On the second sectional H-arithmetic genus of polarized manifolds

Let (X,L) be a polarized variety defined over the complex number field with dim X=n. In this paper we introduce the notion of the i-th sectional H-arithmetic genus ^H_i(X,L) for every integer i with 0in. We expect that this invariant has a property similar to the Euler-Poincaré characteristic of the structure sheaf of i-dimensional varieties. In this paper, we consider the case where X is smooth and i=2, and we study a polarized version of some results in the theory of surfaces.Mathematics Subject Classification (2000): 14C20, 14C17, 14C40, 14J29, 14J30, 14J35, 14J40This research was partially supported by the Grant-in-Aid for Young Scientists (B) (No.14740018), The Ministry of Education, Culture, Sports, Science and Technology, Japan.     

On the sectional invariants of polarized manifolds

Let (X,L) be a polarized manifold of dimension n. In this paper, for any integer i with 0≤i≤n we introduce the notion of the ith sectional invariant of (X,L). We define the ith sectional Euler number e_i(X,L), the ith sectional Betti number b_i(X,L), and the ith sectional Hodge number of type (j,i−j) of (X,L) and we will study some properties of these.     

Algebraic non ruled surfaces with sectional genus equal to seven

Summary Let (X, L) be a pair consisting of a smooth, algebraic, non-ruled surfaceX and a very ample line bundleL on it. In this paper we classifyX in the casesď=2g−i, wherei=2, … 5,g is the sectional genus ofX andď is the degree of the minimal reduction ofX. We also give good estimates of the irregularityq and the geometric genusp _g. In particular we apply our results to the caseg=7.

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Riassunto Sia (X, L) la coppia formata da una superficie algebrica, liscia, non-rigataX e un fibrato di retteL molto ampio suX. In questo lavoro diamo una classificazione diX nel caso in cuiď=2g−i, dovei=2, … 5,q è il genere sezionale diX eď è il grado della riduzione minimale diX. Diamo anche delle stime dell'irregolaritàq e del genere geometricop _g diX. In particolare applichiamo i nostri risultati al casoq=7.

     

Addendum to “On the Sectional Geometric Genus of Quasi-Polarized Varieties,I” by Yoshiaki Fukuma

Let (X, L) be a polarized manifold defined over the complex number field with dim X = n such that L is very ample. In this article, we will improve the classification of (X, L) with g ₂(X, L) = h ²(𝒪 _X ) + 1 which was obtained in Fukuma (Fukuma 2004 Fukuma , Y. ( 2004 ). On the sectional geometric genus of quasi-polarized varieties, I . Comm. Alg. 32 : 1069 – 1100 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], Theorem 3.6), where g ₂(X, L) denotes the second sectional geometric genus of (X, L).     

A Lower Bound for the Sectional Genus of Quasi-Polarized Surfaces

Let (X,L) be a quasi-polarized variety, i.e. X is a smooth projective variety over the complex numbers and L is a nef and big divisor on X. Then we conjecture that g(L) = q(X), whereg(L) is the sectional genus of L and . In this paper, we treat the case . First we prove that this conjecture is true for , and we classify (X,L) withg(L)=q(X), where is the Kodaira dimension of X. Next we study some special cases of .     

Complex surfaces polarized by an ample and spanned line bundle of genus three

Let X be a complex connected projective nonsingular algebraic surface endowed with an ample line bundle L, which is spanned by its global sections. Pairs (X, L) as above, with sectional genus g(X, L)=1+(L·(K _X L))/2=3 are classified by means of the main techniques of adjunction theory.     

A bound on the plurigenera of projective varieties

We exhibit a sharp Castelnuovo bound for the i-th plurigenus of a smooth variety of given dimension n and degree d in the projective space P ^r , and classify the varieties attaining the bound, when n2, r2n+1, d>>r and i>>r. When n=2 and r=5, or n=3 and r=7, we give a complete classification, i.e. for any i1. In certain cases, the varieties with maximal plurigenus are not Castelnuovo varieties, i.e. varieties with maximal geometric genus. For example, a Castelnuovo variety complete intersection on a variety of dimension n+1 and minimal degree in P ^r , with r>(n ² +3n)/(n–1), has not maximal i-th plurigenus, for i>>r. As a consequence of the bound on the plurigenera, we obtain an upper bound for the self-intersection of the canonical bundle of a smooth projective variety, whose canonical bundle is big and nef. Mathematics Subject Classification (2000):Primary 14J99; Secondary 14N99     

Free lateral vibration of beams of variable cross section

Zusammenfassung Es werden die freien Querschwingungen von Balken mit exponentiell veränderlichen Querschnitten und Trägheitsmomenten für verschiedene Randbedingungen untersucht. Die Frequenzengleichungen, Eigenfrequenzen und Schwingungsformen werden angegeben (wobei die Auswertung einfachheitshalber auf die beiden ersten Eigenschwingungen beschränkt bleibt).

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Notation x, y, z Rectangular coordinates - L span of beam - A(x) cross sectional area - I(x) moment of inertia of cross section - X normal function for beam - X _i thei-th normal function - T function of time - P _i circular frequency for thei-th mode     

On Polarized 3-Folds (X,L) Such That <lowercase > h < /lowercase >0(L) = 2 and the Sectional Genus of (X,L) is Equal to the Irregularity of X

Let X be a smooth complex projective variety of dimension 3 and let L be an ample line bundle on X. In this article, we give a characterization of (X, L) with g(X, L) = q(X) and h⁰(L) = 2, where g(X, L) (resp. q(X)) denotes the sectional genus of (X, L) (resp. the irregularity of X).     

Numerical properties of Chern classes of certain covering manifolds

In this paper, we study numerical properties of Chern classes of certain covering manifolds. One of the main results is the following: Let ψ : X → Pⁿ be a finite covering of the n-dimensional complex projective space branched along a hypersurface with only simple normal crossings and suppose X is nonsingular. Let c_i(X) be the i-th Chern class of X. Then (i) if the canonical divisor K_X is numerically effective, then (−1)^kc_k(X) (k ≥ 2) is numerically positive, and (ii) if X is of general type, then (−1)ⁿcⁱ_l (X) cⁱ_r, (X) > 0, where i_l + … + i_r = n. Furthermore we show that the same properties hold for certain Kummer coverings.     

Abel maps of Gorenstein curves

Abstract  For a Gorenstein curve X and a nonsingular point P∈ X, we construct Abel maps and , where J_Xⁱ is the moduli scheme for simple, torsion-free, rank-1 sheaves on X of degree i. The image curves of A and A_P are shown to have the same arithmetic genus of X. Also, A and A_P are shown to be embeddings away from rational subcurves L⊂ X meeting in separating nodes. Finally we establish a connection with Seshadri’s moduli scheme U_X(1) for semistable, torsion-free, rank-1 sheaves on X, obtaining an embedding of A(X) into U_X(1). Keywords Abel map, Torsion-free rank-1 sheaf, Compactified Jacobian, Gorenstein singularity Mathematics Subject Classification (2000) 14H40, 14H60     

Arcs and resolution of singularities

For a certain class of varieties X, including the toric surfaces, we derive a formula for the valuation d_X on the arc space of a smooth ambient space Y, in terms of an embedded resolution of singularities. A simple transformation rule yields a formula for the geometric Poincaré series. Furthermore, we prove that for this class of varieties, the arithmetic and the geometric Poincaré series coincide. We also study the geometric valuation for plane curves.Research Assistant of the Fund for Scientific Research – Flanders (Belgium) (F.W.O.)Mathematics Subject Classification (2000): 14J17, 14E15, 14B20, 14M25, 03C98Acknowledgement The author would like to thank the referee, whose remarks have greatly improved the structure of this paper.     

Manifolds polarized by vector bundles

LetX be a complex projective manifold of dimension n and let ε be an ample vector bundle of rank r. Let also τ = τ (X,ε) = min {t ∈ ℝ : K_X + t det ε is nef} be the nef value of the pair (X, ε). In this paper we classify the pairs (X, ε) such that{ Mathematics Subject Classification (2000)14J60; 14J40; 14E30     

Directed Projection Functions of Convex Bodies

For 1 ≤ i < j < d, a j-dimensional subspace L of and a convex body K in , we consider the projection K|L of K onto L. The directed projection function v _i,j (K;L,u) is defined to be the i-dimensional size of the part of K|L which is illuminated in direction u ∈ L. This involves the i-th surface area measure of K|L and is motivated by Groemer’s [17] notion of semi-girth of bodies in . It is well-known that centrally symmetric bodies are determined (up to translation) by their projection functions, we extend this by showing that an arbitrary body is determined by any one of its directed projection functions. We also obtain a corresponding stability result. Groemer [17] addressed the case i = 1, j = 2, d = 3. For j > 1, we then consider the average of v _1,j (K;L,u) over all spaces L containing u and investigate whether the resulting function determines K. We will find pairs (d,j) for which this is the case and some pairs for which it is false. The latter situation will be seen to be related to some classical results from number theory. We will also consider more general averages for the case of centrally symmetric bodies. The research of the first author was supported in part by NSF Grant DMS-9971202 and that of the second author by a grant from the Volkswagen Foundation.     

Collapsing and Dirac-Type Operators

We analyze the limit of the spectrum of a geometric Dirac-type operator under a collapse with bounded diameter and bounded sectional curvature. In the case of a smooth limit space B, we show that the limit of the spectrum is given by the spectrum of a certain first-order differential operator on B, which can be constructed using superconnections. In the case of a general limit space X, we express the limit operator in terms of a transversally elliptic operator on a G-manifold X/ with X = X//G. As an application, we give a characterization of manifolds which do not admit uniform upper bounds, in terms of diameter and sectional curvature, on the k-th eigenvalue of the square of a Dirac-type operator. We also give a formula for the essential spectrum of a Dirac-type operator on a finite-volume manifold with pinched negative sectional curvature.     

On the iteration of the adjunction process for surfaces of negative Kodaira dimension

Let (X,L) be a pair consisting of a smooth, complex, projective surface X and L a very ample line bundle on it. Suppose that the Kodaira dimension K(X) of X is negative. Then using the results obtained by A.J.Sommese and A.Van de Ven in [11], we find that the sectional genus, g_k=g(L_k), of the successive iterated minimal reductions (X_k,L_k), see 1. for the definition, reach a maximum value and then monotonically decrease to a final value g_n=g(L_n)g=g(L). This result gives a concrete way to express any pair (X,L), with K(X)=–, in terms of a minimal model.     

Décomposition d"une théorie du potentiel en des sous-espaces elliptiques et paraboliques

Let V be a proper kernel on a measurable space (X, B), and E _V the cone of excessive functions generated by V. We give a necessary and sufficient condition to decompose the space (X, E _V ) in an ordered and countable family of subspaces (X _n, E _{V n} ), which are elliptic or parabolic. The X _i 's are finely open and measurable and form a partition of X. The kernel W = V _i on X _i is subordinate to V, and has a triangular matrix. Résumé. Soit V un noyau propre sur un espace mesurable (X, B), et E _V le cône des fonctions excessives engendré par V. Nous donnons une condition nécéssaire et suffisante pour décomposer l"espace (X, E _V ) en une famille dénombrable et ordonnée (X _n , E _{V n} ) de sous-espaces elliptiques ou paraboliques. Les X _i sont des ouverts fins mesurables et forment une partition de X. Le noyau W = V _i sur X _i est subordonné à V, et sa matrice est triangulaire.     

The Braided product of Ockham algebras

In this paper we introduce an algebraic concept of the product of Ockham algebras called the Braided product. We show that ifL _i MS(i=1, 2, ,n) then the Braided product ofL _i(i=1, 2, ,n) exists if and only ifL ₁, ,L _n have isomorphic skeletons.     

The rate function of hypoelliptic diffusions

Let be a hypoelliptic diffusion operator on a compact manifold M. Given an a priori smooth reference measure λ on M, we can then rewrite L as the sum of a λ-symmetric part L⁰ and a first-order drift part Y. The paper investigates the effect of the drift Y on the Donsker-Varadhan rate function corresponding to the large deviations of the empirical measure of the diffusion. When Y is in the linear span of the first and second-order Lie brackets of the X_i's, we derive an affine bound relating the rate functions associated with L and L⁰. As soon as one point exists where Y is not in the linear span of the first and second-order Lie brackets of the X_i's, we show that such an affine bound is impossible. © 1994 John Wiley & Sons, Inc.     

Asymptotic Properties of Self-Consistent Estimators with Mixed Interval-Censored Data

Mixed interval-censored (MIC) data consist of n intervals with endpoints L _i and R _i , i = 1, ..., n. At least one of them is a singleton set and one is a finite non-singleton interval. The survival time X _i is only known to lie between L _i and R _i , i = 1, 2, ..., n. Peto (1973, Applied Statistics, 22, 86–91) and Turnbull (1976, J. Roy. Statist. Soc. Ser. B, 38, 290–295) obtained, respectively, the generalized MLE (GMLE) and the self-consistent estimator (SCE) of the distribution function of X with MIC data. In this paper, we introduce a model for MIC data and establish strong consistency, asymptotic normality and asymptotic efficiency of the SCE and GMLE with MIC data under this model with mild conditions.