[Paper Review] Generalized Futaki Invariant of Almost Fano Toric Varieties, Examples

Mirroslav Yotov|ArXiv.org|Jun 21, 1998

Geometry and complex manifolds8 references7 citations

TL;DR

This paper extends T. Mabuchi's interpretation of the classical Futaki invariant to the generalized Futaki invariant for almost Fano toric varieties, using techniques from algebraic geometry and differential geometry. It proves that the real part of the generalized Futaki invariant is positive for all degenerations of the Fano manifold $V_{38}$, constructed as an intersection of the Veronese embedding of $\mathbb{P}^3 \times \mathbb{P}^2$ with codimension-two hyperplanes.

ABSTRACT

The interpretation, due to T. Mabuchi, of the classical Futaki invariant of Fano toric manifolds is extended to the case of the Generalized Futaki invariant, introduced by W. Ding and G. Tian, of almost Fano toric varieties. As an application it is shown that the real part of the Generalized Futaki invariant is positive for all degenerations of the Fano manifold V_{38}, obtained by intersection of the Veronese embedding of ${\bf P}^3 imes{\bf P}^2 \subset {\bf P}^{11}$ with codimension-two hyperplanes.

Motivation & Objective

• To generalize T. Mabuchi's interpretation of the classical Futaki invariant to the setting of generalized Futaki invariants for almost Fano toric varieties.
• To analyze the behavior of the generalized Futaki invariant under degenerations of Fano manifolds, particularly $V_{38}$.
• To establish a criterion for the positivity of the real part of the generalized Futaki invariant in the context of toric geometry.
• To provide explicit examples of almost Fano toric varieties where the generalized Futaki invariant can be computed and analyzed.
• To contribute to the understanding of K-stability and the existence of Kähler-Einstein metrics on toric Fano varieties via invariants.

Proposed method

• Adapts the framework of generalized Futaki invariants introduced by W. Ding and G. Tian to the setting of almost Fano toric varieties.
• Applies toric geometry techniques, including the use of moment polytopes and the associated Lie algebra of holomorphic vector fields.
• Computes the generalized Futaki invariant using integrals over the moment polytope, leveraging the toric structure of the variety.
• Analyzes degenerations of the Fano manifold $V_{38}$, which arises from the Veronese embedding of $\mathbb{P}^3 \times \mathbb{P}^2$ in $\mathbb{P}^{11}$, intersected with codimension-two hyperplanes.
• Employs algebraic-geometric methods to evaluate the real part of the invariant under these degenerations.
• Uses the positivity of the generalized Futaki invariant as a criterion for potential K-stability or obstruction to Kähler-Einstein metrics.

Experimental results

Research questions

• RQ1 Can the generalized Futaki invariant be meaningfully extended to almost Fano toric varieties, and how does it behave under degenerations?
• RQ2 What is the sign of the real part of the generalized Futaki invariant for degenerations of the Fano manifold $V_{38}$?
• RQ3 Does the positivity of the generalized Futaki invariant imply any geometric or stability properties for the degenerated varieties?
• RQ4 How do the toric structure and moment polytope influence the computation of the generalized Futaki invariant?
• RQ5 What role does the Veronese embedding of $\mathbb{P}^3 \times \mathbb{P}^2$ play in the construction and invariance properties of $V_{38}$?

Key findings

• The generalized Futaki invariant is well-defined and computable for almost Fano toric varieties using toric geometry and moment polytope integration.
• For all degenerations of the Fano manifold $V_{38}$, the real part of the generalized Futaki invariant is strictly positive.
• The positivity of the real part of the invariant is established through explicit computation using the toric structure and hyperplane section data.
• The result provides evidence for the K-stability or non-existence of Kähler-Einstein metrics on certain degenerations of $V_{38}$.
• The method successfully generalizes Mabuchi's framework to the generalized invariant setting, extending its applicability to non-Fano but almost Fano toric varieties.
• The construction of $V_{38}$ as a codimension-two hyperplane section of the Veronese embedding of $\mathbb{P}^3 \times \mathbb{P}^2$ is central to the analysis and enables explicit computation of the invariant.