Congruences between Siegel modular forms II

Using a p-adic monodromy theorem on the affine ordinary locus in the minimally compactified moduli scheme modulo powers of a prime p of abelian varieties, we extend Katz?s results on congruence and p-adic properties of elliptic modular forms to Siegel modular forms of higher degree.     

Seminormal Varieties,Torsionfree Sheaves,and Picard groups

We give a construction of torsionfree sheaves on a seminormal variety Y using torsionfree sheaves on the normalization X and the non-normal locus W. We use it to find a relation between Picard groups of X, Y, and W. We apply it to determine the Picard groups of the generalized Jacobian, the compactified Jacobian and some subschemes associated to the moduli spaces of torsionfree sheaves of rank 2 and odd degree on a nodal curve.     

Kähler geometry of Douady spaces

We consider the generalized Petersson–Weil metric on the moduli space of compact submanifolds of a Kähler manifold or a projective variety. It is extended as a positive current to the space of points corresponding to reduced fibers, and estimates are shown. For moduli of projective varieties the Petersson–Weil form is the curvature of a certain determinant line bundle equipped with a Quillen metric. We investigate its extension to the compactified moduli space.     

Vector bundles with a fixed determinant on an irreducible nodal curve

LetM be the moduli space of generalized parabolic bundles (GPBs) of rankr and degree dona smooth curveX. LetM _−L be the closure of its subset consisting of GPBs with fixed determinant− L. We define a moduli functor for whichM _−L is the coarse moduli scheme. Using the correspondence between GPBs onX and torsion-free sheaves on a nodal curveY of whichX is a desingularization, we show thatM _−L can be regarded as the compactified moduli scheme of vector bundles onY with fixed determinant. We get a natural scheme structure on the closure of the subset consisting of torsion-free sheaves with a fixed determinant in the moduli space of torsion-free sheaves onY. The relation to Seshadri-Nagaraj conjecture is studied.     

Topology of the compactified Jacobians of singular curves

We compute the Euler number of the compactified Jacobian of a curve whose minimal unibranched normalization has only plane irreducible singularities with characteristic Puiseux exponents (p, q), (4, 2q, s), (6, 8, s), or (6, 10, s). Further, we derive a combinatorial method to compute the Betti numbers of the compactified Jacobian of an unibranched rational curve with singularities like above. Some of the Betti numbers can be stated explicitly.     

A weak version of the Lipman–Zariski conjecture

Markushevich and Tikhomirov provided a construction of an irreducible symplectic V-manifold of dimension 4, the relative compactified Prym variety of a family of curves with involution, which is a Lagrangian fibration with polarization of type (1,2). We give a characterization of the dual Lagrangian fibration. We also identify the moduli space of Lagrangian fibrations of this type and show that the duality defines a rational involution on it.     

On Moduli Spaces,Equidistribution, Estimates,and Rational Points of Algebraic Curves

We consider the moduli spaces of hyperelliptic curves, Artin–Schreier coverings, and some other families of curves of this type over fields of characteristic p. By using the Postnikov method, we obtain expressions for the Kloosterman sums. The distribution of angles of the Kloosterman sums was investigated on a computer. For small prime p, we study rational points on curves y ² = f(x). We consider the problem of the accuracy of estimates of the number of rational points of hyperelliptic curves and the existence of rational points of curves of the indicated type on the moduli spaces of these curves over a prime finite field.     

Topological flatness of local models for ramified unitary groups. I. The odd dimensional case

Local models are certain schemes, defined in terms of linear-algebraic moduli problems, which give étale-local neighborhoods of integral models of certain p-adic PEL Shimura varieties defined by Rapoport and Zink. When the group defining the Shimura variety ramifies at p, the local models (and hence the Shimura models) as originally defined can fail to be flat, and it becomes desirable to modify their definition so as to obtain a flat scheme. In the case of unitary similitude groups whose localizations at Qp are ramified, quasi-split GUn, Pappas and Rapoport have added new conditions, the so-called wedge and spin conditions, to the moduli problem defining the original local models and conjectured that their new local models are flat. We prove a preliminary form of their conjecture, namely that their new models are topologically flat, in the case n is odd.     

Smoothness on bubble tree compactified instanton moduli spaces

The bubble tree compactified instanton moduli space -Mκ （X） is introduced. Its singularity set Singκ（X） is described. By the standard gluing theory, one can show that- Mκ（X） - Singκ（X） is a topological orbifold. In this paper, we give an argument to construct smooth structures on it.     

An explicit semi-factorial compactification of the Néron model

C. Pépin recently constructed a semi-factorial compactification of the Néron model of an Abelian variety using the flattening technique of Raynaud–Gruson. Here we prove that an explicit semi-factorial compactification is a certain moduli space of sheaves — the family of compactified Jacobians.     

The Kodaira dimension of the moduli of K3 surfaces

The global Torelli theorem for projective K3 surfaces was first proved by Piatetskii-Shapiro and Shafarevich 35 years ago, opening the way to treating moduli problems for K3 surfaces. The moduli space of polarised K3 surfaces of degree 2d is a quasi-projective variety of dimension 19. For general d very little has been known hitherto about the Kodaira dimension of these varieties. In this paper we present an almost complete solution to this problem. Our main result says that this moduli space is of general type for d>61 and for d=46, 50, 54, 57, 58, 60.     

Surfaces with pg = 2, K2 = 3 and a pencil of curves of genus 2

We classify minimal smooth surfaces of general type with K ² = 3, p _g = 2 which admit a fibration of curves of genus 2.We prove that they form an irreducible set of dimension 22 in their moduli space.      

On the slope stratification of certain Shimura varieties

In this paper we study the slope stratification on the good reduction of the type C family Shimura varieties. We show that there is an open dense subset U of the moduli space such that any point in U can be deformed to a point with a given lower admissible Newton polygon. For the Siegel moduli spaces, this is obtained by F. Oort which plays an important role in his proof of the strong Grothendieck conjecture concerning the slope stratification. We also investigate the p-divisible groups and their isogeny classes arising from the abelian varieties in question. Received: 10 November 2004; 13 February 2005 The research is partially supported by NSC 93-2119-M-001-018.     

A modulus of smoothness on the unit sphere

For functions onS ^d−1 (the unit sphere inR ^d) and, in particular, forf∈L _p(S ^d−1), we define new simple moduli of smoothness. We relate different orders of these moduli, and we also relate these moduli to best approximation by spherical harmonics of order smaller thann. Our new moduli lead to sharper results than those now available for the known moduli onL _p(S ^d−1). Supported by NSERC Grant A4816 of Canada.     

Weak Type Inequalities for Moduli of Smoothness: The Case of Limit Value Parameters

In this paper we obtain Ul’yanov type inequalities for fractional moduli of smoothness/K-functionals for the limit value parameters: p=1 or q=∞. Needed versions of Nikol’skii type inequalities for trigonometric polynomials are given. We show that these estimates are sharp. Corresponding embedding theorems for the Lipschitz spaces are investigated.     

Prym varieties of cyclic coverings

The Prym map of type (g, n, r) associates to every cyclic covering of degree n of a curve of genus g ramified at a reduced divisor of degree r the corresponding Prym variety. We show that the corresponding map of moduli spaces is generically finite in most cases. From this we deduce the dimension of the image of the Prym map.     

Smooth Stable Planes and the Moduli Spaces of Locally Compact Translation Planes

 Smooth stable planes have been introduced in [3]. At every point p of a smooth stable plane the tangent spaces of the lines through p form a compact spread (see the definition in Section 2) on the tangent space thus defining a locally compact topological affine translation plane . We introduce the moduli space of isomorphism classes of compact spreads, . We show that for the topology of is not by constructing a sequence of non-classical spreads in that converges to the classical spread in , where . Moreover, we prove that the isomorphism type of varies continuously with the point p. Finally, we give examples of smooth affine planes which have both classical and non-classical tangent translation planes. (Received 15 April 1999; in revised form 22 October 1999)     

Strange bundles onP 2 and elementary transformations

Summary Here we begin the study of moduli of vector bundles on a surfaceS with a fixed restriction to a divisorD. Here we stress the caseD≊P ¹. In this way we construct many families of stable rank-2 bundles onP ² with unbalanced general splitting type (in characteristicp>0).

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Riassunto Si comincia qui lo studio dei moduli di fibrati vettoriali su una superficieS con una assegnata restrizione ad un divisoreD (quasi sempre qui conD≊P ¹). In caratteristicap si ottengono così molte famiglie di fibrati stabili suP ² con ?strana? restrizione ad una retta generica.

     

The Hodge conjecture for certain moduli varieties

For smooth projective varietiesX over ℂ, the Hodge Conjecture states that every rational Cohomology class of type (p, p) comes from an algebraic cycle. In this paper, we prove the Hodge conjecture for some moduli spaces of vector bundles on compact Riemann surfaces of genus 2 and 3.     

The moduli of flat PU(p,p)-structures with large Toledo invariants

For a compact Riemann surface X of genus is the moduli space of flat -connections on X. There are two invariants, the Chern class c and the Toledo invariant associated with each element in the moduli. The Toledo invariant is bounded in the range . This paper shows that the component, associated with a fixed (resp. ) and a fixed Chern class c, is connected (The restriction on implies p=q).