Generic Torelli and local Schottky theorems for Jacobian elliptic surfaces

In the limit [Formula: see text], we analyze a class of Schrödinger operators [Formula: see text] acting on sections of a vector bundle [Formula: see text] over a Riemannian manifold [Formula: see text] where [Formula: see text] is a Laplace type operator, [Formula: see text] is an endomorphism field and the potential energy [Formula: see text] has a non-degenerate minimum at some point [Formula: see text]. We construct quasimodes of WKB-type near [Formula: see text] for eigenfunctions associated with the low-lying eigenvalues of [Formula: see text]. These are obtained from eigenfunctions of the associated harmonic oscillator [Formula: see text] at [Formula: see text], acting on smooth functions on the tangent space.

We establish new measures of linear independence of logarithms on a commutative algebraic group. Let G be a connected commutative algebraic group over Q−− and let tG be the tangent space at the origin. We consider a vector u∈tG⊗Q−C such that its image by the exponential map of the Lie group G(C) is an algebraic point p∈G(Q−−). Let V be a hyperplane in tG. We obtain lower bounds for the distance d(u,V) between u and V⊗Q−C in the rational case, where V=tH is the tangent space at the origin of an algebraic connected subgroup of G. These lower bounds are the best currently known in terms of the height h(p) of p. They generalize measures of linear forms in logarithms previously obtained by Gaudron. Our approach is based on new arguments which allow us to exclude the so-called periodic case in the demonstration, by revisiting previous work of Bertrand and Philippon. Our proofs also rely on tools from Bost’s slope theory of hermitian vector bundles. Moreover, we present ultrametric analogues of our results, and we deal with the case where V=tH is a linear subspace of any dimension.

Here we give an upper bound for the number of irreducible components of the moduli scheme of stable rank r torsion-free sheaves of fixed degree on the integral curve X. This bound depends only on r, Sing(X),pa(X) and the corresponding number for the rank 1 case. Let X be an integral complete curve of arithmetic genus Pa and geometric genus g. For all integers r, d with r > 1 let M(X, r, d) (or just M(r, d)) be the moduli scheme of stable rank r torsion-free sheaves on X with degree d. For general background on stable vector bundles and sheaves on curves and their moduli schemes, see [N] or [S]. The number of irreducible components of M (r, d) does not depend on d, but only on X and r (see Theorem 1.2). Let n(r, X) (or just n(r)) be the number of irreducible components of M(r, d). Here we want to give upper bounds on n(r) depending only on the singularities of X, the integer r and n(l). This is given by Theorem 0.1 (see the discussion in the last part of Remark 1.4). To state Theorem 0.1 (our main result) we need to introduce a few concepts (e.g. measures for the singularities of X) and fix some notation. Let 7r: C -* X be the normalization (hence g = pa(C)). Set 6 := Pa g. Set o:= Ox. For any x C Sing(X), set Ox := Ox,x and let mx be the maximal ideal of Ox; set kx := Ox/mx. Set fx := dimK(Ext1(kx,Ox)) and f := max{fx}xEsirlg(x). For example, we have fx = 1 if and only if Ox is Gorenstein (see e.g. [Co], Lemma 2.1.4). Let ex be the multiplicity of X at x; set := max(ex) and e := Ex ex. Let F be a rank r torsion-free sheaf on X; the set of all formal isomorphism classes of the germs {FX}xEsirlg(x) will be called the formal isomorphism class of F. The set W of all possible isomorphism classes for rank r torsion-free sheaves around Sing(X) has in a natural way a scheme structure; for any reduced subscheme Q of W, let M(r, d, Q) be the reduced subscheme of M(r, d)r.eci parametrizing sheaves with formal isomorphism type in Q. Let n(r, Q) (or n(r, d, Q)) be the number of irreducible components of M(r, d, Q). By Theorem 1.2 this integer does not depend on d. Our main interest is when Q is an irreducible component of Wr.ec. Here is our main result. Received by the editors November 28, 1994. 1991 Mathematics Subject Classification. Primary 14H60, 14D20, 14B99. This research was partially supported by MURST and GNSAGA of CNR (Italy). The author is a member of Europroj (and its group Vector bundles on curves). ?)1997 Arnericari Mathernatical Society

We find upper bounds on the essential dimension of the moduli stack of parabolic vector bundles over a curve. When there is no parabolic structure, we improve the known upper bound on the essential dimension of the usual moduli stack. Our calculations also give lower bounds on the essential dimension of the semistable locus inside the moduli stack of vector bundles of rank r and degree d without parabolic structure.

Following G. Laumon [12], to a nonramified $\\ell$-adic local system\n$E$ of rank $n$ on a curve $X$ one associates a complex of $\\ell$-adic\nsheaves $\\sb n\\mathscr {K}\\sb E$ on the moduli stack of rank $n$\nvector bundles on $X$ with a section, which is cuspidal and satisfies\nthe Hecke property for $E$. This is a geometric counterpart of the\nwell-known construction due to J. Shalika [19] and\nI. Piatetski-Shapiro [18]. We express the cohomology of the tensor\nproduct $\\sb n\\mathscr {K}\\sb {E\\sb 1}\\otimes \\sb n\\mathscr {K}\\sb\n{E\\sb 2}$ in terms of cohomology of the symmetric powers of $X$. This\nmay be considered as a geometric interpretation of the local part of\nthe classical Rankin-Selberg method for ${\\rm GL}(n)$ in the framework\nof the geometric Langlands program.

Manifolds are abound in mathematics and physics, and increasingly in cybernetics and visualization where they often reflect properties of complex systems and their configurations. Differential topology gives us the tools to study these spaces and extract information about the underlying systems. This book offers a concise and modern introduction to the core topics of differential topology for advanced undergraduates and beginning graduate students. It covers the basics on smooth manifolds and their tangent spaces before moving on to regular values and transversality, smooth flows and differential equations on manifolds, and the theory of vector bundles and locally trivial fibrations. The final chapter gives examples of local-to-global properties, a short introduction to Morse theory and a proof of Ehresmann's fibration theorem. The treatment is hands-on, including many concrete examples and exercises woven into the text, with hints provided to guide the student.

The author describes a canonical structure on a Grassmannian fibration whose fiber is a Grassmann manifold of the tangent spaces of a smooth manifold. This structure generalizes the symplectic structure on the cotangent bundle. This symplectic form takes its values in a vector space or even in a vector bundle. This structure is canonical; it is uniquely defined by a smooth manifold. Bibliography: 5 titles.

In a previous paper, the authors together with L. Vrancken initiated the study of $3$-dimensional CR submanifolds of the nearly K\" ahler homogeneous $\mathbb S^3\times \mathbb S^3$. As is shown by Butruille this is one of only four homogeneous $6$-dimensional nearly K\"ahler manifolds. Besides its almost complex structure $J$ it also admits a canonical almost product structure $P$. Along a $3$-dimensional CR submanifold the tangent space of $\mathbb S^3\times\mathbb S^3$ can be naturally split as the orthogonal sum of three $2$-dimensional vector bundles $\mathcal D_1$, $\mathcal D_2$ and $\mathcal D_3$. We study the CR submanifolds in relation to the behavior of the almost product structure on these vector bundles.

Abstract. We study Riemannian or pseudo-Riemannian mani-folds which carry the space of closed conformal vector flelds of atleast 2-dimension. Subject to the condition that at each point theset of closed conformal vector flelds spans a non-degenerate sub-space of the tangent space at the point, we prove a global and alocal classiflcation theorems for such manifolds. 1. IntroductionConformal mappings and conformal vector flelds are important ingeneral relativity, as is well known since the early 1920’s [6, 18]. In1925, Brinkmann studied conformal mappings between Riemannian orpseudo-Riemannian Einstein spaces [1]. Later conformal vector flelds,or inflnitesimal conformal mappings on Einstein spaces were reducedto the case of gradient vector flelds, leading to a very fruitful theoryof conformal gradient vector flelds in general. Brinkmann’s work hasattracted renewed interest, especially in the context of general relativity[2, 3, 4, 5, 9, 13, 16], and the following local theorems have been shown:Proposition 1.1. [3]

Generic Torelli Theorem for Hypersurfaces in Compact Irreducible Hermitian Symmetric Spaces

We report on the construction of secondary invariants in connection with the Atiyah-Singer index theorem for families, and the Theorem of Riemann-Roch-Grothendieck. The local families index theorem plays an important role in the construction. In complex geometry, the corresponding objects are the analytic torsion forms and the analytic torsion currents. These objects exhibit natural functorial properties with respect to composition of maps. Gillet and Soulé have used these objects to prove a Riemann-Roch theorem in Arakelov geometry. Also we state a Riemann-Roch theorem for flat vector bundles, and report on the construction of corresponding higher analytic torsion forms.

Extension of 2-forms and symplectic varietiesYoshinori Namikawa q(α) := l/2).This form is independent of the choice of ν : Z → Z.(3) Put H := H 2 (U, C).Then there exists a trivialization of the local system H: H ∼ = H × S. Let D := {x ∈ P(H); q(x) = 0, q(x + x) > 0}.Then one has a period map p : S → D and p is a local isomorphism.Note that when Z is a symplectic variety with terminal singularities, the condition codim(Σ ⊂ Z) ≥ 4 is always satisfied ([Na 2]).The stability theorem will be proved by using the following theorem and the fact that a projective variety with rational singularities has Du Bois singularities (cf.[Ko]): Theorem 4..Let X be a Stein open subset of a complex algebraic variety.Assume that X has only rational Gorenstein singularities.Let Σ be the singular locus of X and let f : Y → X be a resolution of singularities such that fU where U := X \ Σ and i : U → X is a natural injection.

On the local torelli theorem, a review of known results

An efficacious realization is presented of finite-gap solutions of the Veselov-Novikov equation expressed in terms of the theta function of Prym varieties of double coverings of algebraic curves with two branch points. For the given Prym mapping equations are obtained which locally solve a problem of Riemann-Schottky type, and a local Torelli theorem is proved.

Oo The purpose of the present notes is to show some properties of Jacobian rings of smooth hypersurfaces of compact irreducible Hermitian