Simplified numerical form of universal finite type invariant of Gauss words

In the present paper, we study the finite type invariants of Gauss words. In the Polyak algebra techniques, we reduce the determination of the group structure to transformation of a matrix into its Smith normal form and we give the simplified form of a universal finite type invariant by means of the isomorphism of this transformation. The advantage of this process is that we can implement it as a computer program. We obtain the universal finite type invariant of degree 4, 5, and 6 explicitly. Moreover, as an application, we give the complete classification of Gauss words of rank 4 and the partial classification of Gauss words of rank 5 where the distinction of only one pair remains.Comment: 12 pages, 3 table

Re-calibration of SDF/SXDS Photometric Catalogs of Suprime-Cam with SDSS Data Release 8

We present photometric recalibration of the Subaru Deep Field (SDF) and Subaru/XMM-Newton Deep Survey (SXDS). Recently, Yamanoi et al. (2012) suggested the existence of a discrepancy between the SDF and SXDS catalogs. We have used the Sloan Digital Sky Survey (SDSS) Data Release 8 (DR8) catalog and compared stars in common between SDF/SXDS and SDSS. We confirmed that there exists a 0.12 mag offset in B-band between the SDF and SXDS catalogs. Moreover, we found that significant zero point offsets in i-band (~ 0.10 mag) and z-band (~ 0.14 mag) need to be introduced to the SDF/SXDS catalogs to make it consistent with the SDSS catalog. We report the measured zero point offsets of five filter bands of SDF/SXDS catalogs. We studied the potential cause of these offsets, but the origins are yet to be understood.Comment: 36 pages, 19 figures(128 EPS files), PASJ accepte

Hyperbolicity and fundamental groups of complex quasi-projective varieties

This paper investigates the relationship between the hyperbolicity of complex quasi-projective varieties $X$ and the (topological) fundamental group $\pi_1(X)$ in the presence of a linear representation $\varrho: \pi_1(X) \to {\rm GL}_N(\mathbb{C})$. We present our main results in three parts. Firstly, we show that if $\varrho$ is bigand the Zariski closure of $\varrho(\pi_1(X))$ semisimple, then for any $X^\sigma:=X\times_\sigma\mathbb{C}$ where $\sigma\in {\rm Aut}(\mathbb{C}/\mathbb{Q})$, there exists a proper Zariski closed subset $Z \subsetneqq X^\sigma$ such that any closed irreducible subvariety $V$ of $X^\sigma$ not contained in $Z$ is of log general type, and any holomorphic map from the punctured disk $\mathbb{D}^*$ to $X^\sigma$ with image not contained in $Z$ does not have an essential singularity at the origin. In particular, all entire curves in $X^\sigma$ lie on $Z$. We provide examples to illustrate the optimality of this condition. Secondly, assuming that $\varrho$ is big and reductive, we prove the generalized Green-Griffiths-Lang conjecture for $X^\sigma$. Furthermore, if $\varrho$ is large, we show that the special subsets of $X^\sigma$ that capture the non-hyperbolicity locus of $X^\sigma$ from different perspectives are equal, and this subset is proper if and only if $X$ is of log general type. Lastly, we prove that if $X$ is a special quasi-projective manifold in the sense of Campana or $h$-special, then $\varrho(\pi_1(X))$ is virtually nilpotent. We provides examples to demonstrate that this result is sharp and thus revise Campana's abelianity conjecture for smooth quasi-projective varieties. To prove these theorems, we develop new features in non-abelian Hodge theory, geometric group theory, and Nevanlinna theory. Some byproducts are obtained.Comment: v2, 98 pages, add new results on generalized Green-Griffiths-Lang conjecture and hyperbolicity of Galois conjugate varieties; v3, 99 pages, added results on orbifold base in quasi-projective setting, submitte