[Paper Review] Asymptotically conical Calabi-Yau metrics on quasi-projective varieties

Ronan J. Conlon, Hans‐Joachim Hein|arXiv (Cornell University)|Jan 22, 2013

Geometry and complex manifolds50 references4 citations

TL;DR

This paper establishes the existence and uniqueness of asymptotically conical Calabi-Yau metrics on quasi-projective varieties $X \setminus D$, where $X$ is a compact Kähler orbifold and $D$ is a Fano suborbifold divisor with $-pK_X = q[D]$ and $q > p$. It proves that each Kähler class on $X \setminus D$ admits a unique Ricci-flat Kähler metric asymptotic to a cone metric at infinity with rate $O(r^{-1-\varepsilon})$ when $D$ is Kähler-Einstein, and extends this to a non-Kähler-Einstein case with rate $O(r^{-0.0128})$ using irregular Sasaki-Einstein structures, yielding the first example of an affine Calabi-Yau manifold with irregular tangent cone and Euclidean volume growth.

ABSTRACT

Let X be a compact Kahler orbifold without \C-codimension-1 singularities. Let D be a suborbifold divisor in X such that D \supset Sing(X) and -pK_X = q[D] for some p, q \in \N with q > p. Assume that D is Fano. We prove the following two main results. (1) If D is Kahler-Einstein, then, applying results from our previous paper, we show that each Kahler class on X\D contains a unique asymptotically conical Ricci-flat Kahler metric, converging to its tangent cone at infinity at a rate of O(r^{-1-ε}) if X is smooth. This provides a definitive version of a theorem of Tian and Yau. (2) We introduce new methods to prove an analogous statement (with rate O(r^{-0.0128})) when X = Bl_{p}P^3 and D = Bl_{p_1,p_2}P^2 is the strict transform of a smooth quadric through p in P^3. Here D is no longer Kahler-Einstein, but the normal S^1-bundle to D in X admits an irregular Sasaki-Einstein structure which is compatible with its canonical CR structure. This provides the first example of an affine Calabi-Yau manifold of Euclidean volume growth with irregular tangent cone at infinity.

Motivation & Objective

• To generalize and optimize the Tian-Yau theorem on the existence of complete Ricci-flat Kähler metrics on quasi-projective varieties $X \setminus D$.
• To establish sharp asymptotic decay rates for curvature and metric convergence to a tangent cone at infinity.
• To construct a new class of Calabi-Yau metrics when $D$ is not Kähler-Einstein, using irregular Sasaki-Einstein structures on the normal bundle.
• To provide the first example of an affine Calabi-Yau manifold with Euclidean volume growth and an irregular tangent cone at infinity.
• To unify and extend previous results on asymptotically conical Calabi-Yau metrics using orbifold and algebraic geometry techniques.

Proposed method

• Use of the Calabi ansatz to construct a model Ricci-flat Kähler cone metric on the total space of the line bundle $K_D^{-p}$ via the adjunction isomorphism $N_D^{q-p} \cong K_D^{-p}$.
• Application of the exponential map $\exp: N_D \to X$ to pull back the metric from the cone to a neighborhood of $D$ in $X$, enabling asymptotic analysis.
• Proof of existence and uniqueness of Ricci-flat Kähler metrics in each Kähler class on $X \setminus D$ via a continuity method and weighted Sobolev estimates.
• Introduction of new geometric techniques to handle the non-Kähler-Einstein case, particularly leveraging the existence of an irregular Sasaki-Einstein structure on the unit normal bundle to $D$ in $X$.
• Use of explicit coordinate charts and meromorphic volume forms to verify global holomorphic structure and canonical bundle triviality on $X \setminus \{p_9\}$.
• Analysis of the asymptotic behavior of the metric using weighted $L^2$-estimates and decay estimates for curvature and metric differences from the model cone metric.

Experimental results

Research questions

• RQ1 Can the Tian-Yau theorem on asymptotically conical Calabi-Yau metrics be strengthened to include sharp decay rates for curvature and metric convergence?
• RQ2 What happens to the asymptotic geometry of Calabi-Yau metrics on $X \setminus D$ when $D$ is not Kähler-Einstein, but admits an irregular Sasaki-Einstein structure?
• RQ3 Is it possible to construct a Calabi-Yau metric on a quasi-projective variety with an irregular tangent cone at infinity and Euclidean volume growth?
• RQ4 How does the existence of a Kähler-Einstein metric on $D$ influence the asymptotic behavior of the Ricci-flat metric on $X \setminus D$?
• RQ5 What role does the orbifold structure of $X$ and the suborbifold divisor $D$ play in the existence and uniqueness of such metrics?

Key findings

• Each Kähler class on $X \setminus D$ admits a unique Ricci-flat Kähler metric $\omega_c$ that converges to its tangent cone at infinity with rate $O(r^{-1-\varepsilon})$ when $D$ is Kähler-Einstein and $X$ is smooth.
• The metric $\omega_c$ satisfies the decay estimate $|\nabla^{k}_{g_0}(\exp^*(g_c) - c g_0)|_{g_0} \leq C(k) r^{-\lambda - k}$ with $\lambda = \min\{2 - \varepsilon, \frac{n}{\alpha - 1}\}$ for any $\varepsilon > 0$, where $\alpha = q/p > 1$.
• When $X = \mathrm{Bl}_p \mathbb{P}^3$ and $D = \mathrm{Bl}_{p_1,p_2} \mathbb{P}^2$ is the strict transform of a smooth quadric through $p$, the metric exists with decay rate $O(r^{-0.0128})$ despite $D$ not being Kähler-Einstein.
• The normal $\mathbb{S}^1$-bundle to $D$ in $X$ admits an irregular Sasaki-Einstein structure compatible with its canonical CR structure, enabling the construction in the non-Kähler-Einstein case.
• This construction yields the first known example of an affine Calabi-Yau manifold with Euclidean volume growth and an irregular tangent cone at infinity.
• The metric $\omega_c$ is invariant under automorphisms of $(X,D)$ preserving the Kähler class and the model cone metric, and scales as $\omega_{tc,t\mathfrak{k}} = t \omega_{c,\mathfrak{k}}$ for all $t > 0$.