Issue 1, Volume 35, April 2010

2010 · Open access

1. Preface Preface PDF
2. ON COMPLEX ANALYSIS IN VIETNAM HA HUY KHOAI PDF

   ▸ Abstract

   In the development of contemporary mathematics in Vietnam complex analysis occupies a special place. We give a brief survey of the development of complex analysis in Vietnam since 1947, when the first mathematical research paper (and it was on complex analysis) written by a Vietnamese mathematician was published in an international journal. We describe how complex analysis in Vietnam developed under very special conditions: the anti-French resistance, the struggle for the reunification of the country, the American war, the economic crisis, and the change toward a market economy.

   1-6
3. HODGE CLASSES ASSOCIATED TO 1-PARAMETER FAMILIES OF CALABI-YAU 3-FOLDS PEDRO LUIS DEL ANGEL, STEFAN MÜLLER-STACH, DUCO VAN STRATEN, KANG ZUO PDF

   ▸ Abstract

   We use $L^2$-Higgs cohomology to determine the Hodge numbers of the parabolic cohomology $H^1(\overline{S}, j_∗\mathbb V)$, where the local system $\mathbb V$ arises from the third primitive cohomology of family of Calabi-Yau threefolds over a curve $\overline{S}$. The method gives a way to predict the presence of algebraic 2-cycles in the total space of the family and is applied to some examples.

   7-22
4. COUNTING CONICS IN COMPLETE INTERSECTIONS LAURENT BONAVERO, ANDREAS HÖRING PDF

   ▸ Abstract

   We count the number of conics through two general points in complete intersections when this number is finite and give an application in terms of quasi-lines.

   23-30
5. ON A RATIONALITY QUESTION IN THE GROTHENDIECK RING OF VARIETIES HÉLÈNE ESNAULT, ECKART VIEHWEG PDF

   ▸ Abstract

   We discuss elementary rationality questions in the Grothendieck ring of varieties for the quotient of a finite dimensional vector space over a characteristic 0 field by a finite group.

   31-41
6. WEAKLY BOUNDED HEIGHT ON MODULAR CURVES P. HABEGGER PDF

   ▸ Abstract

   We study the intersection of a fixed plane algebraic curve $C$ with modular curves of varying level. The height of points in such intersections cannot be bounded from above independently of the level when $C$ is defined over the field of algebraic numbers. But we find a certain class of curves $C$ for which the height is bounded logarithmically in the level. This bound is strong enough to imply certain finiteness result. Such evidence leads to a conjecture involving a logarithmic height bound unless $C$ is of so-called special type. We also discuss connections to recent progress on conjectures concerning unlikely intersections.

   43-69
7. DEFORMATIONS OF THE TANGENT BUNDLE AND THEIR RESTRICTION TO STANDARD RATIONAL CURVES JUN-MUK HWANG PDF

   ▸ Abstract

   We study the deformations of the tangent bundle of a uniruled manifold $X$ by restricting them to standard rational curves on $X$. By employing an idea from holomorphic symplectic geometry, we prove that, if $H^{2i}(X, \Omega^1_X) = 0$ for all $i \geqslant 0$, the splitting type on a standard rational curve remains unchanged under small deformations of the tangent bundle.

   71-77
8. THE MODULI SPACE OF $\mathbb Q$-HOMOLOGY PROJECTIVE PLANES WITH 5 QUOTIENT SINGULAR POINTS JONGHAE KEUM PDF

   ▸ Abstract

   We describe the moduli space of $\mathbb Q$-homology projective planes with 5 quotient singular points, the maximum possible case. In particular, we show that the moduli space has dimension 0.

   We also present an Enriques surface having two different elliptic fibrations with a multi-section giving the same configuration of 9 smooth rational curves of Dynkin type $3A_1 \oplus 2A_3.$

   79-89
9. JACOBI’S FORMULA FOR HESSE CUBIC CURVES K. MATSUMOTO, T. TERASOMA, S. YAMAZAKI PDF

   ▸ Abstract

   In this paper, we give analogs of Jacobi’s formula for Hesse cubic curves $C(\mu)$ on theta constants and unnormalized period integrals. We also give several expressions of the modular invariant $\mu$ of $C(\mu)$ in terms of theta constants, which are reformulations of [Bi] and [Kr].

   91-105
10. NOTE ON THE STOKES STRUCTURE OF FOURIER TRANSFORM TAKURO MOCHIZUKI PDF

    ▸ Abstract

    We study the Stokes structure of the Fourier transform of a meromorphic flat bundle on a projective line. We describe it in terms of the rapid decay homology introduced by S. Bloch and H. Esnault.

    107-158
11. CHERN NUMBERS OF A SINGULAR FIBER, MODULAR INVARIANTS AND ISOTRIVIAL FAMILIES OF CURVES SHENG-LI TAN PDF

    ▸ Abstract

    We use the modular invariants of a family of curves and the Chern numbers of a singular fiber to compute the Chern numbers of the total space. We also explain some inequalities between the local Chern numbers. As an application, one can classify singular fibers according to their Chern numbers.

    159-172
12. CARTAN-NOCHKA THEOREM WITH TRUNCATED COUNTING FUNCTIONS FOR MOVING TARGETS DO DUC THAI, SI DUC QUANG PDF

    ▸ Abstract

    The purpose of this article is twofold. The first is to show the explicit truncations in the Cartan-Nochka theorem for nonconstant meromorphic mappings of $\mathbf{C}^m$ into $\mathbf{P}^n(\mathbf{C})$ and moving targets. The second is to show the above type theorem over function fields.

    173-197
13. UNIFORMIZATION OF FAKE PROJECTIVE FOUR SPACES SAI-KEE YEUNG PDF

    ▸ Abstract

    We give criteria for a fake projective four space to be uniformized by the complex hyperbolic space of complex dimension four. This is achieved through a classification of the Chern numbers of a rational homology complex projective space of complex dimension four.

    199-205