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Shigetaka Fukuda

Publications and source records attributed to Shigetaka Fukuda.

14 recordsLinked to original sources
arXiv2019

Note on the semiampleness theorem for the log canonical divisors on klt varieties which are nef and abundant

We give another alternative proof to the Kawamata semiampleness theorem for the log canonical divisors on klt varieties which are nef and abundant. After the first version of this article was posted to the e-print Arxiv, Prof. Fujino notified the author that the quick and essential proof ([Fujino. On Kawamata's theorem.(EMS 2011), Rem 2.7]) is already known. The author would like to thank him. More precisely, Prof. Fujino already gave the quick and essential proof ([Fujino. On Kawamata's thm.(EMS 2011), Rem 2.7], [Fujino. Finite generation of the lc ring in dim 4. (Kyoto J. Math. 50 (2010)), Rem 3.15]) from the finite generation thm (Birkar-Cascini-Hacon-McKernan [BCHM]) of the lc rings for klt pairs and from the fact (cf. Mourougane-Russo [MoRu, C.R.A.S. Math. 325 (1997)]) that a nef and abundant $\mathbf{Q}$-divisor $D$ is semiample if its graded ring is finitely generated:"For a nef and abundant lc divisor which is klt, the lc ring is finitely generated, thus it is semiample." [BCHM] first proved that the minimal model program runs for big klt lc divisors and next implied the finite generation of the lc rings for klt lc divisors which are not necessarily big from the Fujino-Mori lc bdle formula ([FM, J. Differential Geom., 56 (2000)]). Mourougane-Russo [MoRu] implies the semiampleness of a nef and abundant $\mathbf{Q}$-divisor whose graded ring is finitely generated, using the Kawamata numerically trivial fibrations ([Kawamata. Pluricanonical systems. Invent. Math. 79 (1985)]). Consequently the author withdraw the article.

math.AG↗
arXiv2014

Algebraic fiber space whose generic fiber and base space are of almost general type

We assume that the existence and termination conjecture for flips holds. A complex projective manifold is said to be {\it of almost general type} if the intersection number of the canonical divisor with every very general curve is strictly positive. Let $f$ be an algebraic fiber space from $X$ to $Y$. Then the manifold $X$ is of almost general type if every very general fiber $F$ and the base space $Y$ of $f$ are of almost general type.

math.AG↗
arXiv2010

A note on the projective varieties of almost general type

A $\mathbf{Q}$-Cartier divisor $D$ on a projective variety $M$ is {\it almost nup}, if $(D , C) > 0$ for every very general curve $C$ on $M$. An algebraic variety $X$ is of {\it almost general type}, if there exists a projective variety $M$ with only terminal singularities such that the canonical divisor $K_M$ is almost nup and such that $M$ is birationally equivalent to $X$. We prove that a complex algebraic variety is of almost general type if and only if it is neither uniruled nor covered by any family of varieties being birationally equivalent to minimal varieties with numerically trivial canonical divisors, under the minimal model conjecture. Furthermore we prove that, for a projective variety $X$ with only terminal singularities, $X$ is of almost general type if and only if the canonical divisor $K_X$ is almost nup, under the minimal model conjecture.

math.AG↗
arXiv2005

On the projective fourfolds with almost numerically positive canonical divisors

Let $X$ be a four-dimensional projective variety defined over the field of complex numbers with only terminal singularities. We prove that if the intersection number of the canonical divisor $K$ with every very general curve is positive ($K$ is almost numerically positive) then every very general proper subvariety of $X$ is of general type in the viewpoint of geometric Kodaira dimension. We note that the converse does not hold for simple abelian varieties.

math.AG↗
arXiv2004

On Numerically Effective Log Canonical Divisors

Let $(X,Δ)$ be a 4-dimensional log variety which is proper over the field of complex numbers and with only divisorial log terminal singularities. The log canonical divisor $K_X+Δ$ is semi-ample, if it is nef (numerically effective) and the Iitaka dimension $κ(X,K_X+Δ)$ is strictly positive. For the proof, we use Fujino's abundance theorem for semi log canonical threefolds.

math.AG↗
arXiv2004

On log canonical divisors that are log quasi-numerically positive

Let $(X, Δ)$ be a four-dimensional log variety that is projective over the field of complex numbers. Assume that $(X, Δ)$ is not Kawamata log terminal (klt) but divisorial log terminal (dlt). First we introduce the notion of "log quasi-numerically positive", by relaxing that of "numerically positive". Next we prove that, if the log canonical divisor $K_X + Δ$ is log quasi-numerically positive on $(X, Δ)$ then it is semi-ample.

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arXiv2003

A note on the ampleness of numerically positive log canonical and anti-log canonical divisors

In this short note, we consider the conjecture that the log canonical divisor (resp. the anti-log canonical divisor) $K_X + Δ$ (resp. $-(K_X + Δ)$) on a pair $(X, Δ)$ consisting of a complex projective manifold $X$ and a reduced simply normal crossing divisor $Δ$ on $X$ is ample if it is numerically positive. More precisely, we prove the conjecture for $K_X + Δ$ with $Δ\neq 0$ in dimension 4 and for $-(K_X + Δ)$ with $Δ\neq 0$ in dimension 3 or 4.

math.AG↗
arXiv1998

A Base Point Free Theorem of Reid Type, II

Let $X$ be a complete algebraic variety over {\bf C}. We consider a log variety $(X,Δ)$ that is weakly Kawamata log terminal. We assume that $K_X+Δ$ is a {\bf Q}-Cartier {\bf Q}-divisor and that every irreducible component of $\lfloor Δ\rfloor$ is {\bf Q}-Cartier. A nef and big Cartier divisor $H$ on $X$ is called {\it nef and log big} on $(X,Δ)$ if $H |_B$ is nef and big for every center $B$ of non-"Kawamata log terminal" singularities for $(X,Δ)$. We prove that, if $L$ is a nef Cartier divisor such that $aL-(K_X+Δ)$ is nef and log big on $(X,Δ)$ for some $a \in$ {\bf N}, then the complete linear system $| mL |$ is base point free for $m \gg 0$.

math.AG↗