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Katsutoshi Yamanoi

Publications and source records attributed to Katsutoshi Yamanoi.

11 recordsLinked to original sources
arXiv2025

Hyperbolicity and fundamental groups of complex quasi-projective varieties (III): applications

This paper is Part III of a series of three. We begin by introducing the notion of $h$-special varieties, which can be seen as varieties "chain-connected by the Zariski closures of entire curves." We prove that if $X$ is either a special complex quasi-projective variety in the sense of Campana or an $h$-special variety, then for any linear representation $\varrho:π_1(X)\to \mathrm{GL}_N(\mathbb{C})$, the image $\varrho(π_1(X))$ is virtually nilpotent. We also provide examples showing that this result is sharp, leading to a revised form of Campana's abelianity conjecture for smooth quasi-projective varieties. In addition, we prove a structure theorem for quasi-projective varieties with big and semisimple representations of the fundamental groups, thereby addressing a conjecture by Kollár in 1995. We also construct several examples of quasi-projective varieties that are special and $h$-special, highlighting certain atypical properties of the non-compact case in contrast with the projective setting.

math.AG↗
arXiv2025

Hyperbolicity and fundamental groups of complex quasi-projective varieties (II): via non-abelian Hodge theories

This is Part II of a series of three papers. We studies the hyperbolicity of complex quasi-projective varieties $X$ in the presence of a big and reductive representation $\varrho: π_1(X)\to {\rm GL}_N(\mathbb{C})$. For any Galois conjugate variety $X^σ$ with $σ\in {\rm Aut}(\mathbb{C}/\mathbb{Q})$, we prove the generalized Green-Griffiths-Lang conjecture. When $\varrho$ is furthermore large, we show that the special subsets of $X^σ$ describing the non-hyperbolicity locus coincide, and that this locus is proper exactly when $X$ is of log general type. Moreover, if the Zariski closure of $ρ(π_1(X))$ is semisimple, we prove that there exists a proper Zariski closed subset $Z \subsetneq X^σ$ such that every subvariety not contained in $Z$ is of log general type and all entire curves in $X^σ$ are contained in $Z$. This result extends the theorems of the third author (2010) and of Campana-Claudon-Eyssidieux (2015) from projective to quasi-projective varieties, and yields stronger conclusions even in the projective case.

math.AG↗
arXiv2025

Hyperbolicity and fundamental groups of complex quasi-projective varieties (I): Maximal quasi-Albanese dimension by Nevanlinna theory

This is the first part of a series of three papers. In this paper, we establish a Big Picard type theorem for holomorphic maps $f:Y \to X$, where $Y$ is a ramified covering of the punctured disc $\mathbb{D}^*$ with small ramification and $X$ is a complex quasi-projective variety of log-general type and of maximal quasi-Albanese dimension. As a byproduct, we prove the generalized Green-Griffiths-Lang conjecture for such $X$. This paper summarizes the parts of the three-paper series that are based primarily on Nevanlinna theory.

math.AG↗
arXiv2025

Linear Shafarevich Conjecture in positive characteristic, Hyperbolicity and Applications

Given a complex quasi-projective normal variety $X$ and a linear representation $\varrho:π_1(X)\to {\rm GL}_{N}(K)$ with $K$ any field of positive characteristic, we mainly establish the following results: 1. the construction of the Shafarevich morphism ${\rm sh}_\varrho:X\to {\rm Sh}_\varrho(X)$ associated with $\varrho$. 2. In cases where $X$ is projective, $\varrho$ is faithful and the $Γ$-dimension of $X$ is at most two (e.g. $\dim X=2$), we prove that the Shafarevich conjecture holds for $X$. 3. In cases where $\varrho$ is big, we prove that the Green-Griffiths-Lang conjecture holds for $X$. 4. When $\varrho$ is big and the Zariski closure of $\varrho(π_1(X))$ is a semisimple algebraic group, we prove that $X$ is pseudo Picard hyperbolic, and strongly of log general type. 5. If $X$ is special or $h$-special, then $\varrho(π_1(X))$ is virtually abelian. We also prove Claudon-Höring-Kollár's conjecture for complex projective manifolds with linear fundamental groups of any characteristic.

math.AG↗
arXiv2024

Reductive Shafarevich Conjecture

In this paper, we prove the holomorphic convexity of the covering of a complex projective {normal} variety $X$, which corresponds to the intersection of kernels of reductive representations $ρ:π_1(X)\to {\rm GL}_{N}(\mathbb{C})$, therefore answering a question by Eyssidieux, Katzarkov, Pantev, and Ramachandran in 2012. It is worth noting that Eyssidieux had previously proven this result in 2004 when $X$ is smooth. While our approach follows the general strategy employed in Eyssidieux's proof, it introduces several improvements and simplifications. Notably, it avoids the necessity of using the reduction mod $p$ method in Eyssidieux's original proof. Additionally, we construct the Shafarevich morphism for complex reductive representations of fundamental groups of complex quasi-projective varieties unconditionally, and proving its algebraic nature at the function field level.

math.AG↗
arXiv2024

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 $π_1(X)$ in the presence of a linear representation $\varrho: π_1(X) \to {\rm GL}_N(\mathbb{C})$. We present our main results in three parts. Firstly, we show that if $\varrho$ is bigand the Zariski closure of $\varrho(π_1(X))$ semisimple, then for any $X^σ:=X\times_σ\mathbb{C}$ where $σ\in {\rm Aut}(\mathbb{C}/\mathbb{Q})$, there exists a proper Zariski closed subset $Z \subsetneqq X^σ$ such that any closed irreducible subvariety $V$ of $X^σ$ not contained in $Z$ is of log general type, and any holomorphic map from the punctured disk $\mathbb{D}^*$ to $X^σ$ with image not contained in $Z$ does not have an essential singularity at the origin. In particular, all entire curves in $X^σ$ 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 $X^σ$. Furthermore, if $\varrho$ is large, we show that the special subsets of $X^σ$ that capture the non-hyperbolicity locus of $X^σ$ 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 $\varrho(π_1(X))$ 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.

math.AG↗
arXiv2023

Bloch's principle for holomorphic maps into subvarieties of semi-abelian varieties

We generalize a fundamental theorem in higher dimensional value distribution theory about entire curves in subvarieties $X$ of semi-abelian varieties to the situation of the sequences of holomorphic maps from the unit disc into $X$. This generalization implies, among other things, that subvarieties of log general type in semi-abelian varieties are pseudo-Kobayashi hyperbolic. As another application, we improve a classical theorem due to Cartan in 1920's about the system of nowhere vanishing holomorphic functions on the unit disc satisfying Borel's identity.

math.CV↗
arXiv2005

The Second Main Theorem for Holomorphic Curves into Semi-Abelian Varieties II

We establish the second main theorem with the best truncation level one for an entire holomorphic curve $f:\C \to A$ into a semi-abelian variety $A$ and an arbitrary effective reduced divisor $D$ on $A$; the low truncation level is important for applications. We will actually prove this for the jet lifts of $f$. Finally we give some applications, including the solution of a problem posed by Mark Green.

math.CV↗
arXiv2005

Degeneracy of Holomorphic Curves into Algebraic Varieties

Applying the Second Main Theorem we deal with the algebraic degeneracy of entire holomorphic curves from the complex plane into a complex algebraic normal variety of positive log Kodaira dimension that admits a finite proper morphism to a semi-abelian variety. We will also discuss applications to the Kobayashi hyperbolicity problem.

math.CV↗
arXiv1999

The Second Main Theorem for Holomorphic Curves into Semi-Abelian Varieties

Let f:C -> A be an entire holomorphic curve into a semi-Abelian variety A. Then the Zariski closure of f(C) is a translate of a semi-Abelian subvariety of A (logarithmic Bloch-Ochiai's theorem). The purpose of the present paper is to establish a quantitative version of the above result for such f i.e., the second main theorem and the defect relation.

math.CV↗