety X has Bott vanishing if for every ample line bundle L, i > 0 and j H(X,ΩjX ⊗ L) = 0 The purpose of this paper is to show that Bott vanishing is a simple consequence of a very specific condition on the Frobenius morphism in prime characteristic p. Recall that the absolute Frobenius morphism F : X → X on X, where X is a variety over Z/p...

This paper describes a method of interpreting three dimensional motion of an object by making use of r i g i d i t y assumption and orientation of i t s edge. We employ a vanishing point to determine the orientat ion. We propose a new idea of using cross ra t i o , i .e . one of the most fundamental concepts in projective geometry to f ind a vanishing point of a l ine . This allows to calculate...

The planar polynomial vector fields with a center at the origin can be written as an scalar differential equation, for example Abel equation. If the coefficients of an Abel equation satisfy the composition condition, then the Abel equation has a center at the origin. Also the composition condition is sufficient for vanishing the first order moments of the coefficients. The composition conjectur...

Horn recursion is a term used to describe when non-vanishing products of Schubert classes in the cohomology of flag varieties G/P are characterized by inequalities parameterized by similar non-vanishing products in the cohomology of “smaller” flag varieties. We consider the partial flag variety SLn/P and find that H ∗(SLn/P ) exhibits Horn recursion on a certain deformation of the cup product. ...