Candelas–De la Ossa Metric Overview

• The Candelas–De la Ossa metric is a Ricci-flat Kähler metric on the resolved conifold that models conifold singularities and transitions within Calabi–Yau spaces.
• It is constructed using a generalized Calabi ansatz, yielding an asymptotically conical geometry that converges to a nodal cone at infinity.
• Recent rigidity results demonstrate its uniqueness up to scaling and diffeomorphism within its quasi-isometry class, ensuring standard AC behavior.

The Candelas–De la Ossa metric is a Ricci–flat Kähler metric on the six-dimensional noncompact complex manifold

$$\mathcal{O}_{\mathbb{P}^1}(-1)\oplus \mathcal{O}_{\mathbb{P}^1}(-1)\longrightarrow \mathbb{P}^1,$$

commonly called the conifold in the resolved sense. In the form recalled in recent work, its Kähler form is

$$\omega_{\mathrm{co}}= i\,\partial\bar\partial\bigl(f(r^3)\bigr)+4\,\pi^*\omega_{FS},$$

where $\pi$ is the bundle projection, $\omega_{FS}$ is the Fubini–Study form on $\mathbb{P}^1$, and $f$ solves a certain ODE; in complex dimension $3$ the metric is asymptotic to the nodal cone Calabi–Yau metric on

$$C=\Bigl\{z\in\mathbb{C}^{4}:\sum_{i=1}^{4}z_i^2=0\Bigr\}.$$

Within the literature considered here, it appears as a local model for conifold singularities and transitions, as a boundary limit of a generalized Calabi ansatz on $K_{\mathbb{CP}^1\times\mathbb{CP}^1}$, and as an asymptotically conical geometry that is unique up to scaling and diffeomorphism within its quasi-isometry class (Benabida, 2 Jun 2026, Song, 2012, Achmed-Zade et al., 2019).

1. Definition and ambient geometric setting

In the asymptotically conical framework, the model at infinity is a Calabi–Yau cone. A Riemannian cone over a closed connected Riemannian manifold $(L,g_L)$ is

$$\omega_{\mathrm{co}}= i\,\partial\bar\partial\bigl(f(r^3)\bigr)+4\,\pi^*\omega_{FS},$$0

with radial function $$\omega_{\mathrm{co}}= i\,\partial\bar\partial\bigl(f(r^3)\bigr)+4\,\pi^*\omega_{FS},$$1, and a Calabi–Yau cone is a simply connected Riemannian cone $$\omega_{\mathrm{co}}= i\,\partial\bar\partial\bigl(f(r^3)\bigr)+4\,\pi^*\omega_{FS},$$2 together with a $$\omega_{\mathrm{co}}= i\,\partial\bar\partial\bigl(f(r^3)\bigr)+4\,\pi^*\omega_{FS},$$3-parallel complex structure $$\omega_{\mathrm{co}}= i\,\partial\bar\partial\bigl(f(r^3)\bigr)+4\,\pi^*\omega_{FS},$$4 such that $$\omega_{\mathrm{co}}= i\,\partial\bar\partial\bigl(f(r^3)\bigr)+4\,\pi^*\omega_{FS},$$5 is $$\omega_{\mathrm{co}}= i\,\partial\bar\partial\bigl(f(r^3)\bigr)+4\,\pi^*\omega_{FS},$$6-Kähler and Ricci-flat. For the nodal cone,

$$\omega_{\mathrm{co}}= i\,\partial\bar\partial\bigl(f(r^3)\bigr)+4\,\pi^*\omega_{FS},$$7

Stenzel constructed the Ricci–flat Kähler cone metric

$$\omega_{\mathrm{co}}= i\,\partial\bar\partial\bigl(f(r^3)\bigr)+4\,\pi^*\omega_{FS},$$8

and when $$\omega_{\mathrm{co}}= i\,\partial\bar\partial\bigl(f(r^3)\bigr)+4\,\pi^*\omega_{FS},$$9 this is the cone to which both the Stenzel metric on $\pi$0 and the Candelas–De la Ossa metric on $\pi$1 are asymptotic (Benabida, 2 Jun 2026).

The underlying complex manifold of the Candelas–De la Ossa metric is the small crepant resolution of the nodal cone for $\pi$2,

$\pi$3

In the formulation used in the Liouville theorem, the metric is an asymptotically conical Calabi–Yau metric of the form

$\pi$4

The paper recalling this formula does not reproduce the explicit ODE from the original construction; it uses only that $\pi$5 is Ricci-flat and asymptotic to the nodal cone metric with some positive rate $\pi$6 in the sense that, for a suitable diffeomorphism $\pi$7,

$\pi$8

This places the metric squarely in the standard AC Calabi–Yau category (Benabida, 2 Jun 2026).

2. Explicit ansatz and boundary-limit construction

A detailed differential-geometric realization of the Candelas–De la Ossa metric is given through a generalized Calabi ansatz on the total space of the canonical bundle

$\pi$9

With homogeneous coordinates $\omega_{FS}$0 on the first $\omega_{FS}$1, $\omega_{FS}$2 on the second, and fiber coordinate $\omega_{FS}$3, the Kähler potential is taken to be

$\omega_{FS}$4

with

$\omega_{FS}$5

Introducing the Legendre variable $\omega_{FS}$6, the resulting Ricci-flat metric can be written as

$\omega_{FS}$7

where

$\omega_{FS}$8

and Ricci-flatness becomes the first-order ODE

$\omega_{FS}$9

This is the Pando Zayas–Tseytlin family, and the Candelas–De la Ossa metric appears as a special boundary limit inside it (Achmed-Zade et al., 2019).

The boundary regime is characterized by

$\mathbb{P}^1$0

Assuming $\mathbb{P}^1$1, one has a second-order zero of the relevant polynomial at $\mathbb{P}^1$2, so that

$\mathbb{P}^1$3

With $\mathbb{P}^1$4, the metric near $\mathbb{P}^1$5 takes the form

$\mathbb{P}^1$6

where $\mathbb{P}^1$7. For fixed $\mathbb{P}^1$8, the bracket is the standard round metric on $\mathbb{P}^1$9, provided $f$0 has period $f$1. The resulting manifold is then

$f$2

and the metric obtained in this limit is identified with the Candelas–De la Ossa metric. In this description, the metric is Kähler, Ricci-flat, cohomogeneity-one, and asymptotically conical; the construction is simultaneously a desingularization mechanism and a boundary-limit operation in Kähler moduli (Achmed-Zade et al., 2019).

3. Local conifold model and analytic formulation

In the projective conifold setting, the local singularity is the ordinary double point

$f$3

A small contraction

$f$4

from a smooth projective Calabi–Yau threefold to a singular Calabi–Yau threefold with ordinary double points has exceptional curves

$f$5

with normal bundle

$f$6

This is the small resolution of an ODP and provides the local complex-geometric arena for the Candelas–De la Ossa model (Song, 2012).

The local Ricci–flat Kähler model is constructed on

$f$7

by Calabi’s ansatz. Writing

$f$8

with

$f$9

the Ricci-flat condition becomes

$3$0

equivalently

$3$1

At $3$2,

$3$3

and the resulting Ricci-flat metric is

$3$4

In that local setting, this is the Ricci–flat Calabi–Yau model associated with the resolved conifold; the discussion identifies $3$5 as the Candelas–De la Ossa metric in the local conifold picture. The analytic significance is that the metric is produced directly from a radial Monge–Ampère reduction to an ODE, and the geometric significance is that the same bundle $3$6 governs the local behavior of conifold contractions (Song, 2012).

4. Conifold transitions, flops, and metric degeneration

A central role of the Candelas–De la Ossa metric is as the local metric model for conifold flops and conifold transitions. In the projective setting, the canonical singular Ricci-flat Kähler metric on a normal Calabi–Yau variety $3$7 with ordinary double points is obtained by solving the degenerate complex Monge–Ampère equation

$3$8

with $3$9. The metric completion of $$C=\Bigl\{z\in\mathbb{C}^{4}:\sum_{i=1}^{4}z_i^2=0\Bigr\}.$$0 is then a compact length space $$C=\Bigl\{z\in\mathbb{C}^{4}:\sum_{i=1}^{4}z_i^2=0\Bigr\}.$$1 homeomorphic to the projective variety $$C=\Bigl\{z\in\mathbb{C}^{4}:\sum_{i=1}^{4}z_i^2=0\Bigr\}.$$2 itself, and if $$C=\Bigl\{z\in\mathbb{C}^{4}:\sum_{i=1}^{4}z_i^2=0\Bigr\}.$$3 denotes the family of smooth Ricci-flat Kähler metrics on the small resolution $$C=\Bigl\{z\in\mathbb{C}^{4}:\sum_{i=1}^{4}z_i^2=0\Bigr\}.$$4, one has

$$C=\Bigl\{z\in\mathbb{C}^{4}:\sum_{i=1}^{4}z_i^2=0\Bigr\}.$$5

This is the metric-analytic form of the conjecture of Candelas and de la Ossa for conifold flops and transitions (Song, 2012).

The local estimates near the exceptional $$C=\Bigl\{z\in\mathbb{C}^{4}:\sum_{i=1}^{4}z_i^2=0\Bigr\}.$$6 make the collapse mechanism precise. On the neighborhood

$$C=\Bigl\{z\in\mathbb{C}^{4}:\sum_{i=1}^{4}z_i^2=0\Bigr\}.$$7

there exists $$C=\Bigl\{z\in\mathbb{C}^{4}:\sum_{i=1}^{4}z_i^2=0\Bigr\}.$$8 such that

$$C=\Bigl\{z\in\mathbb{C}^{4}:\sum_{i=1}^{4}z_i^2=0\Bigr\}.$$9

and the diameter of the exceptional curve satisfies

$K_{\mathbb{CP}^1\times\mathbb{CP}^1}$0

These inequalities show that the exceptional curves shrink to points in the Gromov–Hausdorff limit, while away from the exceptional locus the convergence is smooth. For a conifold flop

$K_{\mathbb{CP}^1\times\mathbb{CP}^1}$1

and for a conifold transition

$K_{\mathbb{CP}^1\times\mathbb{CP}^1}$2

the metrics on the resolution side and on the smoothing side both converge to the same compact length metric space $K_{\mathbb{CP}^1\times\mathbb{CP}^1}$3. This situates the Candelas–De la Ossa metric as the local model behind the continuity of Ricci-flat geometry through ordinary double point degeneration (Song, 2012).

5. Liouville rigidity and uniqueness in the quasi-isometry class

A recent Liouville theorem gives a strong rigidity statement for the Candelas–De la Ossa metric. In general, let $K_{\mathbb{CP}^1\times\mathbb{CP}^1}$4 be a complete Ricci-flat Kähler manifold modeled on a Calabi–Yau cone $K_{\mathbb{CP}^1\times\mathbb{CP}^1}$5 in the sense that there exists a diffeomorphism

$K_{\mathbb{CP}^1\times\mathbb{CP}^1}$6

with

$K_{\mathbb{CP}^1\times\mathbb{CP}^1}$7

for some $K_{\mathbb{CP}^1\times\mathbb{CP}^1}$8, and suppose

$K_{\mathbb{CP}^1\times\mathbb{CP}^1}$9

Then $(L,g_L)$0 is asymptotically conical with tangent cone at infinity given by $(L,g_L)$1. The proof proceeds through blow-down sequences, convergence of rescaled complex structures, the Nirenberg–Newlander theorem with parameter, Evans–Krylov and Schauder estimates for the complex Monge–Ampère equation, Klemmensen’s Liouville theorem on the cone, and a quadratic-curvature-decay argument, after which Sun–Zhang’s theorem yields the AC conclusion (Benabida, 2 Jun 2026).

Specialized to $(L,g_L)$2 with $(L,g_L)$3, the corollary states that if $(L,g_L)$4 is another Ricci-flat Kähler metric on $(L,g_L)$5 such that

$(L,g_L)$6

then

$(L,g_L)$7

for some diffeomorphism $(L,g_L)$8 and some $(L,g_L)$9. Thus the Candelas–De la Ossa metric is unique up to scaling and diffeomorphism among Ricci-flat Kähler metrics in its global quasi-isometry class. The geometric consequence is that no exotic non-AC ends occur under the stated hypotheses, and the asymptotic nodal cone completely determines the metric within that class (Benabida, 2 Jun 2026).

6. Generalizations and arithmetic shadow

The Candelas–De la Ossa metric serves as the prototype for a broader family of explicit Ricci-flat metrics on vector bundles over flag manifolds. A generalized Calabi-type construction produces metrics on

$$\omega_{\mathrm{co}}= i\,\partial\bar\partial\bigl(f(r^3)\bigr)+4\,\pi^*\omega_{FS},$$00

and, in a boundary limit analogous to the conifold case, on rank-$$\omega_{\mathrm{co}}= i\,\partial\bar\partial\bigl(f(r^3)\bigr)+4\,\pi^*\omega_{FS},$$01 vector bundles

$$\omega_{\mathrm{co}}= i\,\partial\bar\partial\bigl(f(r^3)\bigr)+4\,\pi^*\omega_{FS},$$02

when $$\omega_{\mathrm{co}}= i\,\partial\bar\partial\bigl(f(r^3)\bigr)+4\,\pi^*\omega_{FS},$$03 exists. The general metric has the form

$$\omega_{\mathrm{co}}= i\,\partial\bar\partial\bigl(f(r^3)\bigr)+4\,\pi^*\omega_{FS},$$04

with

$$\omega_{\mathrm{co}}= i\,\partial\bar\partial\bigl(f(r^3)\bigr)+4\,\pi^*\omega_{FS},$$05

The Candelas–De la Ossa case is recovered when $$\omega_{\mathrm{co}}= i\,\partial\bar\partial\bigl(f(r^3)\bigr)+4\,\pi^*\omega_{FS},$$06 and $$\omega_{\mathrm{co}}= i\,\partial\bar\partial\bigl(f(r^3)\bigr)+4\,\pi^*\omega_{FS},$$07; in that sense it is the rank-$$\omega_{\mathrm{co}}= i\,\partial\bar\partial\bigl(f(r^3)\bigr)+4\,\pi^*\omega_{FS},$$08, $$\omega_{\mathrm{co}}= i\,\partial\bar\partial\bigl(f(r^3)\bigr)+4\,\pi^*\omega_{FS},$$09 prototype for a hierarchy of explicit asymptotically conical Ricci-flat Kähler metrics (Achmed-Zade et al., 2019).

A different line of work gives an arithmetic and mirror-symmetry context. The study of local Weil zeta-functions for families of Calabi–Yau $$\omega_{\mathrm{co}}= i\,\partial\bar\partial\bigl(f(r^3)\bigr)+4\,\pi^*\omega_{FS},$$10-folds with singular fibers is explicitly built on the arithmetic/mirror-symmetry program initiated by Candelas and de la Ossa and, in particular, on the Candelas–de la Ossa–Rodrigues-Villegas approach for computing zeta-functions. That paper does not discuss the Ricci-flat Candelas–De la Ossa metric on the deformed conifold itself, but it analyzes degenerations such as conifolds and relates singularity data, including Milnor numbers, to the drop in degree of $$\omega_{\mathrm{co}}= i\,\partial\bar\partial\bigl(f(r^3)\bigr)+4\,\pi^*\omega_{FS},$$11-class factors in the zeta-function. Conceptually, this places the Candelas–De la Ossa picture in a second domain: on the complex-geometric side, conifold degenerations are modeled by Ricci-flat Kähler metrics and vanishing cycles; on the arithmetic side, the same degenerations are reflected in changes in the local zeta-function and in the combinatorics of strong $$\omega_{\mathrm{co}}= i\,\partial\bar\partial\bigl(f(r^3)\bigr)+4\,\pi^*\omega_{FS},$$12-classes (Frühbis-Krüger et al., 2011).