[Paper Review] Two observations about normal functions

Christian Schnell|ArXiv.org|Apr 1, 2009

Algebraic Geometry and Number Theory8 references3 citations

TL;DR

This paper establishes two fundamental results about normal functions: (1) If a normal function associated to a primitive Hodge class vanishes on an algebraic curve, it must be singular at a point where the curve meets the dual variety's discriminant locus; (2) For sufficiently ample embeddings (degree ≥3), the intersection cohomology of the dual variety contains the cohomology of the original variety as a direct summand. These results connect singularity theory of normal functions with the geometry of dual varieties.

ABSTRACT

Two simple observations are made: (1) If the normal function associated to a Hodge class has a zero locus of positive dimension, then it has a singularity. (2) The intersection cohomology of the dual variety contains the cohomology of the original variety, if the degree of the embedding is large.

Motivation & Objective

• To clarify the relationship between the vanishing locus of a normal function and its singularities.
• To investigate how the cohomology of a smooth projective variety embeds into the intersection cohomology of its dual variety under ample embeddings.
• To provide elementary but non-trivial observations that may clarify or extend known results in Hodge theory and normal function singularities.
• To establish conditions under which the intersection cohomology of the dual variety contains the cohomology of the original variety as a direct summand.

Proposed method

• Use the Leray spectral sequence to relate the Hodge class η to the cohomology class [η] of the normal function ν.
• Apply the compatibility of spectral sequences to show that the pullback of η to a curve in the zero locus of ν vanishes on the smooth locus of the family.
• Use Poincaré duality and the intersection pairing to relate the vanishing of η on the smooth part to the non-triviality of the class α in the relative cohomology group.
• Analyze the fiberwise singular locus of the family π: X → P to determine the codimension of strata in the dual variety X∨.
• Apply results from Dimca and Saito on the discriminant locus to show that for d ≥ 3, the map φ: Xsing → X∨ is a small resolution of singularities.
• Use the fact that small resolutions induce isomorphisms on intersection cohomology to conclude that IH*(X∨, ℚ) ≅ H*(Xsing, ℚ), and hence H*(X, ℚ) embeds into IH*(X∨, ℚ).

Experimental results

Research questions

• RQ1 Under what conditions does the vanishing of a normal function on an algebraic curve imply the existence of a singularity at the boundary of the base space?
• RQ2 When does the intersection cohomology of the dual variety of a smooth projective variety contain the cohomology of the original variety as a direct summand?
• RQ3 What is the precise relationship between the singularities of a normal function and the geometry of the discriminant locus in the dual variety?
• RQ4 How does the degree of the very ample line bundle affect the resolution of singularities of the dual variety and the cohomological embedding of the original variety?

Key findings

• If the zero locus of a normal function ν associated to a non-torsion primitive Hodge class η contains an algebraic curve, then ν is singular at some point where the closure of the curve meets the discriminant locus X∨.
• For a very ample line bundle H = dA with d ≥ 3, the map φ: Xsing → X∨ is a small resolution of singularities.
• As a consequence, the intersection cohomology of the dual variety satisfies IH*(X∨, ℚ) ≅ H*(Xsing, ℚ).
• Since Xsing is a projective bundle over X, H*(X, ℚ) is a direct summand of H*(Xsing, ℚ), and hence of IH*(X∨, ℚ).
• The condition d ≥ 3 ensures that the codimension of the singular locus of fibers is large enough to guarantee that the resolution is small.
• The results are valid under the assumption that the vanishing cohomology of the smooth fibers is nontrivial, which holds for d ≥ 3 by Dimca and Saito.