[Paper Review] Regularity on abelian varieties III: further applications

Giuseppe Pareschi, Mihnea Popa|ArXiv.org|Jun 5, 2003

Algebraic Geometry and Number Theory27 references3 citations

TL;DR

This paper extends M-regularity theory on abelian varieties to establish new bounds on Seshadri constants via M-regularity indices, prove vanishing theorems for Picard bundles on Jacobians, and derive numerical criteria for global generation and surjectivity of multiplication maps on semihomogeneous vector bundles. A key contribution is a numerical criterion for surjectivity of multiplication maps on semihomogeneous bundles using Theta regularity and Chern class conditions.

ABSTRACT

In the present sequel to our previous two papers on regularity on abelian varieties, we give a number of new applications of the theory of $M$-regularity to the study of Seshadri constants, Picard bundles, pluricanonical maps on irregular varieties, and semihomogeneous vector bundles.

Motivation & Objective

• To establish a connection between Seshadri constants and M-regularity indices on abelian varieties.
• To prove vanishing theorems for higher cohomology of twisted Picard bundles on Jacobians.
• To develop numerical criteria for global generation and surjectivity of multiplication maps on semihomogeneous vector bundles.
• To extend effective results on pluricanonical maps to irregular varieties of maximal Albanese dimension.
• To provide a cohomological criterion for surjectivity of multiplication maps using skew-Pontrjagin products and Theta regularity.

Proposed method

• Uses the Fourier-Mukai transform and derived category techniques to analyze cohomological properties of sheaves on abelian varieties.
• Applies the M-regularity criterion from [PP1] and techniques from [PP2] §3 to bound Seshadri constants via M-regularity indices.
• Employs the Eagon-Northcott resolution for determinantal varieties to compute strong Theta regularity of Picard bundles.
• Utilizes skew-Pontrjagin products of sheaves to reduce surjectivity of multiplication maps to cohomological vanishing conditions.
• Applies the Index Theorem and isogeny techniques to analyze higher cohomology vanishing of tensor products of semihomogeneous bundles.
• Uses the abelian Castelnuovo-Mumford Lemma and generalized Theorem 2.3 from [PP1] to prove surjectivity of global section maps.

Experimental results

Research questions

• RQ1 How can Seshadri constants of ample line bundles on abelian varieties be bounded using M-regularity indices?
• RQ2 What cohomological properties govern the behavior of Picard bundles when twisted by theta divisors?
• RQ3 What numerical conditions ensure the surjectivity of multiplication maps on global sections of semihomogeneous vector bundles?
• RQ4 Can effective bounds on pluricanonical maps be derived for irregular varieties of maximal Albanese dimension using M-regularity?
• RQ5 What is the role of Theta regularity in characterizing global generation and normal generation of vector bundles on abelian varieties?

Key findings

• The Seshadri constant of a polarization L on an abelian variety is bounded below by the M-regularity index of L, establishing a direct link between local positivity and cohomological regularity.
• Twisting Picard bundles by the theta divisor or its translates causes all higher cohomology to vanish, proving a strong vanishing theorem for these bundles.
• For semihomogeneous bundles E and F on (X, Θ) satisfying I.T. with index 0, the multiplication map H⁰(E) ⊗ H⁰(tₓ*F) → H⁰(E ⊗ tₓ*F) is surjective if (1/r_F)·c₁(F(−Θ)) + (1/r′_E)·φ*Θc₁(Ẽ(−Θ)) > 0.
• The surjectivity of multiplication maps on (−1)-Θ-regular vector bundles is guaranteed by a general criterion involving the Castelnuovo-Mumford Lemma and cohomological vanishing.
• Every (−1)-Θ-regular vector bundle is normally generated, extending results on projective normality of ample line bundles and Verlinde bundles.
• The proof of surjectivity for multiplication maps relies on reducing the problem to vanishing of cohomology of tensor products of pullbacks via isogenies, using numerical positivity of Chern classes.