Indecomposable Elements in K1 of a Smooth Projective Variety

Let X be a smooth projective variety over the complex numbers. We consider the cohomology of the sheaves and arising from Deligne–Beilinson cohomology and the Hodge filtration on the singular cohomology of X. We show that one can identify with the image of the truncated regulator map c¯2,1. In particular, this implies that is countable. Since this group is a direct summand of coker , this gives a partial answer to Voisin's conjecture that cocker() is countable. In the case of X a surface, we prove that the Albanese kernel T(X) is isomorphic to the group of global sections of if and only if pg=0.     

Indecomposable higher Chow cycles on Jacobians

We construct some natural indecomposable elements of with trivial regulator, and in particular, prove that is uncountable for C a generic curve or a generic hyperelliptic curve of genus . Received: 10 October 2000 / in final form: 12 June 2001 / Published online: 1 February 2002     

Elliptic functions and equations of modular curves

Let be a prime. We show that the space of weight one Eisenstein series defines an embedding into ${mathbb P}^{(p-3)/2}X_1(p)$ for the congruence group that is scheme-theoretically cut out by explicit quadratic equations. Received: 8 November 2000 / Published online: 17 August 2001     

Log-Syntomic Regulators and p-Adic Polylogarithms

The purpose of this paper is to show that under the syntomic regulator map from the motivic cohomology to the syntomic cohomology of the integer ring of a cyclotomic field, the image of Beilinson's element is a special value of the p-adic polylogarithm.     

On the fundamental group of the complement¶of certain affine hypersurfaces

Let and be two non-constant weighted homogeneous polynomials. Suppose that f and g are not powers of polynomials. Let p and q be two positive integers. In this paper, we calculate the fundamental group of the complement of the affine hypersurface in . In particular, we prove that the fundamental group is determined by p and q. Received: 29 January 1998     

Explicit valuations of division polynomials¶of an elliptic curve

In this paper, we estimate valuations of division polynomials and compute them explicitely at singular primes. We show that ν_?(ψ _m (M)) is asymptotically equal to ν_?(m) for a non-torsion point M such that M mod ? is non-zero and non-singular, and it is asymptotically equal to c ₁ m ¹ for some constant c ₁ for a non-torsion point M such that M mod ? is either singular or zero. Furthermore, we show that the common factors of φ _m (M) and ψ _m ²(M) have valuations at ? asymptotically equal to c ₂ m ² for some constant c ₂ when M mod ? is singular, which is a generalization of M. Ayad's result. Received: 10 July 1997 / Revised version: 11 May 1998     

On closed anti de Sitter spacetimes

This article deals with the structure of the fundamental group of compact anti de Sitter spacetimes, i.e. Lorentz manifolds with constant negative curvature. Algebraically such a manifold is the quotient of the universal cover of the homogeneous space by a discrete group acting properly and co-compactly on it. This exists if and only if is even. Indeed, as this was observed by Kulkarni, is contained in , and acts properly transitively on . It then suffices to take as a co-compact lattice in . The results of the present article give evidence to the question: in dimension , are all compact anti de Sitter spacetimes constructed in this way? Received: 18 May 1996 / Revised version: 3 January 1997     

On complex manifolds polarized by an ample line bundle of sectional genus q ( X ) + 2

Let (X,L) be a polarized manifold with dim X = n. In this paper, we classify (X,L) with n = 3, , and g(L)=q(X) + 2. Moreover we also classify (X,L) with , g(L)=q(x) + 2, and . Received February 12, 1999     

On the base point freeness of adjoint bundles¶on normal surfaces

We introduce some invariants of singularities which represent the anti-freeness of the adjoint linear systems. The invariants indicate that if either the singularities or the boundaries are worse then the adjoint linear systems are much global generative. Using these invariants, we prove effective global generation of adjoint linear systems on normal log surfaces. Received: 14 May 1999/ Revised version: 2 August 1999     

Complexity of Chow varieties and number of morphisms on surfaces of general type

We provide a general estimate for the number of irreducible components of a Chow variety, the variety that parametrizes algebraic cycles of given dimension and degree contained in a projective variety. The result is then applied to obtain an upper bound for the finite number of surfaces of general type that are images of a fixed surface. Received: 29 January 1998 / Revised version: 24 June 1998     

On a class of stochastic partial differential equations related to turbulent transport

Summary. We consider the Cauchy problem for the mass density ρ of particles which diffuse in an incompressible fluid. The dynamical behaviour of ρ is modeled by a linear, uniformly parabolic differential equation containing a stochastic vector field. This vector field is interpreted as the velocity field of the fluid in a state of turbulence. Combining a contraction method with techniques from white noise analysis we prove an existence and uniqueness result for the solution ρ∈C ^1,2([0,T]×ℝ ^d ,(S)^*), which is a generalized random field. For a subclass of Cauchy problems we show that ρ actually is a classical random field, i.e. ρ(t,x) is an L ²-random variable for all time and space parameters (t,x)∈[0,T]×ℝ ^d . Received: 27 March 1995 / In revised form: 15 May 1997     

On singular fibres of Lagrangian fibrations over holomorphic symplectic manifolds

We classify singular fibres over general points of the discriminant locus of projective Lagrangian fibrations over 4-dimensional holomorphic symplectic manifolds. The singular fibre F is the following either one: F is isomorphic to the product of an elliptic curve and a Kodaira singular fibre up to finite unramified covering or F is a normal crossing variety consisting of several copies of a minimal elliptic ruled surface of which the dual graph is Dynkin diagram of type or . Moreover, we show all types of the above singular fibres actually occur. Received: 10 March 2000 / Revised version: 29 September 2000 / Published online: 24 September 2001     

On the flatness of models of certain Shimura varieties of PEL-type

Consider a PEL-Shimura variety associated to a unitary group that splits over an unramified extension of . Rapoport and Zink have defined a model of the Shimura variety over the ring of integers of the completion of the reflex field at a place lying over p, with parahoric level structures at p. We show that this model is flat, as conjectured by Rapoport and Zink, and that its special fibre is reduced. Received: 11 September 2000 / Published online: 24 September 2001     

Elliptic equations for measures on infinite dimensional spaces and applications

We introduce and study a new concept of a weak elliptic equation for measures on infinite dimensional spaces. This concept allows one to consider equations whose coefficients are not globally integrable. By using a suitably extended Lyapunov function technique, we derive a priori estimates for the solutions of such equations and prove new existence results. As an application, we consider stochastic Burgers, reaction-diffusion, and Navier-Stokes equations and investigate the elliptic equations for the corresponding invariant measures. Our general theorems yield a priori estimates and existence results for such elliptic equations. We also obtain moment estimates for Gibbs distributions and prove an existence result applicable to a wide class of models. Received: 23 January 2000 / Revised version: 4 October 2000 / Published online: 5 June 2001