Extending the Torelli map to toroidal compactifications of Siegel space

It has been known since the 1970s that the Torelli map M  _g →A  _g , associating to a smooth curve its Jacobian, extends to a regular map from the Deligne–Mumford compactification [`(\operatorname M)]<sub>g</sub>\overline {\operatorname {M}}_{g} to the 2nd Voronoi compactification [`(\operatorname A)]_g^vor\overline {\operatorname {A}}_{g}^{\mathrm {vor}}. We prove that the extended Torelli map to the perfect cone (1st Voronoi) compactification [`(\operatorname A)]<sub>g</sub><sup>perf</sup>\overline {\operatorname {A}}_{g}^{\mathrm {perf}} is also regular, and moreover [`(\operatorname A)]_g^vor\overline {\operatorname {A}}_{g}^{\mathrm {vor}} and [`(\operatorname A)]<sub>g</sub><sup>perf</sup>\overline {\operatorname {A}}_{g}^{\mathrm {perf}} share a common Zariski open neighborhood of the image of [`(\operatorname M)]_g\overline {\operatorname {M}}_{g}. We also show that the map to the Igusa monoidal transform (central cone compactification) is not regular for g≥9; this disproves a 1973 conjecture of Namikawa.     

The second cohomology with symmplectic coefficients of the moduli space of smooth projcetive curves

Each finite dimensional irreducible rational representation V of the symplectic group Sp_2g(Q) determines a generically defined local system V over the moduli space M_g of genus g smooth projective curves. We study H² (M_g; V) and the mixed Hodge structure on it. Specifically, we prove that if g 6, then the natural map IH²(M~_g; V) H²(M_g; V) is an isomorphism where M~_{_g} is tfhe Satake compactification of M_g. Using the work of Saito we conclude that the mixed Hodge structure on H²(M_g; V) is pure of weight 2+r if V underlies a variation of Hodge structure of weight r. We also obtain estimates on the weight of the mixed Hodge structure on H²(M_g; V) for 3 g < 6. Results of this article can be applied in the study of relations in the Torelli group T_g.     

Uniform boundedness of level structures on abelian varieties over complex function fields

Let X = Ω/Γ be a smooth quotient of a bounded symmetric domain Ω by an arithmetic subgroup . We prove the following generalization of Nadel's result: for any non-negative integer g, there exists a finite étale cover X_g = Ω/Γ(g) of X determined by a subgroup depending only on g, such that for any compact Riemann surface R of genus g and any non-constant holomorphic map f : R → X_g^* from R into the Satake-Baily-Borel compactification X_g^* of X_g, the image f(R) lies in the boundary ∂X_g: = X^*_g\X_g. Nadel proved it for g = 0 or 1. Moreover, for any positive integer n and any non-negative integer g≥0, we show that there exists a positive number a(n,g) depending only on n and g with the following property: a principally polarized non-isotrivial n-dimensional abelian variety over a complex function field of genus g does not have a level-N structure for N≥a(n,g). This was proved by Nadel for g = 0 or 1, and by Noguchi for arbitrary g under the additional hypothesis that the abelian variety has non-empty singular fibers.     

Sub-Riemannian constant mean curvature surfaces in the Heisenberg group as limits

The 3-dimensional Heisenberg group H together with its standard sub-Riemannian metric g₀ is viewed as the limit of a family of Riemannian manifolds, (H,g_u), u>0. For each u>0, we consider some invariant surfaces with constant mean curvature in (H,g_u). These surfaces of (H,g_u) have very nice limits as u0. We then define the mean curvature of a hypersurface in (H,g₀) to be the limit of its mean curvature in (H,g_u). We show that in a more general case, this definition is appropriate. Mathematics Subject Classification (2000) Principal 53C17; Secondary 22E25     

Compactification of symmetric varieties

The symmetric varieties considered in this paper are the quotientsG/H, whereG is an adjoint semi-simple group over a fieldk of characteristic 2, andH is the fixed point group of an involutorial automorphism ofG which is defined overk. In the casek=C, De Concini and Procesi (1983) constructed a wonderful compactification ofG/H. We prove the existence of such a compactification for arbitraryk. We also prove cohomology vanishing results for line bundles on the compactification. Dedicated to the memory of C. Chevalley     

Products of Hankel operators

This paper studies products of Hankel operators on the Hardy space. We show thatH _f ^(1)* H _f(2) H _f ^(3)* =0 for all permutation if and only if eitherH _f1 orH _f2 orH _f3 is zero. Using Douglas' localization theorem and Izuchi's theorem on Sarason's three functions problem, we show that

Non-completeness of the Arakelov-induced metric on moduli space of curves

Let X be a compact Riemann surface of genus g>1. We study two different, naturally defined metric forms on X: The hyperbolic metric form μ_hyp,X, arising from hyperbolic geometry, and the Arakelov metric form μ_Ar,X, arising from arithmetic algebraic geometry. Now consider a sequence X_t of Riemann surfaces approaching the Deligne-Mumford boundary of the moduli space of compact Riemann surfaces of genus g. We prove here that As a corollary of this result, we prove that the Weil-Petersson metric on the moduli space induced from the Arakelov metric is not complete, i.e., certain boundary components of the Deligne-Mumford compactification are at finite distance. The first author acknowledges support from grants from the NSF and PSC-CUNY. The second author thanks the Centre de Recerca Matemàtica (CRM) in Barcelona for its support and hospitality.     

Euler Characteristic of Spaces of Real Meromorphic Functions

For any connected component H ₀ of the space of real meromorphic functions we construct a compactification N(H ₀). Then we express the Euler characteristics of the spaces H ₀ and N(H ₀) in terms of topological invariants of functions in H ₀.     

Generalized Hadamard Matrices

The paper studies a generalized Hadamard matrix H = (g _{i j}) of order n with entries gi j from a group G of order n. We assume that H satisfies: (i) For m k, G = {g _{m i} g k i ^-1 i = 1,...., n} (ii) g _1i = g _i1 = 1 for each i; (iii) g _ij ^-1 = g _ji for all i, j. Conditions (i) and (ii) occur whenever G is a(P, L) -transitivity for a projective plane of order n. Condition (iii) holds in the case that H affords a symmetric incidence matrix for the plane. The paper proves that G must be a 2-group and extends previous work to the case that n is a square.     

Curves in the double plane

We study in detail locally Cohen-Macaulay curves in P⁴ which are contained in a double plane 2H, thus completing the classification of curves lying on surfaces of degree two. We describe the irreducible components of the Hilbert schemes H _d,g(2H) of lo-cally Cohen-Macaulay curves in 2H of degree d and arithmetic genus g, and we show that H _d,g(2H) is connected. We also discuss the Rao module of these curves and liaison and biliaison equiva-lence classes.     

EQUIVARIANT SELF EQUIVALENCES OF PRINCIPAL FIBRE BUNDLES

Let E be a compact Lie group, G a closed subgroup of E, and H a closed normal sub-group of G. For principal fibre bundle (E,p, E,/G;G) tmd (E/H,p‘,E/G;G/H), the relation between auta(E) (resp. autce (E)) and autG/H(E/H) (resp. autGe/H(E/H)) is investigated by using bundle map theory and transformation group theory. It will enable us to compute the group JG(E) (resp. SG(E)) while the group J G/u(E/H) is known.     

On the Hartley – Fourier sine generalized convolution

In this paper, we construct and study a new generalized convolution (f * g)(x) of functions f,g for the Hartley (H₁,H₂) and the Fourier sine (F_s) integral transforms. We will show that these generalized convolutions satisfy the following factorization equalities: We prove the existence of this generalized convolution on different function spaces, such as . As examples, applications to solve a type of integral equations and a type of systems of integral equations are presented. Copyright © 2013 John Wiley & Sons, Ltd.     

The Deligne–Mumford compactification of the real multiplication locus and Teichmüller curves in genus 3

In the moduli space M \mathcal{M} _g of genus-g Riemann surfaces, consider the locus RM_O \mathcal{R}{\mathcal{M}_{\mathcal{O}}} of Riemann surfaces whose Jacobians have real multiplication by the order O \mathcal{O} in a totally real number field F of degree g. If g = 3, we compute the closure of RM_O \mathcal{R}{\mathcal{M}_{\mathcal{O}}} in the Deligne–Mumford compactification of M \mathcal{M} _g and the closure of the locus of eigenforms over RM_O \mathcal{R}{\mathcal{M}_{\mathcal{O}}} in the Deligne–Mumford compactification of the moduli space of holomorphic 1-forms. For higher genera, we give strong necessary conditions for a stable curve to be in the boundary of RM_O \mathcal{R}{\mathcal{M}_{\mathcal{O}}} . Boundary strata of RM_O \mathcal{R}{\mathcal{M}_{\mathcal{O}}} are parameterized by configurations of elements of the field F satisfying a strong geometry of numbers type restriction.     

Platonic surfaces

If S _O is a Riemann surface with a complete metric of finite area and constant curvature -1, let S _C denote the conformal compactification of S _O. We show that, under the assumption that the cusps of S _O are large, there is a close relationship between the hyperbolic metrics on S _O and S _C. We use this relationship to show that , where the Platonic surface P _k is the conformal compactification of the modular surface S _k. Received: November, 1996; revised: February, 1998     

On Conformal Infinity and Compactifications of the Minkowski Space

Using the standard Cayley transform and elementary tools it is reiterated that the conformal compactification of the Minkowski space involves not only the “cone at infinity” but also the 2-sphere that is at the base of this cone. We represent this 2-sphere by two additionally marked points on the Penrose diagram for the compactified Minkowski space. Lacks and omissions in the existing literature are described, Penrose diagrams are derived for both, simple compactification and its double covering space, which is discussed in some detail using both the U(2) approach and the exterior and Clifford algebra methods. Using the Hodge *{\star} operator twistors (i.e. vectors of the pseudo-Hermitian space H _2,2) are realized as spinors (i.e., vectors of a faithful irreducible representation of the even Clifford algebra) for the conformal group SO(4, 2)/Z ₂. Killing vector fields corresponding to the left action of U(2) on itself are explicitly calculated. Isotropic cones and corresponding projective quadrics in H _p,q are also discussed. Applications to flat conformal structures, including the normal Cartan connection and conformal development has been discussed in some detail.     

On schur rings over cyclic groups

In this paper, we introduce the notion of wedge product of Schur rings. We show that for any nontrivial Schur ringS over a cyclic groupG, if there is a subgroupH such that Σ _g ε _H g Σ _g∈H g ∉S, thenS is either a dot product or wedge product for some Schur rings over smaller cyclic groups.     

On Some Moduli Spaces of Bundles on K3 Surfaces

We give infinitely many examples in which the moduli space of rank 2 H-stable sheaves on a K3 surface S endowed by a polarization H of degree 2g – 2, with Chern classes c₁ = H and c₂ = g – 1, is birationally equivalent to the Hilbert scheme S[g – 4] of zero dimensional subschemes of S of length g – 4. We get in this way a partial generalization of results from [5] and [1].