Log mirror symmetry and local mirror symmetry

We study Mirror Symmetry of log Calabi-Yau surfaces. On one hand, we consider the number of ``affine lines'' of each degree in the complement of a smooth cubic in the projective plane. On the other hand, we consider coefficients of a certain expansion of a function obtained from the integrals of dxdy/xy over 2-chains whose boundaries lie on B_\phi where {B_\phi} is a family of smooth cubics. Then, for small degrees, they coincide. We discuss the relation between this phenomenon and local mirror symmetry for projective plane in a Calabi-Yau 3-fold by Chiang-Klemm-Yau-Zaslow.Comment: 6 page

<Original>Preparation and Characterization of the Antibody against the Aldobiouronic acid Unit of Xylan

この論文は国立情報学研究所の学術雑誌公開支援事業により電子化されました。The aldobiouronic acid, 2-O-(4-O-methyl-a-D-glucuronic acid)-D-xylose, which was obtained by partial acid hydrolysis of xylan was coupled to methylated bovine serum albumin (BSA). The resulting sugar-protein conjugate was used as an antigen. Antibodies were raised by immunizing a rabbit with the antigen. The specificities for the antiserum obtained were examined by immunodiffusion analysis and enzyme-linked immunosorbent assay (ELISA). These results indicate that the antiserum contains antibodies against both BSA and the antigen. After removal of anti-BSA antibody by affinity chromatography, the specificity of the purified antiserum to wood cell wall polysaccharides and their hydrolysates was investigated by ELISA. The antiserum preferentially reacted with the conjugates containing xylan and aldobiouronic acid, which suggests that the antiserum recognizes the aldobiouronic acid unit of xylan

Log BPS numbers of log Calabi-Yau surfaces

Let $(S,E)$ be a log Calabi-Yau surface pair with $E$ a smooth divisor. We define new conjecturally integer-valued counts of $\mathbb{A}^1$-curves in $(S,E)$. These log BPS numbers are derived from genus 0 log Gromov-Witten invariants of maximal tangency along $E$ via a formula analogous to the multiple cover formula for disk counts. A conjectural relationship to genus 0 local BPS numbers is described and verified for del Pezzo surfaces and curve classes of arithmetic genus up to 2. We state a number of conjectures and provide computational evidence.Comment: 49 pages, 2 figure

Sheaves of maximal intersection and multiplicities of stable log maps

A great number of theoretical results are known about log Gromov-Witten invariants, but few calculations are worked out. In this paper we restrict to surfaces and to genus 0 stable log maps of maximal tangency. We ask how various natural components of the moduli space contribute to the log Gromov-Witten invariants. The first such calculation by Gross-Pandharipande-Siebert deals with multiple covers over rigid curves in the log Calabi-Yau setting. As a natural continuation, in this paper we compute the contributions of non-rigid irreducible curves in the log Calabi-Yau setting and that of the union of two rigid curves in general position. For the former, we construct and study a moduli space of "logarithmic" 1-dimensional sheaves and compare the resulting multiplicity with tropical multiplicity. For the latter, we explicitly describe the components of the moduli space and work out the logarithmic deformation theory in full, which we then compare with the deformation theory of the analogous relative stable maps.Comment: Added two example sections including a comparison with tropical multiplicity. 53 pages, 4 figure