Generalized complete intersections with linear resolutions

We determine the simplicial complexes Δ whose Stanley-Reisner ideals I _Δ have the following property: for all n ≥ 1 the powers I _Δ ⁿ have linear resolutions and finite length local cohomologies. Received: 10 July 2007     

Pull-back of currents by holomorphic maps

We define the pull-back operator, associated to a meromorphic transform, on several types of currents. We also give a simple proof to a version of a classical theorem on the extension of currents.     

Perfect difference sets constructed from Sidon sets

A set of positive integers is a perfect difference set if every nonzero integer has a unique representation as the difference of two elements of . We construct dense perfect difference sets from dense Sidon sets. As a consequence of this new approach we prove that there exists a perfect difference set such that

Computing residue currents of monomial ideals using comparison formulas

Given a free resolution of an ideal a$\mathfrak{a}$ of holomorphic functions, one can construct a vector-valued residue current R  , which coincides with the classical Coleff–Herrera product if a$\mathfrak{a}$ is a complete intersection ideal and whose annihilator ideal is precisely a$\mathfrak{a}$.     

Log canonical thresholds of binomial ideals

We prove that the log canonical thresholds of a large class of binomial ideals, such as complete intersection binomial ideals and the defining ideals of space monomial curves, are computable by linear programming. Dedicated to Professor Toshiyuki Katsura on the occasion of his sixtieth birthday     

Some algebraic invariants related to mixed product ideals

We compute some algebraic invariants (e.g. depth, Castelnuovo-Mumford regularity) for a special class of monomial ideals, namely the ideals of mixed products. As a consequence, we characterize the Cohen-Macaulay ideals of mixed products. Received: 25 October 2007     

Cohen-Macaulay intersections

We study when the intersection of irreducible monomial ideals of height 2 with Cohen-Macaulay radicals are themselves Cohen-Macaulay. Received: 4 September 2008     

An inequality for mixed Monge–Ampère measures

We generalize an inequality for mixed Monge–Ampère measures from Kołodziej (Indiana Univ. Math. J. 43, 1321–1338, 1994). We also give an example that shows that our assumptions are sharp. The corresponding result in the setting of compact K?hler manifold is also discussed.     

On automorphisms of the symmetrized bidisc

We characterize the group Aut for the symmetrized bidisc

Holomorphic vector bundles on non-algebraic tori of dimension 2

We describe the Chern classes of holomorphic vector bundles on non-algebraic complex torus of dimension 2.     

On surfaces of prescribed weighted mean curvature

Utilising a weight matrix we study surfaces of prescribed weighted mean curvature which yield a natural generalisation to critical points of anisotropic surface energies. We first derive a differential equation for the normal of immersions with prescribed weighted mean curvature, generalising a result of Clarenz and von der Mosel. Next we study graphs of prescribed weighted mean curvature, for which a quasilinear elliptic equation is proved. Using this equation, we can show height and boundary gradient estimates. Finally, we solve the Dirichlet problem for graphs of prescribed weighted mean curvature.     

p-convergence in measure of a sequence of measurable functions and corresponding minimal elements of c 0

We characterize convergence in measure of a sequence (f_n)_n of measurable functions to a measurable function f by elements of c₀, which express the quality of convergence of (f_n)_n to f. This characterization motivates the introduction of a new notion of convergence, called “p-convergence in measure” (p > 0), which is stronger than convergence in measure. We prove the existence of “minimal” elements in c₀ which characterize the convergence in measure of (f_n)_n to f.      

A unified theory for modifications of<Emphasis Type="Italic">g</Emphasis>-closed sets

We introduce a new set calledmg-closed which is defined on a family of sets satisfying some minimal conditions. This set enables us to unify certain kind of modifications of generalized closed sets due to Levine [17].     

Which spaces for design?

We determine the largest class of spaces of sufficient regularity which are suitable for design in the sense that they do possess blossoms. It is the class of all spaces containing constants of which the spaces derived under differentiation are Quasi Extended Chebyshev spaces, i.e., they permit Hermite interpolation, Taylor interpolation excepted. It is also the class of all spaces which possess Bernstein bases, or of all spaces for which any associated spline space does possess a B-spline basis. Note that blossoms guarantee that such bases are normalised totally positive bases. They even are the optimal ones.     

Function classes for double exponential integration formulas

The double exponential (DE) formulas for numerical integration are known to be highly efficient, more efficient than the single exponential (SE) formulas in many cases. Function classes suited to the SE formulas have already been investigated in the literature through rigorous mathematical analysis, whereas this is not the case with the DE formulas. This paper identifies function classes suited to the DE formulas in a way compatible with the existing theoretical results for the SE formulas. The DE formulas are good for more restricted classes of functions, but more efficient for such functions. Two concrete examples demonstrate the subtlety in the behavior of the DE formulas that is revealed by our theoretical analysis.     

Fourier–Mukai transform on abelian surfaces

We study moduli spaces of stable sheaves on abelian surfaces whose Mukai vectors are related by a cohomological Fourier-Mukai transform. We show that there is a Fourier-Mukai transform inducing a birational map between them.     

Calabi–Yau varieties with semi-stable fibre structures

Motivated by the Strominger–Yau–Zaslow conjecture, we study Calabi–Yau varieties with semi-stable fibre structures. We use Hodge theory to study the higher direct images of wedge products of relative cotangent sheaves of certain semi-stable families over higher dimensional quasi-projective bases, and obtain some results on positivity. We then apply these results to study non-isotrivial Calabi–Yau varieties fibred by semi-stable Abelian varieties (or hyperkähler varieties).     

Pointwise convergence for semigroups in vector-valued <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

Suppose that {T _t  : t  ≥  0} is a symmetric diffusion semigroup on L ²(X) and denote by its tensor product extension to the Bochner space , where belongs to a certain broad class of UMD spaces. We prove a vector-valued version of the Hopf–Dunford–Schwartz ergodic theorem and show that this extends to a maximal theorem for analytic continuations of on . As an application, we show that such continuations exhibit pointwise convergence.     

Constant mean curvature spheres in Riemannian manifolds

We prove the existence of embedded spheres with large constant mean curvature in any compact Riemannian manifold (M, g). This result partially generalizes a result of R. Ye which handles the case where the scalar curvature function of the ambient manifold (M, g) has non-degenerate critical points.