Calabi-Yau manifolds with isolated conical singularities

Abstract

Let X be a complex projective variety with only canonical singularities and with trivial canonical bundle. Let L be an ample line bundle on X. Assume that the pair (X, L) is the flat limit of a family of smooth polarized Calabi-Yau manifolds. Assume that for each singular point x ∈ X there exist a Kähler-Einstein Fano manifold Z and a positive integer q dividing KZ such that − 1 q KZ is very ample and such that the germ (X, x) is locally analytically isomorphic to a neighborhood of the vertex of the blow-down of the zero section of 1 q KZ. We prove that up to biholomorphism, the unique weak Ricci-flat Kähler metric representing 2πc1(L) on X is asymptotic at a polynomial rate near x to the natural Ricci-flat Kähler cone metric on 1 q KZ constructed using the Calabi ansatz. In particular, our result applies if (X, O(1)) is a nodal quintic threefold in P 4 . This provides the first known examples of compact Ricci-flat manifolds with non-orbifold isolated conical singularities.

$$\hbox {G}_2$$ Manifolds with Nodal Singularities Along Circles

“…In this section, Theorem 1.1 is proved as a combination of [43] and [41]. There are several major technical problems in this process.…”

“…There are several major technical problems in this process. Firstly, it is necessary to find a good substitute for the finiteness of diameter property in [43]. Secondly, it is not clear how to get the generalization of [33] on the existence of weak solutions because the weak solution in the sense of current is too weak to apply the standard analysis for the asymptotically cylindrical manifolds.…”

“…The third problem is to control neck length in the gluing construction. This paper starts from the solution of the first problem by combing the Theorem 1.4 of Hein-Sun's work [43] with Theorem D of Haskins-Hein-Nordström's work [41] and proving the following theorem: Theorem 1.1. Let i : Z ⊂ CP N1 × ∆ be a flat family of projective varieties over disc ∆.…”

A gluing construction of collapsing Calabi–Yau metrics on K3 fibred 3-folds

“…Remark 8 explains the special choice of power t • When t 6 14+τ ≤ y ≤ 2t 6 14+τ , the metric ω t starts to receive contribution from the nodal fibre X 0 (here the diffeomorphism G 0 is well defined on the support of the cutoff functions and is used to graft the potential on X 0 to X y ), but the fluctuation effect of ω C 3 is not yet significant. The expression inside χ 0 plays the same role as the potential of the approximate metric ω y on X y as in (7).…”

“…By adjunction, the fibres are Calabi-Yau varieties in their own right, and admit (possibly singular) Calabi-Yau metrics. The result of Hein and Sun [7] says in particular that the CY metrics on the singular fibres are modelled on the Stenzel metric near the nodal point, with polynomial rate of convergence. By standard gluing argument, the CY metrics on the smoothing fibres are modelled on the stenzel metric in the region close to the vanishing cycles.…”