[Paper Review] Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs

Caucher Birkar, De‐Qi Zhang|arXiv (Cornell University)|Oct 3, 2014

Algebraic Geometry and Number Theory17 references4 citations

TL;DR

This paper establishes effective bounds for pluricanonical systems on smooth projective varieties by proving that the pluricanonical system $|mK_W|$ defines an Iitaka fibration for some $m$ depending only on the dimension $d$, the minimal index $b_F$ of the general fiber, and the middle Betti number $\beta_{\widetilde{F}}$ of its canonical cover. The result is achieved via a generalized theory of polarized pairs and effective birationality for adjoint divisors with bounded invariants.

ABSTRACT

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.

Motivation & Objective

• To resolve the effective Iitaka fibration conjecture under bounded invariants of general fibers.
• To establish uniform bounds on the pluricanonical system $|mK_W|$ that define Iitaka fibrations.
• To develop a generalized theory of pairs, singularities, and log canonical thresholds for polarized adjoint divisors.
• To prove effective birationality for log canonical pairs with bounded coefficients and nef parts.
• To show that the existence of a universal constant $m(d, b_F, \beta_{\widetilde{F}})$ ensures the pluricanonical system defines the Iitaka fibration.

Proposed method

• Introduce generalized polarized pairs with boundary $B$ and nef part $M$, where $K_X + B + M$ is big and $M$ has bounded Cartier index.
• Use the canonical bundle formula to relate $|mK_W|$ on $W$ to $|m(K_X + B + M)|$ on the base $X$, preserving birationality.
• Apply the DCC (descending chain condition) for generalized log canonical thresholds and global ACC to control coefficients of $B$ and $M$.
• Construct a universal constant $m(\Lambda, d, r)$ depending on a DCC set $\Lambda$, dimension $d$, and Cartier index $r$ such that $|m(K_X + B + M)|$ defines a birational map.
• Use Euler’s totient function to define $N(\beta)$, bounding the index of the canonical cover of the fiber, and define the DCC set $A(b, N)$ for coefficients of $B$.
• Combine these tools to prove that $m(d, b_F, \beta_{\widetilde{F}}) = b \cdot m(\Lambda, d, r)$ suffices for $|mK_W|$ to define the Iitaka fibration.

Experimental results

Research questions

• RQ1 Can the effective Iitaka fibration conjecture be proven under bounded invariants of the general fiber of the fibration?
• RQ2 Is there a uniform bound $m(d, b_F, \beta_{\widetilde{F}})$ such that $|mK_W|$ defines the Iitaka fibration for all $m$ divisible by it?
• RQ3 How do the invariants $b_F$ and $\beta_{\widetilde{F}}$ of the canonical cover of the general fiber control the effectiveness of pluricanonical systems?
• RQ4 Can effective birationality be established for log canonical pairs with bounded coefficients and nef parts?
• RQ5 What is the role of the generalized pair structure in controlling the birationality of pluricanonical systems?

Key findings

• There exists a universal constant $m(d, b_F, \beta_{\widetilde{F}})$ depending only on the dimension $d$, the minimal index $b_F$ of the general fiber, and the middle Betti number $\beta_{\widetilde{F}}$ such that $|mK_W|$ defines an Iitaka fibration for all $m$ divisible by it.
• The constant $m(d, b_F, \beta_{\widetilde{F}})$ is constructed as $m = b \cdot m(\Lambda, d, r)$, where $\Lambda = A(b, N)$ and $r = Nb$, with $N = \operatorname{lcm}\{m \in \mathbb{N} \mid \varphi(m) \leq \beta_{\widetilde{F}}\}$.
• The coefficients of the boundary divisor $B$ lie in a DCC set $A(b, N)$, ensuring boundedness of singularities and effective control of the canonical bundle formula.
• The nef part $M$ has bounded Cartier index $Nb$, which is crucial for effective birationality and descent of linear systems.
• The result reduces the effective Iitaka fibration problem to effective birationality of $|m(K_X + B + M)|$ on the base, which is established via global ACC and DCC techniques.
• When the base $X$ is of general type, the bound depends only on $d$, as $b_F$ and $\beta_{\widetilde{F}}$ become irrelevant.