Let p be normal singularity of the 2-dimensional Stein space V . Let : M V be a minimal good resolution of V , such that the irreducible components A i of A = -1 (p) are nonsingular and have only normal crossings. Associated to A is weighted dual graph which, along with the genera of the A i , fully describes the topology and differentiable structure of A and the topological and differentiable nature of the embedding of A in M . It is well known that the simplest Gorenstein non-complete intersection singularities of dimension two are exactly those minimal elliptic singularities with fundamental cycle self intersection number -5. In this paper we classify all weighted dual graphs of these singularities. In particular, we prove that there is no integral homology link structure in the class of simplest Gorenstein non-complete intersection singularities of dimension two.

Let M S n+1 be a complete orientable hypersurface with constant Gauss-Kronecker curvature G. For any v R n+2 , let us define the following two real functions lv, fv : M R on M by lv(x) = x, v and fv(x) = (x), v with : M S n+1 a Gauss map of M . In this paper, we show that if n = 3, lv = fv for some nonzero vector v R 5 and some real number , then M is either totally umbilical (a Euclidean sphere) or M is a cartesian product of Euclidean spheres. We will also show with an example that the completeness condition is needed in the result we just mentioned. We also show that if n = 4, lv = fv for some nonzero vector v R 6 and some real number and ( 2 -1) 2 + (G -1) 2 = 0, then M is either totally umbilical (a Euclidean sphere) or M is a cartesian product of Euclidean spheres. Moreover, we will give an example of a complete hypersurface in S 5 with constant Gauss-Kronecker curvature that satisfies the condition lv = fv for some non zero v, which is neither a totally umbilical hypersurface nor a cartesian product of Euclidean spheres.

In this paper, we prove the vanishing of a map between the absolute Hodge cohomology spaces of the boundaries of the Baily-Borel compactifications of the product of two modular curves on one side and of the Siegel threefold on the other side. As an application, we construct some 1-extensions of mixed Hodge structures between the trivial Hodge structure and the middledegree interior cohomology of the Siegel threefold, which come from motivic cohomology. These are conjecturally related to non-critical values of the degree 4 L-function of some cuspidal automorphic representations of GSp 4 .

We prove that the second Betti number of a compact Riemannian manifold vanishes under certain Ricci curved restriction. As consequences we obtain an interesting curved restriction for compact Kahler-Einstein manifolds and a homology sphere theorem in ${\rm dim}=4,5$.

For any nonzero complex number q, there is a Lie algebra of Block type, denoted by B(q).In this paper, a complete classification of irreducible quasifinite modules is given.More precisely, an irreducible quasifinite module is a highest weight or lowest weight module, or a module of intermediate series.As a consequence, a classification for uniformly bounded modules over another class of Lie algebras, the semi-direct product of the Virasoro algebra and a module of intermediate series, is also obtained.Our method is conceptional, instead of computational.

We prove a wall crossing formula of Donaldson-Thomas type in- variants without Chern-Simons functionals.

We construct Yang-Mills connections on SO(n)-bundles over spheres equipped with the Euclidean metric. We use a cohomogeneity one group action on the bundle to reduce the Yang-Mills-equation to a system of ordinary differential equations. The system is shown to have solutions by variational methods, using ideas from harmonic map theory. Examples include Yang-Mills connections on each of the countably many principal SO(6)-bundles over $S^6$, and countably many Yang-Mills connections on $TS^n$ for $n\in\{5,...,9\}$.

Ideas and methods of Clemens C. H., Griffiths Ph. The intermediate Jacobian of a cubic threefold are applied to a Fano threefold $X$ of genus 6 — intersection of $G(2, 5) \subset P^9$ with $P^7$ and a quadric. Main results: 1. The Fano surface $F(X)$ of $X$ is smooth and irreducible. Hodge numbers and some other invariants of $F(X)$ are calculated. 2. Tangent bundle theorem for $X$ is proved, and its geometric interpretation is given. It is shown that $F(X)$ defines $X$ uniquely. 3. The Abel-Jacobi map $\Phi : \operatorname{Alb} F(X) \to J^3(X)$ is an isogeny. 4. As a necessary step of calculation of $h^{1,0}(F(X))$ we describe a special intersection of 3 quadrics in $P^6$ (having 1 double point) whose Hesse curve is a smooth plane curve of degree 6. 5. $\operatorname{im} \Phi(F(X)) \subset J^3(X)$ is algebraically equivalent to $\frac{2\Theta^8}{8!}$ where $\Theta \subset J^3(X)$ is a Poincare divisor (a sketch of the proof).

The link between Frobenius manifolds and singularity theory is well known, with the simplest examples coming from the simple hypersurface singularities. Associated with any such manifold is a function known as the $G$-function. This plays a role in the construction of higher-genus terms in various theories. For the simple singularities the G-function is known explicitly: G=0. The next class of singularities, the unimodal hypersurface or elliptic hypersurface singularities consists of three examples, \widetilde{E}_6,\widetilde{E}_7,\widetilde{E}_8 (or equivalently P_8, X_9,J_10). Using a result of Noumi and Yamada on the flat structure on the space of versal deformations of these singularities the $G$-function is explicitly constructed for these three examples. The main property is that the function depends on only one variable, the marginal (dimensionless) deformation variable. Other examples are given based on the foldings of known Frobenius manifolds. Properties of the $G$-function under the action of the modular group is studied, and applications within the theory of integrable systems are discussed.

Minimal affine Kac-Moody groups act on affine twin buildings by isometries. However there is no way to extend this action to any completion of the Kac-Moody groups. To remedy this, we introduce in this paper affine twin cities, a new class of objects, whose elements behave like completions of twin buildings. Twin cities are defined as special arrays of affine buildings connected among themselves by twinnings. Corresponding to completed affine Kac-Moody groups they are characterized by the type of the affine buildings and by some kind of regularity conditions describing the completion. The isometry groups of affine twin cities are (completions of) affine Kac-Moody groups. We study applications of cities in infinite dimensional differential geometry by proving infinite dimensional versions of classical differential geometric results: For example, we show that points in an isoparametric submanifold in a Hilbert space correspond to all chambers in a city. In two sequels we will describe the theory of twin cities for formal completions.