Asymptotically conical Calabi–Yau metrics on quasi-projective varieties

Abstract

Let X be a compact Kähler orbifold without C-codimension-1 singularities. Let D be a suborbifold divisor in X such that D ⊃ Sing(X) and −pKX = q[D] for some p, q ∈ N with q > p. Assume that D is Fano. We prove the following two main results. (1) If D is Kähler-Einstein, then, applying results from our previous paper [15], we show that each Kähler class on X \ D contains a unique asymptotically conical Ricci-flat Kähler 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 [54]. (2) We introduce new methods to prove an analogous statement (with rate O(r −0.0128 )) when X = BlpP 3 and D = Blp 1 ,p 2 P 2 is the strict transform of a smooth quadric through p in P 3 . Here D is no longer Kähler-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.Date: July 4, 2018. 1 arXiv:1301.5312v3 [math.DG] 8 Dec 2014 Comparing the two theorems and their proofs. Many of the refinements in Theorem A (no neat or almost ample condition, all Kähler classes, the parameter c, uniqueness and symmetry) are due to an improvement of general technique in [15], partly based on important earlier contributions by van Coevering [55], whereas asymptotics of the form (1.1) are already implicit in Tian-Yau [54]. Let us point out one useful consequence of our explicit estimate (1.2).Corollary B. If X is smooth, then the best possible convergence rate λ of the Ricci-flat metrics of Theorem A to their tangent cones at infinity is always strictly greater than 1.Proof. Since N q−p D = K −p D and Pic(D) is torsion-free because π 1 (D) = 0, there exists a line bundle L with L p = N D and L q−p = K −1 D . Thus, by [32, p. 32, Corollary], q − p n with equality if and only if D = P n−1 , so that α − 1 n and λ 1 with equality if and only if D = P n−1 , N D = O(1). But in the latter case, the cone, and hence (X \ D, g c ) itself, must be isometric to flat C n .It seems reasonable to expect that α − 1 n even if X is singular. Moreover, equality should still imply that the cone is C n /Γ (see Remark A.2 for some examples where Γ = {1}), so that λ 2n by 1 Cristiano Spotti pointed out to us that this phenomenon was first observed in [41] for the classical elliptic modular curve H/PSL(2, Z), which is isomorphic to C as a variety but whose π orb 1 and Pic orb are nontrivial.

Degenerations and Moduli Spaces in Kähler Geometry

“…In higher dimensions, by the volume non-collapsing condition we know Z is asymptotically conical. There has been extensive study on these spaces in the case when the tangent cone at infinity is smooth, see for example van Coevering [2011], Conlon and Hein [2015], C. Li [2015].…”

On the degeneration of asymptotically conical Calabi–Yau metrics

“…The authors are also grateful to R. Conlon and H.-J. Hein for explaining aspects of their papers [15][16][17].…”

“…Following Yau's resolution of the Calabi Conjecture [63] the study of Ricci-flat Kähler metrics has played a central role in geometric analysis. Subsequently, motivated by questions in differential geometry, mathematical physics, and algebraic geometry there has been a great deal of interest in extensions of Yau's theorem to the complete, non-compact setting [2,3,5,11,[15][16][17][18]25,28,31,32,35,53,54,61,62], the degeneration of Calabi-Yau metrics (see, for example, the surveys [56][57][58] and the references there in), and the existence of Calabi-Yau metrics on singular spaces (see for example [24,51]). In this paper we initiate the study of degenerations of non-compact Calabi-Yau manifolds, and the existence of Calabi-Yau metrics on certain non-compact singular varieties.…”

“…The first analytic construction of asymptotically conical Calabi-Yau manifolds was given in [2,3,54], and the construction has been further refined by the work of many authors, see [15][16][17]25,28,32,35,61,62] and the references therein. One nice improvement given by these refinements is that, in analogy with Yau's theorem in the compact case [63], one is able to produce an asymptotically conical Ricci-flat Kähler metric in every suitable Kähler class on an asymptotically conical Kähler manifold X .…”

Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry, II

“…By our assumption Z is also endowed with a Ricci-flat Kähler metric. When Z is smooth, it is an asymptotically conical Calabi-Yau manifold, which has been well-studied recently (see for example [12]). Theorem 1.4 can also be compared with [25], where a similar result is proved for complete Kähler manifolds with non-negative bisectional curvature and maximal volume growth.…”