Iitaka Fibration Research Articles

Boundedness of elliptic Calabi–Yau threefolds

We show that elliptic Calabi–Yau threefolds form a bounded family. We also show that the same result holds for minimal terminal threefolds of Kodaira dimension 2 , upon fixing the rate of growth of pluricanonical forms and the degree of a multisection of the Iitaka fibration. Both of these hypotheses are necessary to prove the boundedness of such a family.

On Effective Log Iitaka Fibrations and Existence of Complements

Abstract We study the relationship between Iitaka fibrations and the conjecture on the existence of complements, assuming the good minimal model conjecture. In one direction, we show that the conjecture on the existence of complements implies the effective log Iitaka fibration conjecture. As a consequence, the effective log Iitaka fibration conjecture holds in dimension $3$. In the other direction, for any Calabi-Yau type variety $X$ such that $-K_{X}$ is nef, we show that $X$ has an $n$-complement for some universal constant $n$ depending only on the dimension of $X$ and two natural invariants of a general fiber of an Iitaka fibration of $-K_{X}$. We also formulate the decomposable Iitaka fibration conjecture, a variation of the effective log Iitaka fibration conjecture which is closely related to the structure of ample models of pairs with non-rational coefficients, and study its relationship with the forestated conjectures.

Higher-order estimates of long-time solutions to the Kähler-Ricci flow

In this article, we study the higher-order regularity of the Kähler-Ricci flow on compact Kähler manifolds with semi-ample canonical line bundle. By developing sharp new parabolic Schauder estimates on cylinders, we proved that when the generic fibers of the Iitaka fibration are biholomorphic to each other, the flow converges in Cloc∞-topology away from singular fibers to a negative Kähler-Einstein metric on the base manifold. In particular, we proved that the Ricci curvature of the flow is uniformly bounded on any compact subsets away from singular fibers when the generic fibers are biholomorphic to each other.

Threefolds of Kodaira dimension one

We prove that for any smooth complex projective threefold of Kodaira dimension one, the m-th pluricanonical map is birational to the Iitaka fibration for every m

Iitaka dimensions of vector bundles

Let X be a projective variety. If L is a line bundle on X, for each positive integer m in \({\mathbf {N}}(L)=\{m\in {\mathbb {N}}\mid H^0(X,L^{\otimes m})\ne 0\}\), the global sections of \(L^{\otimes m}\) define a rational map $$\begin{aligned} \phi _m:X\dashrightarrow Y_m\subseteq {\mathbb {P}}\bigl (H^0(X,L^{\otimes m})\bigr ), \end{aligned}$$ where \(Y_m\) is the closure of \(\phi _m(X)\). It is well-known that for all sufficiently large \(m\in {\mathbf {N}}(L)\), the rational maps \(\phi _m:X\dashrightarrow Y_m\) are birationally equivalent to a fixed fibration (the Iitaka fibration), and \(\kappa (L):=\dim Y_m\) is called the Iitaka dimension of L. In a recent paper titled “Iitaka fibrations for vector bundles”, Mistretta and Urbinati generalized this to a vector bundle E on X. Let \({\mathbf {N}}(E)\) be the set of positive integers m such that the evaluation map \(H^0(X,S^m E)\rightarrow S^m E_x\) is surjective for all points x in some nonempty open subset of X. For each \(m\in {\mathbf {N}}(E)\), the global sections of \(S^m E\) define a rational map $$\begin{aligned} \varphi _m:X\dashrightarrow Y_m\subseteq {\mathbb {G}}(H^0(X,S^m E),{{\mathrm{rank}}}S^m E), \end{aligned}$$ where \({\mathbb {G}}(H^0(X,S^m E),{{\mathrm{rank}}}S^m E)\) is the Grassmannian of \({{\mathrm{rank}}}S^m E\)-dimensional quotients of \(H^0(X,S^m E)\). Mistretta and Urbinati showed that for every \(m\in {\mathbf {N}}(E)\), the rational maps \(\varphi _{km}\) are birationally equivalent for sufficiently large k, and called \(\kappa (E):=\dim Y_{km}\) the Iitaka dimension of E. Here we first slightly improve Mistretta and Urbinati’s result to show that the rational maps \(\varphi _{m}\) are birationally equivalent for all sufficiently large \(m\in {\mathbf {N}}(E)\). Then we show that $$\begin{aligned} \kappa (E)\ge \kappa \bigl ({\mathcal {O}}_{{\mathbb {P}}(E)}(1)\bigr )-{{\mathrm{rank}}}E+1. \end{aligned}$$ An immediate corollary of this inequality is that if E is big then \(\kappa (E)=\dim X\), which answers a question of Mistretta and Urbinati. Another corollary is that if E is big then \(\det E\) is big, provided that \({\mathbf {N}}(E)\ne \emptyset \).

Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs

For every smooth complex projective variety $W$ of dimension $d$ and nonnegative Kodaira dimension, we show the existence of a universal constant $m$ depending only on $d$ and two natural invariants of the very general fibres of an Iitaka fibration of $W$ such that the pluricanonical system $|mK_W|$ defines an Iitaka fibration. This is a consequence of a more general result on polarized adjoint divisors. In order to prove these results we develop a generalized theory of pairs, singularities, log canonical thresholds, adjunction, etc.

Boundedness of log Calabi–Yau pairs of Fano type

We prove a boundedness result for klt pairs $(X,B)$ such that $K_X+B\equiv 0$ and $B$ is big. As a consequence we obtain a positive answer to the Effective Iitaka Fibration Conjecture for klt pairs with big boundary.

Uniform bounds for the Iitaka fibration

We give effective bounds for the uniformity of the Iitaka fibration. These bounds follow from an effective theorem on the birationality of some adjoint linear series. In particular we derive an effective version of the main theorem in [19].

The cohomological support locus of pluri-canonical sheaves and the Iitaka fibration

The cohomological support locus of pluri-canonical sheaves and the Iitaka fibration

On the pluricanonical maps of varieties of intermediate Kodaira dimension

In this paper we will prove a uniformity result for the Iitaka fibration $f:X \rightarrow Y$, provided that the generic fiber has a good minimal model and the variation of $f$ is zero or that $\kappa(X)=\rm{dim}(X)-1$.

On the Iitaka Fibration of Varieties of Maximal Albanese Dimension

We prove that the tetracanonical map of a variety X of maximal Albanese dimension induces the Iitaka fibration. Moreover, if X is of general type, then the tricanonical map is birational.

Varieties fibered by good minimal models

Let f:X->Y be an algebraic fiber space such that the general fiber has a good minimal model. We show that if f is the Iitaka fibration or if f is the Albanese map of relative dimension no more than three, then X has a good minimal model.

Effective log Iitaka fibrations for surfaces and threefolds

We prove an analogue of Fujino and Mori’s “bounding the denominators” (Theorem 3.1 of Fujino and Mori, J Diff Geom 50:167–188, 2000) in the log canonical bundle formula (see also Theorem 8.1 of Prokhorov and Shokurov, J Algebraic Geom 18(1):151–199, 2009) for Kawamata log terminal pairs of relative dimension one. As an application we prove that for a klt pair (X,Δ) of Kodaira codimension one and dimension at most three such that the coefficients of Δ are in a DCC set \({\mathcal{A}}\), there is a natural number N that depends only on \({\mathcal{A}}\) for which \({\lfloor{N(K_X+\Delta)\rfloor}}\) induces the Iitaka fibration. We also prove a birational boundedness result for klt surfaces of general type.

Effectiveness of the log Iitaka fibration for 3-folds and 4-folds

We prove the effectiveness of the log Iitaka fibration in Kodaira codimension two for varieties of dimension ≤4. In particular, we finish the proof of effective log Iitaka fibration in dimension two. Also, we show that for the log Iitaka fibration, if the fiber is of dimension two, the denominator of the moduli part is bounded.

Effective Iitaka fibrations

For every n n -dimensional projective manifold X X of Kodaira dimension 2 2 we show that Φ | M K X | \Phi _{|M K_X|} is birational to an Iitaka fibration for a computable positive integer M = M ( b , B n − 2 ) M = M(b, B_{n-2}) , where b > 0 b > 0 is minimal with | b K F | ≠ ∅ |bK_F| \ne \emptyset for a general fibre F F of an Iitaka fibration of X X , and where B n − 2 B_{n-2} is the Betti number of a smooth model of the canonical Z / ( b ) \mathbb {Z}/(b) -cover of the ( n − 2 ) (n-2) -fold F F . In particular, M M is a universal constant if the dimension n ≤ 4 n \le 4 .

On the uniformity of the Iitaka fibration

We study pluricanonical systems on smooth projective varieties of positive Kodaira dimension, following the approach of Hacon-McKernan, Takayama and Tsuji succesfully used in the case of varieties of general type. We prove a uniformity result for the Iitaka fibration X 99K Iitaka(X) of smooth projective varieties of positive Kodaira dimension, provided that Iitaka(X) is not uniruled, the variation of the fibration is maximal, and the general fiber has a good minimal model.

On the Irregularity of the Image of the Iitaka Fibration#

We characterize the irregularity of the image of the Iitaka fibration in terms of the dimension of certain cohomological support loci.

Iitaka’s fibrations via multiplier ideals

We give a new characterization of Iitaka's fibration of algebraic varieties associated to line bundles. Introducing an intersection number of line bundles and curves by using the notion of multiplier ideal sheaves, Iitaka's fibration can be regarded as a numerically trivial fibration in terms of this intersection theory.

Compact complex manifolds with numerically effective cotangent bundles

We prove that a projective manifold of dimension n=2 or 3 and Kodaira dimension 1 has a numerically effective cotangent bundle if and only if the Iitaka fibration is almost smooth, i.e. the only singular fibres are multiples of smooth elliptic curves ( n=2 ) resp. multiples of smooth Abelian or hyperelliptic surfaces ( n=3 ). In the case of a threefold which is fibred over a rational curve the proof needs an extra assumption concerning the multiplicities of the singular fibres. Furthermore, we prove the following theorem: let X be a complex manifold which is hyberbolic with respect to the Carathéodory-Reiffen-pseudometric, then any compact quotient of X has a numerically effective cotangent bundle.