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51. CMB 2007 (vol 50 pp. 486)

Cynk, S.; Hulek, K.
Higher-Dimensional Modular\\Calabi--Yau Manifolds
We construct several examples of higher-dimensional Calabi--Yau manifolds and prove their modularity.

Categories:14G10, 14J32, 11G40

52. CMB 2007 (vol 50 pp. 567)

Joshi, Kirti
Exotic Torsion, Frobenius Splitting and the Slope Spectral Sequence
In this paper we show that any Frobenius split, smooth, projective threefold over a perfect field of characteristic $p>0$ is Hodge--Witt. This is proved by generalizing to the case of threefolds a well-known criterion due to N.~Nygaard for surfaces to be Hodge-Witt. We also show that the second crystalline cohomology of any smooth, projective Frobenius split variety does not have any exotic torsion. In the last two sections we include some applications.

Keywords:threefolds, Frobenius splitting, Hodge--Witt, crystalline cohomology, slope spectral sequence, exotic torsion
Categories:14F30, 14J30

53. CMB 2007 (vol 50 pp. 427)

Mejía, Israel Moreno
On the Image of Certain Extension Maps.~I
Let $X$ be a smooth complex projective curve of genus $g\geq 1$. Let $\xi\in J^1(X)$ be a line bundle on $X$ of degree $1$. Let $W=\Ext^1(\xi^n,\xi^{-1})$ be the space of extensions of $\xi^n$ by $\xi^{-1}$. There is a rational map $D_{\xi}\colon G(n,W)\rightarrow SU_{X}(n+1)$, where $G(n,W)$ is the Grassmannian variety of $n$-linear subspaces of $W$ and $\SU_{X}(n+1)$ is the moduli space of rank $n+1$ semi-stable vector bundles on $X$ with trivial determinant. We prove that if $n=2$, then $D_{\xi}$ is everywhere defined and is injective.

Categories:14H60, 14F05, 14D20

54. CMB 2007 (vol 50 pp. 196)

Fernández, Julio; González, Josep; Lario, Joan-C.
Plane Quartic Twists of $X(5,3)$
Given an odd surjective Galois representation $\varrho\from \G_\Q\to\PGL_2(\F_3)$ and a positive integer~$N$, there exists a twisted modular curve $X(N,3)_\varrho$ defined over $\Q$ whose rational points classify the quadratic $\Q$-curves of degree $N$ realizing~$\varrho$. This paper gives a method to provide an explicit plane quartic model for this curve in the genus-three case $N=5$.

Categories:11F03, 11F80, 14G05

55. CMB 2007 (vol 50 pp. 243)

Langlands, Robert P.
Un nouveau point de repère dans la théorie des formes automorphes
Dans le papier Beyond Endoscopy une id\'ee pour entamer la fonctorialit\'e en utilisant la formule des traces a \'et\'e introduite. Maints probl\`emes, l'existence d'une limite convenable de la formule des traces, est eqquiss\'ee dans cette note informelle mais seulement pour $GL(2)$ et les corps des fonctions rationelles sur un corps fini et en ne pas resolvant bon nombre de questions.

Categories:32N10, 14xx

56. CMB 2007 (vol 50 pp. 215)

Kloosterman, Remke
Elliptic $K3$ Surfaces with Geometric Mordell--Weil Rank $15$
We prove that the elliptic surface $y^2=x^3+2(t^8+14t^4+1)x+4t^2(t^8+6t^4+1)$ has geometric Mordell--Weil rank $15$. This completes a list of Kuwata, who gave explicit examples of elliptic $K3$-surfaces with geometric Mordell--Weil ranks $0,1,\dots, 14, 16, 17, 18$.

Categories:14J27, 14J28, 11G05

57. CMB 2007 (vol 50 pp. 161)

Arapura, Donu; Kang, Su-Jeong
Functoriality of the Coniveau Filtration
It is shown that the coniveau filtration on the cohomology of smooth projective varieties is preserved up to shift by pushforwards, pullbacks and products.


58. CMB 2007 (vol 50 pp. 105)

Klep, Igor
On Valuations, Places and Graded Rings Associated to $*$-Orderings
We study natural $*$-valuations, $*$-places and graded $*$-rings associated with $*$-ordered rings. We prove that the natural $*$-valuation is always quasi-Ore and is even quasi-commutative (\emph{i.e.,} the corresponding graded $*$-ring is commutative), provided the ring contains an imaginary unit. Furthermore, it is proved that the graded $*$-ring is isomorphic to a twisted semigroup algebra. Our results are applied to answer a question of Cimpri\v c regarding $*$-orderability of quantum groups.

Keywords:$*$--orderings, valuations, rings with involution
Categories:14P10, 16S30, 16W10

59. CMB 2007 (vol 50 pp. 126)

Ongay, Fausto
$\varphi$-Dialgebras and a Class of Matrix ``Coquecigrues''
Starting with the Leibniz algebra defined by a $\varphi$-dialgebra, we construct examples of ``coquecigrues,'' in the sense of Loday, that is to say, manifolds whose tangent structure at a distinguished point coincides with that of the Leibniz algebra. We discuss some possible implications and generalizations of this construction.

Keywords:Leibniz algebras, dialgebras

60. CMB 2006 (vol 49 pp. 592)

Sarti, Alessandra
Group Actions, Cyclic Coverings and Families of K3-Surfaces
In this paper we describe six pencils of $K3$-surfaces which have large Picard number ($\rho=19,20$) and each contains precisely five special fibers: four have A-D-E singularities and one is non-reduced. In particular, we characterize these surfaces as cyclic coverings of some $K3$-surfaces described in a recent paper by Barth and the author. In many cases, using 3-divisible sets, resp., 2-divisible sets, of rational curves and lattice theory, we describe explicitly the Picard lattices.

Categories:14J28, 14L30, 14E20, 14C22

61. CMB 2006 (vol 49 pp. 560)

Luijk, Ronald van
A K3 Surface Associated With Certain Integral Matrices Having Integral Eigenvalues
In this article we will show that there are infinitely many symmetric, integral $3 \times 3$ matrices, with zeros on the diagonal, whose eigenvalues are all integral. We will do this by proving that the rational points on a certain non-Kummer, singular K3 surface are dense. We will also compute the entire Néron-Severi group of this surface and find all low degree curves on it.

Keywords:symmetric matrices, eigenvalues, elliptic surfaces, K3 surfaces, Néron--Severi group, rational curves, Diophantine equations, arithmetic geometry, algebraic geometry, number theory
Categories:14G05, 14J28, 11D41

62. CMB 2006 (vol 49 pp. 464)

Ravindra, G. V.
A Note on Detecting Algebraic Cycles
The purpose of this note is to show that the homologically trivial cycles contructed by Clemens and their generalisations due to Paranjape can be detected by the technique of spreading out. More precisely, we associate to these cycles (and the ambient varieties in which they live) certain families which arise naturally and which are defined over $\bbC$ and show that these cycles, along with their relations, can be detected in the singular cohomology of the total space of these families.


63. CMB 2006 (vol 49 pp. 296)

Sch"utt, Matthias
On the Modularity of Three Calabi--Yau Threefolds With Bad Reduction at 11
This paper investigates the modularity of three non-rigid Calabi--Yau threefolds with bad reduction at 11. They are constructed as fibre products of rational elliptic surfaces, involving the modular elliptic surface of level 5. Their middle $\ell$-adic cohomology groups are shown to split into two-dimensional pieces, all but one of which can be interpreted in terms of elliptic curves. The remaining pieces are associated to newforms of weight 4 and level 22 or 55, respectively. For this purpose, we develop a method by Serre to compare the corresponding two-dimensional 2-adic Galois representations with uneven trace. Eventually this method is also applied to a self fibre product of the Hesse-pencil, relating it to a newform of weight 4 and level 27.

Categories:14J32, 11F11, 11F23, 20C12

64. CMB 2006 (vol 49 pp. 270)

Occhetta, Gianluca
A Characterization of Products of Projective Spaces
We give a characterization of products of projective spaces using unsplit covering families of rational curves.

Keywords:Rational curves, Fano varieties
Categories:14J40, 14J45

65. CMB 2006 (vol 49 pp. 196)

Chernousov, Vladimir
Another Proof of Totaro's Theorem on $E_8$-Torsors
We give a short proof of Totaro's theorem that every$E_8$-torsor over a field $k$ becomes trivial over a finiteseparable extension of $k$of degree dividing $d(E_8)=2^63^25$.

Categories:11E72, 14M17, 20G15

66. CMB 2006 (vol 49 pp. 11)

Bevelacqua, Anthony J.; Motley, Mark J.
Going-Down Results for $C_{i}$-Fields
We search for theorems that, given a $C_i$-field $K$ and a subfield $k$ of $K$, allow us to conclude that $k$ is a $C_j$-field for some $j$. We give appropriate theorems in the case $K=k(t)$ and $K = k\llp t\rrp$. We then consider the more difficult case where $K/k$ is an algebraic extension. Here we are able to prove some results, and make conjectures. We also point out the connection between these questions and Lang's conjecture on nonreal function fields over a real closed field.

Keywords:$C_i$-fields, Lang's Conjecture
Categories:12F, 14G

67. CMB 2006 (vol 49 pp. 72)

Dwilewicz, Roman J.
Additive Riemann--Hilbert Problem in Line Bundles Over $\mathbb{CP}^1$
In this note we consider $\overline\partial$-problem in line bundles over complex projective space $\mathbb{CP}^1$ and prove that the equation can be solved for $(0,1)$ forms with compact support. As a consequence, any Cauchy-Riemann function on a compact real hypersurface in such line bundles is a jump of two holomorphic functions defined on the sides of the hypersurface. In particular, the results can be applied to $\mathbb{CP}^2$ since by removing a point from it we get a line bundle over $\mathbb{CP}^1$.

Keywords:$\overline\partial$-problem, cohomology groups, line bundles
Categories:32F20, 14F05, 32C16

68. CMB 2005 (vol 48 pp. 622)

Vénéreau, Stéphane
Hyperplanes of the Form ${f_1(x,y)z_1+\dots+f_k(x,y)z_k+g(x,y)}$ Are Variables
The Abhyankar--Sathaye Embedded Hyperplane Problem asks whe\-ther any hypersurface of $\C^n$ isomorphic to $\C^{n-1}$ is rectifiable, {\em i.e.,} equivalent to a linear hyperplane up to an automorphism of $\C^n$. Generalizing the approach adopted by Kaliman, V\'en\'ereau, and Zaidenberg which consists in using almost nothing but the acyclicity of $\C^{n-1}$, we solve this problem for hypersurfaces given by polynomials of $\C[x,y,z_1,\dots, z_k]$ as in the title.

Keywords:variables, Abhyankar--Sathaye Embedding Problem
Categories:14R10, 14R25

69. CMB 2005 (vol 48 pp. 547)

Fehér, L. M.; Némethi, A.; Rimányi, R.
Degeneracy of 2-Forms and 3-Forms
We study some global aspects of differential complex 2-forms and 3-forms on complex manifolds. We compute the cohomology classes represented by the sets of points on a manifold where such a form degenerates in various senses, together with other similar cohomological obstructions. Based on these results and a formula for projective representations, we calculate the degree of the projectivization of certain orbits of the representation $\Lambda^k\C^n$.

Keywords:Classes of degeneracy loci, 2-forms, 3-forms, Thom polynomials, global singularity theory
Categories:14N10, 57R45

70. CMB 2005 (vol 48 pp. 428)

Miyamoto, Roland; Top, Jaap
Reduction of Elliptic Curves in Equal Characteristic~3 (and~2)
and fibre type for elliptic curves over discrete valued fields of equal characteristic~3. Along the same lines, partial results are obtained in equal characteristic~2.

Categories:14H52, 14K15, 11G07, 11G05, 12J10

71. CMB 2005 (vol 48 pp. 473)

Zeron, E. S.
Logarithms and the Topology of the Complement of a Hypersurface
This paper is devoted to analysing the relation between the logarithm of a non-constant holomorphic polynomial $Q(z)$ and the topology of the complement of the hypersurface defined by $Q(z)=0$.

Keywords:Logarithm, homology groups and periods
Categories:32Q55, 14F45

72. CMB 2005 (vol 48 pp. 414)

Kaveh, Kiumars
Vector Fields and the Cohomology Ring of Toric Varieties
Let $X$ be a smooth complex projective variety with a holomorphic vector field with isolated zero set $Z$. From the results of Carrell and Lieberman there exists a filtration $F_0 \subset F_1 \subset \cdots$ of $A(Z)$, the ring of $\c$-valued functions on $Z$, such that $\Gr A(Z) \cong H^*(X, \c)$ as graded algebras. In this note, for a smooth projective toric variety and a vector field generated by the action of a $1$-parameter subgroup of the torus, we work out this filtration. Our main result is an explicit connection between this filtration and the polytope algebra of $X$.

Keywords:Toric variety, torus action, cohomology ring, simple polytope,, polytope algebra
Categories:14M25, 52B20

73. CMB 2005 (vol 48 pp. 180)

Cynk, Sławomir; Meyer, Christian
Geometry and Arithmetic of Certain Double Octic Calabi--Yau Manifolds
We study Calabi--Yau manifolds constructed as double coverings of $\mathbb{P}^3$ branched along an octic surface. We give a list of 87 examples corresponding to arrangements of eight planes defined over $\mathbb{Q}$. The Hodge numbers are computed for all examples. There are 10 rigid Calabi--Yau manifolds and 14 families with $h^{1,2}=1$. The modularity conjecture is verified for all the rigid examples.

Keywords:Calabi--Yau, double coverings, modular forms
Categories:14G10, 14J32

74. CMB 2005 (vol 48 pp. 237)

Kimura, Kenichiro
Indecomposable Higher Chow Cycles
Let $X$ be a projective smooth variety over a field $k$. In the first part we show that an indecomposable element in $CH^2(X,1)$ can be lifted to an indecomposable element in $CH^3(X_K,2)$ where $K$ is the function field of 1 variable over $k$. We also show that if $X$ is the self-product of an elliptic curve over $\Q$ then the $\Q$-vector space of indecomposable cycles $CH^3_{ind}(X_\C,2)_\Q$ is infinite dimensional. In the second part we give a new definition of the group of indecomposable cycles of $CH^3(X,2)$ and give an example of non-torsion cycle in this group.

Categories:14C25, 19D45

75. CMB 2005 (vol 48 pp. 203)

de Quehen, Victoria E.; Roberts, Leslie G.
Non-Cohen--Macaulay Projective Monomial Curves with Positive ${h}$-Vector
We find an infinite family of projective monomial curves all of which have $h$-vector with no negative values and are not Cohen-Macaulay.

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