[Paper Review] Comments on the del Pezzo cone

Dmitri Bykov|arXiv (Cornell University)|May 9, 2014

Geometric Analysis and Curvature Flows12 references3 citations

TL;DR

This paper develops a general framework for constructing Ricci-flat metrics on the anticanonical cone over the del Pezzo surface of degree 8 (dP₁), using asymptotic expansion in moment polytope coordinates. It shows that such metrics depend on at most two parameters, generalizing the known orthotoric metric and providing a systematic method to compute higher-order corrections via regularity conditions and perturbation theory.

ABSTRACT

We describe a framework for constructing the general Ricci-flat metric on the anticanonical cone over the del Pezzo surface of rank one.

Motivation & Objective

• To construct the most general Ricci-flat metric on the anticanonical cone over the del Pezzo surface of rank one (dP₁), which is the blow-up of ℂℙ² at one point.
• To provide a systematic method for solving the Ricci-flatness condition on noncompact Calabi-Yau cones with U(1)² isometries.
• To determine the maximal number of free parameters in the Ricci-flat metric through analysis of perturbation theory and regularity conditions at the polytope edges.
• To embed the known orthotoric metric as a special case within the general solution and construct a first-order deformation corresponding to the second parameter.
• To give a geometric interpretation of the second parameter in terms of the asymptotic behavior and topological data of the cone.

Proposed method

• Formulates the Ricci-flatness condition on the cone over dP₁ as a nonlinear PDE for a potential function G, using a U(1)²-invariant ansatz.
• Introduces a moment polytope for the U(1)² action, identifying the topological data (normal bundles of embedded ℂℙ¹s) that constrain the leading-order asymptotic behavior.
• Performs an asymptotic expansion of the metric potential G at infinity, with leading order corresponding to a real cone over an Einstein-Sasaki manifold.
• Imposes a regularity condition on the potential near the edges of the moment polytope, which restricts the form of coefficient functions in the expansion.
• Analyzes the linear inhomogeneous equation at each order of perturbation theory, showing that only two orders admit solutions compatible with regularity.
• Uses the orthotoric metric as a seed solution and constructs a first-order deformation with the correct asymptotic behavior, confirming the existence of a second parameter.

Experimental results

Research questions

• RQ1 What is the maximal number of free parameters in a Ricci-flat metric on the anticanonical cone over dP₁ with U(1)² isometries?
• RQ2 How can the general Ricci-flat metric on this cone be systematically constructed using asymptotic expansion and regularity constraints?
• RQ3 In what way does the known orthotoric metric fit into the broader family of Ricci-flat metrics on the dP₁ cone?
• RQ4 What is the geometric meaning of the second parameter in the general solution beyond the orthotoric case?
• RQ5 How do the coefficient functions in the asymptotic expansion of the metric potential behave at higher orders, and what constraints do regularity conditions impose?

Key findings

• The general Ricci-flat metric on the anticanonical cone over dP₁ depends on at most two free parameters, as determined by the solvability of the linear inhomogeneous equation in perturbation theory.
• The leading-order asymptotic behavior of the metric potential is fixed by topological data — specifically, the normal bundles of the two ℂℙ¹s embedded in the del Pezzo surface.
• The known orthotoric metric is recovered as a special case when the two parameters are related in a specific way, confirming consistency with prior constructions.
• A first-order deformation of the orthotoric metric is explicitly constructed, with asymptotic behavior matching the predictions of the general framework.
• Coefficient functions in the asymptotic expansion (e.g., S₃(y), S₄(y), ..., S₈(y)) are polynomials in y, with increasing degree at higher orders, indicating that orthotoric coordinates are ill-suited for the full general solution.
• The coefficient S₈(y) contains a degree-5 polynomial term in y, and the coefficient of (β + α/(2ξ₀))² in its expansion is explicitly computed, confirming the growth of complexity in higher-order terms.