[Paper Review] On Eckl's pseudo-effective reduction map

Brian Lehmann|arXiv (Cornell University)|Mar 5, 2011

Algebraic Geometry and Number Theory12 references4 citations

TL;DR

This paper constructs and refines two numerical reduction maps—L-trivial and pseudo-effective reduction maps—for pseudo-effective divisors on complex projective varieties, using algebraic techniques to correct and clarify earlier analytic approaches. The key contribution is showing that the pseudo-effective reduction map captures the numerical core of the Iitaka fibration and enables vanishing theorems for abundant divisors, generalizing results for big divisors.

ABSTRACT

Suppose that X is a complex projective variety and L is a pseudo-effective divisor. A numerical reduction map is a quotient of X by all subvarieties along which L is numerically trivial. We construct two variants: the L-trivial reduction map and the pseudo-effective reduction map of [Eckl05]. We show that these maps capture interesting geometric properties of L and use them to analyze abundant divisors.

Motivation & Objective

• To develop a birational and numerical framework for understanding the geometry of pseudo-effective divisors via reduction maps.
• To correct and re-derive Eckl's pseudo-effective reduction map using algebraic techniques instead of analytic methods.
• To show that the pseudo-effective reduction map identifies the numerical part of the Iitaka fibration when the divisor is abundant.
• To establish a vanishing theorem for abundant divisors by relating the support of effective divisors to the negative base locus.
• To define and characterize the L-trivial and pseudo-effective reduction maps via numerical dimension and restricted numerical dimension.

Proposed method

• Constructs the L-trivial reduction map by quotienting X by subvarieties where L is numerically trivial, ensuring the numerical dimension of L vanishes on general fibers.
• Introduces the pseudo-effective reduction map via the positive part Pσ(L), ensuring Pσ(L)|F ≡ 0 on general fibers F of the map.
• Uses birational models φ: Y → X and morphisms π: Y → Z to define the maps up to birational equivalence, depending only on the numerical class of L.
• Applies the restricted numerical dimension νX|C(L) to characterize the curves contracted by the pseudo-effective reduction map.
• Employs divisorial Zariski decomposition and the theory of restricted base loci to relate the geometry of L to the structure of the reduction map.
• Applies Theorem 6.6 (Kollár’s injectivity) to prove cohomological injectivity under assumptions on the support of D relative to the negative base locus.

Experimental results

Research questions

• RQ1 How can the pseudo-effective reduction map be algebraically constructed to correct gaps in Eckl’s original analytic proof?
• RQ2 In what way does the pseudo-effective reduction map reflect the numerical behavior of a divisor, especially in relation to the Iitaka fibration?
• RQ3 What conditions ensure that a divisor D induces an injective map on cohomology groups twisted by the multiplier ideal of |mL|?
• RQ4 How does the L-trivial reduction map relate to the numerical dimension of L on fibers?
• RQ5 Can the theory of reduction maps be extended to abundant divisors to yield vanishing theorems analogous to those for big divisors?

Key findings

• The L-trivial reduction map exists as a birational quotient of X by subvarieties where L|V ≡ 0, and satisfies ν(φ*L|F) = 0 for general fibers F.
• The pseudo-effective reduction map is the maximal quotient such that Pσ(φ*L)|F ≡ 0 on general fibers, and it factors the Iitaka fibration when κ(L) ≥ 0.
• The pseudo-effective reduction map is compatible with birational maps and divisorial Zariski decomposition, and is determined by curves C with νX|C(L) = 0.
• For abundant divisors, there exists a morphism g: W → T birationally equivalent to the pseudo-effective reduction map such that R(T, mD) ≅ R(X, mL) for some big D on T.
• The cohomology map H^i(X, O(K_X + L) ⊗ J(‖L‖)) → H^i(X, O(K_X + L + D) ⊗ J(‖L‖)) is injective for i > 0 under the condition that Supp(D) does not dominate Z under the Iitaka fibration.
• The proof relies on showing that φ*D has no g-horizontal components, which follows from the assumption that Supp(D) ∩ B_−(L) does not dominate Z.