[Paper Review] Projective duality and K-energy asymptotics

Sean Timothy Paul|ArXiv.org|Nov 16, 2008

Geometry and complex manifolds11 references3 citations

TL;DR

This paper establishes a link between projective duality and the asymptotic behavior of the Mabuchi K-energy on smooth, linearly normal subvarieties of projective space. It shows that the log-norm of the dual discriminant polynomial under group action corresponds to an energy functional on the variety, reducing to standard Kähler action functionals for smooth plane curves, and proposes a conjectural decomposition of the Mabuchi energy in terms of weights of group representations.

ABSTRACT

Let X be a smooth, linearly normal n dimensional complex projective variety. Assume that the projective dual of X has codimension one with defining polynomial D(X). In this paper the log of the norm of D(X) is expressed as the restriction to the Bergman metrics of an energy functional on X. We show how, for smooth plane curves, this energy functional reduces to the standard action functionals of Kahler geometry.

Motivation & Objective

• To relate the geometry of projective embeddings to the restriction of the Mabuchi K-energy functional on the space of Kähler potentials.
• To understand the singular term in Tian’s formula for the Mabuchi energy on hypersurfaces via projective duality.
• To generalize the Aubin energy formula for hypersurfaces to higher codimension subvarieties using the X-resultant and projective duality.
• To propose a conjectural representation-theoretic decomposition of the Mabuchi energy in terms of group representations associated with the variety.
• To investigate the properness of the Mabuchi energy via weight polytopes and one-parameter subgroups, linking to existence of canonical metrics.

Proposed method

• Expresses the log-norm of the transformed dual discriminant $ orm{ au ullet riangle_X} $ as a restriction of an energy functional on the variety $ X $, using Bergman metrics.
• Applies results from Kähler geometry, including Bott–Chern forms and the complex $ K^ullet $-resolution, to analyze curvature and energy functionals.
• Uses the Cayley–Chow form and the $ X $-resultant $ R_X $ to define a group-invariant energy functional on $ X $, generalizing the Aubin energy formula.
• Analyzes the asymptotic behavior of the Mabuchi energy along one-parameter subgroups $ au(t) $ via weight polytopes and the minimal weight $ w_ au(v) $ of a vector $ v $ in a representation.
• Derives an expansion $ u_ ho( ho_{ au(t)}) = ( ho_1 w_ au(v_1) - ho_2 w_ au(v_2)) ho(|t|^2) + O(1) $ as $ |t| o 0 $, linking energy growth to representation weights.
• Introduces the notion of scaled weight polytope dominance to characterize the properness of the Mabuchi energy in terms of $ rac{ ho_2}{ ho_1} $-scaling of $ v_2 $'s polytope relative to $ v_1 $'s.

Experimental results

Research questions

• RQ1 How can the singular term $ ho_B $ in Tian’s Mabuchi energy formula for hypersurfaces be interpreted geometrically via projective duality?
• RQ2 Can the Mabuchi energy on higher codimension subvarieties be expressed as a combination of log-norms of group-transformed invariants, such as the $ X $-resultant?
• RQ3 What is the role of the dual variety and its defining polynomial $ riangle_X $ in encoding the asymptotic behavior of K-energy functionals?
• RQ4 Under what conditions is the Mabuchi energy proper along one-parameter subgroups, and how does this relate to the weight polytopes of the relevant group representations?
• RQ5 Is there a representation-theoretic decomposition of the Mabuchi energy into positive and negative terms involving $ v_1 $ and $ v_2 $, as conjectured?

Key findings

• The log-norm of $ au ullet riangle_X $ is shown to equal the restriction of a Kähler energy functional on $ X $, linking projective duality to K-energy asymptotics.
• For smooth plane curves, the energy functional reduces to standard action functionals in Kähler geometry, confirming consistency with known results.
• The Aubin energy on $ X $ is expressed via the $ X $-resultant $ R_X $, with $ -(n+1) ext{deg}(X)F^0_ ho( ho_ au) = ext{log}(|| au ullet R_X||^2 / ||R_X||^2) $, generalizing Tian’s formula.
• The conjectural decomposition $ u_ ho( ho_ au) = ho_1 ext{log}(|| au ullet v_1||^2 / ||v_1||^2) - ho_2 ext{log}(|| au ullet v_2||^2 / ||v_2||^2) $ is proposed, with $ ho_j eq 0 $ rational and positive.
• The Mabuchi energy is proper along all one-parameter subgroups $ au $ if and only if $ ho_1 w_ au(v_1) - ho_2 w_ au(v_2) + rac{C}{ ext{deg}(X)(n+1)} e( au; X) eq 0 $, with $ C > 0 $, under the assumption $ ext{Aut}(X) = 0 $.
• The scaled weight polytope of $ v_2 $ strictly dominates that of $ v_1 $ by a factor $ rac{ ho_2}{ ho_1} $, providing a geometric criterion for properness of the Mabuchi energy.