In this paper we study the following question: Is it true that a generic hypersurface X of degree d in P$^n+1$, where (d,n)≠ (3,1), (2,2), (3,2), does not admit a non-trivial, non-isomorphic surjective map to another smooth variety Y, except of course P$^n$? It is easy to see that it is true for n=1,2. We try to prove this for n=3 and can exclude all possibilities for Y except Y= G(1,4)∩ P$^6$and Y=V$_22^s$, a special Fano threefold of type V$_22$ found by Mukai and Umemura in [MU].

We study a ramification theory for a division algebra D of the following type: The center of D is a complete discrete valuation field K with the imperfect residue field F of certain type, and the residue algebra of D is commutative and purely inseparable over F. Using Swan conductors of the corresponding element of Br(K), we define Herbrand‘s ψ-function of D/K, and it describes the action of the reduced norm map on the filtration of D$^*$. We also gives a relation among the Swan conductors and the ’ramification number‘ of D, which is defined by the commutator group of D$^*$.

J. Murre has conjectured that every smooth projective variety X of dimension d admits a decomposition of the diagonal δ=p$_0$+…+p$_2d$ ∈ CH$^d$(X × X) [otimes] Q such that the cycles p$_i$ are orthogonal projectors which lift the Künneth components of the identity map in étale cohomology. If this decomposition induces an intrinsic filtration on the Chow groups of X, we call it a Murre decomposition. In this paper we propose candidates for such projectors on 3-folds by using fiber structures. Using Mori theory, we prove that every smooth uniruled complex 3-fold admits a Murre decomposition.

We describe a method for complete solution of the superelliptic Diophantine equation ay$^p$=f(x). The method is based on Baker‘s theory of linear forms in the logarithms. The characteristic feature of our approach (as compared with the classical method) is that we reduce the equation directly to the linear forms in logarithms, without intermediate use of Thue and linear unit equations. We show that the reduction method of Baker and Davenport [3] is applicable for superelliptic equations, and develop a very efficient method for enumerating the solutions below the reduced bound. The method requires computing the algebraic data in number fields of degree pn(n-1)/2 at most; in many cases this number can be reduced. Two examples with p=3 and n=4 are given.

In this article we prove a Chern–Lashof inequality for immersions of manifolds with H-spherical ends. Related to this inequality we discuss different types of tightness. In particular we shall prove that an immersion of a manifold with at least two H-spherical ends is tightly immersed only if the ends are of a certain geometric type (Wintgen immersion).

We study a ramification theory for a division algebra D of the following type: The center of D is a complete discrete valuation field K with the imperfect residue field F of certain type, and the residue algebra of D is commutative and purely inseparable over F. Using Swan conductors of the corresponding element of Br(K), we define ψ-function of D/K, and it describe the action of the reduced norm map on the filtration of D$^*$. We also gives a relation among the Swan conductors and the ’ramification number‘ of D, which is defined by the commutator group of D$^*$.

To an arbitrary intersection of exceptional varieties of an embedded resolution we associate a finite number of congruences between naturally occurring multiplicities. This theory generalizes previous results concerning just one exceptional variety. Moreover we describe precise equalities which imply the congruences and we give some applications on the poles of Igusa‘s local zeta function.

In this paper we prove that an isometric stable minimal immersion of a complete oriented surface into a hyperkähler 4-manifold is holomorphic with respect to an orthogonal complex structure, if it satisfies a Bernstein-type assumption on the Gauss-lift. This result generalizes a theorem of Micallef for minimal surfaces in the euclidean 4-space. An example found by Atiyah and Hitchin shows that the assumption on the Gauss-lift is necessary.

We extend some results about shifted Schur functions to the general context of shifted Macdonald polynomials. We strengthen some theorems of F. Knop and S. Sahi and give two explicit formulas for these polynomials: a q-integral representation and a combinatorial formula. Our main tool is a q-integral representation for ordinary Macdonald polynomial. We also discuss duality for shifted Macdonald polynomials and Jack degeneration of these polynomials.

We prove a lower bound for the number of singular fibres of a non-isotrivial semi-stable family of curves over the projective line, in positive characteristic. The result is the same obtained previously by Beauville in characteristic zero, namely that the number is at least 4. Furthermore some consequences are discussed in the case that the number is exactly 4.

Let G be a complex semi-simple group, and X a compact Riemann surface. The moduli space of principal G-bundles on X, and in particular the holomorphic line bundles on this space and their global sections, play an important role in the recent applications of Conformal Field Theory to algebraic geometry. In this paper we determine the Picard group of this moduli space when G is of classical or G$_2$ type (we consider both the coarse moduli space and the moduli stack).

Given an irreducible, modular, mod p representation p, we analyse the local components at p of newforms f which give rise to it.

We show that, for certain types of rigid analytic varieties X and constructible l-adic sheaves (F$_n$)$_n$ on, one has H$^p_c$ (X,(F$_n$)$_n$) ${\stackrel{\sim}{\longrightarrow}} {\displaystyle\lim_{\stackrel{\scriptstyle{\longleftarrow}} {\scriptstyle{n}}}}$ H$^p_c$ (X,F$_n$). As an application we obtain that, for an algebraic variety X and associated rigid analytic variety X$^rig$, the l-adic cohomology of X and X$^rig$ agree.

We calculate l-adic nearby cycles in the étale cohomology for families with log smooth reduction using log étale cohomology. In particular, nearby cycles for log smooth families coincide with tame nearby cycles, as L. Illusie expected, and nearby cycles for semistable families depend only on the first infinitesimal neighborhood of the special fiber.

We construct a correspondence between the set of partitions of a finite set M and the set of pairs of walks to the same vertex on a graph giving the Bratteli diagram of the partition algebra on M. This is the precise analogue of the correspondence between the set of permutations of a finite set and the set of pairs of Young tableaux of the same shape, called the Robinson–Schensted correspondence.

This note investigates compact complex manifolds X of dimension 3 with second Betti number b$_2$(X) = 0. If X admits a non-constant meromorphic function, then we prove that either b$_1$(X) = 1 and b$_3$(X) = 0 or that b$_1$(X) = 0 and b$_3$(X) = 2. The main idea is to show that c$_3$(X) = 0 by means of a vanishing theorem for generic line bundles on X. As a consequence a compact complex threefold homeomorphic to the 6-sphere S$^6$ cannot admit a non-constant meromorphic function. Furthermore we investigate the structure of threefolds with b$_2$(X) = 0 and algebraic dimension 1, in the case when the algebraic reduction X → P$_1$ is holomorphic.

The main purpose of this work is to introduce noncommutative relative schemes and establish some of basic properties of schemes and scheme morphisms. In particular, we prove an analogue of the canonical bijection: Hom$_schemes/k$((X, O), Spec(A))[bsime ] Hom$_k-alg$ (A,Γ (X, O)). We define a noncommutative version of the Čech cohomology of an affine cover and show that the Čech cohomology can be used to compute higher direct images. This fact is applied here to compute cohomology of invertible sheaves on skew projective spaces and in [LR3] to study D-modules on quantum flag varieties.