Notions of numerical Iitaka dimension do not coincide

Let X be a smooth projective variety. The Iitaka dimension of a divisor D is an important invariant, but it does not only depend on the numerical class of D. However, there are several definitions of “numerical Iitaka dimension”, depending only on the numerical class. In this note, we show that there exists a pseudoeffective R-divisor for which these invariants take different values. The key is the construction of an example of a pseudoeffective R-divisor D+ for which h 0 (X, bmD+c + A) is bounded above and below by multiples of m3/2 for any sufficiently ample A.

First published in Journal of Algebraic Geometry on 2021-02-02, published by the American Mathematical Society. © 2021 American Mathematical Society.

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Work Title Notions of numerical Iitaka dimension do not coincide
Access
Open Access
Creators
  1. John Lesieutre
Keyword
  1. Iitaka dimension
License CC BY-NC-ND 4.0 (Attribution-NonCommercial-NoDerivatives)
Work Type Article
Publisher
  1. American Mathematical Society (AMS)
Publication Date February 2, 2021
Publisher Identifier (DOI)
  1. 10.1090/jag/763
Source
  1. Journal of Algebraic Geometry
Deposited May 23, 2022

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Version 1
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    Keyword
    • Iitaka dimension
    Description
    • <p>Let <inline-formula content-type="math/mathml">
    • <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X">
    • <mml:semantics>
    • <mml:mi>X</mml:mi>
    • <mml:annotation encoding="application/x-tex">X</mml:annotation>
    • </mml:semantics>
    • </mml:math>
    • </inline-formula> be a smooth projective variety. The Iitaka dimension of a divisor <inline-formula content-type="math/mathml">
    • <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D">
    • <mml:semantics>
    • <mml:mi>D</mml:mi>
    • <mml:annotation encoding="application/x-tex">D</mml:annotation>
    • </mml:semantics>
    • </mml:math>
    • </inline-formula> is an important invariant, but it does not only depend on the numerical class of <inline-formula content-type="math/mathml">
    • <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D">
    • <mml:semantics>
    • <mml:mi>D</mml:mi>
    • <mml:annotation encoding="application/x-tex">D</mml:annotation>
    • </mml:semantics>
    • </mml:math>
    • </inline-formula>. However, there are several definitions of “numerical Iitaka dimension”, depending only on the numerical class. In this note, we show that there exists a pseuodoeffective <inline-formula content-type="math/mathml">
    • <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R">
    • <mml:semantics>
    • <mml:mrow class="MJX-TeXAtom-ORD">
    • <mml:mi mathvariant="double-struck">R</mml:mi>
    • </mml:mrow>
    • <mml:annotation encoding="application/x-tex">\mathbb {R}</mml:annotation>
    • </mml:semantics>
    • </mml:math>
    • </inline-formula>-divisor for which these invariants take different values. The key is the construction of an example of a pseudoeffective <inline-formula content-type="math/mathml">
    • <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R">
    • <mml:semantics>
    • <mml:mrow class="MJX-TeXAtom-ORD">
    • <mml:mi mathvariant="double-struck">R</mml:mi>
    • </mml:mrow>
    • <mml:annotation encoding="application/x-tex">\mathbb {R}</mml:annotation>
    • </mml:semantics>
    • </mml:math>
    • </inline-formula>-divisor <inline-formula content-type="math/mathml">
    • <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D Subscript plus">
    • <mml:semantics>
    • <mml:msub>
    • <mml:mi>D</mml:mi>
    • <mml:mo>+</mml:mo>
    • </mml:msub>
    • <mml:annotation encoding="application/x-tex">D_+</mml:annotation>
    • </mml:semantics>
    • </mml:math>
    • </inline-formula> for which <inline-formula content-type="math/mathml">
    • <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h Superscript 0 Baseline left-parenthesis upper X comma left floor m upper D Subscript plus Baseline right floor plus upper A right-parenthesis">
    • <mml:semantics>
    • <mml:mrow>
    • <mml:msup>
    • <mml:mi>h</mml:mi>
    • <mml:mn>0</mml:mn>
    • </mml:msup>
    • <mml:mo stretchy="false">(</mml:mo>
    • <mml:mi>X</mml:mi>
    • <mml:mo>,</mml:mo>
    • <mml:mrow>
    • <mml:mo>⌊</mml:mo>
    • <mml:mrow class="MJX-TeXAtom-ORD">
    • <mml:mi>m</mml:mi>
    • <mml:msub>
    • <mml:mi>D</mml:mi>
    • <mml:mo>+</mml:mo>
    • </mml:msub>
    • </mml:mrow>
    • <mml:mo>⌋</mml:mo>
    • </mml:mrow>
    • <mml:mo>+</mml:mo>
    • <mml:mi>A</mml:mi>
    • <mml:mo stretchy="false">)</mml:mo>
    • </mml:mrow>
    • <mml:annotation encoding="application/x-tex">h^0(X,\left \lfloor {m D_+}\right \rfloor +A)</mml:annotation>
    • </mml:semantics>
    • </mml:math>
    • </inline-formula> is bounded above and below by multiples of <inline-formula content-type="math/mathml">
    • <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m Superscript 3 slash 2">
    • <mml:semantics>
    • <mml:msup>
    • <mml:mi>m</mml:mi>
    • <mml:mrow class="MJX-TeXAtom-ORD">
    • <mml:mn>3</mml:mn>
    • <mml:mrow class="MJX-TeXAtom-ORD">
    • <mml:mo>/</mml:mo>
    • </mml:mrow>
    • <mml:mn>2</mml:mn>
    • </mml:mrow>
    • </mml:msup>
    • <mml:annotation encoding="application/x-tex">m^{3/2}</mml:annotation>
    • </mml:semantics>
    • </mml:math>
    • </inline-formula> for any sufficiently ample <inline-formula content-type="math/mathml">
    • <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A">
    • <mml:semantics>
    • <mml:mi>A</mml:mi>
    • <mml:annotation encoding="application/x-tex">A</mml:annotation>
    • </mml:semantics>
    • </mml:math>
    • </inline-formula>.</p>
    • Let X be a smooth projective variety. The Iitaka dimension of a divisor D is an important invariant, but it does not only depend on the numerical class of D. However, there are several definitions of “numerical Iitaka dimension”, depending only on the numerical class. In this note, we show that there exists a pseudoeffective R-divisor for which these invariants take different values. The key is the construction of an example of a pseudoeffective R-divisor D+ for which h 0 (X, bmD+c + A) is bounded above and below by multiples of m3/2 for any sufficiently ample A.
  • Updated