Continuity of extremal transitions and flops for Calabi-Yau manifolds

Abstract

In this paper, we study the behavior of Ricci-flat Kähler metrics on Calabi-Yau manifolds under algebraic geometric surgeries: extremal transitions or flops. We prove a version of Candelas and de la Ossa's conjecture: Ricci-flat Calabi-Yau manifolds related by extremal transitions and flops can be connected by a path consisting of continuous families of Ricci-flat Calabi-Yau manifolds and a compact metric space in the Gromov-Hausdorff topology. In an essential step of the proof of our main result, the convergence of Ricci-flat Kähler metrics on Calabi-Yau manifolds along a smoothing is established, which can be of independent interests.is an extremal transition among Calabi-Yau manifolds, then there exists a family of Ricci-flat Kähler metricsḡ s , s ∈ (0, 1), on M, and a family of Ricci-flat Kähler metricsg t on M t satisfying that {(M,ḡ s )} and {(M t ,g t )} converge to a single compact metric space (X, d X ) in the Gromov-Hausdorff topology,ii) IfM 1 → M 0 M 2 is a flop between two Calabi-Yau manifolds, then there are families of Ricci-flat Kähler metricsḡ i,s , s ∈ (0, 1) onM i (i = 1, 2) such that for a single compact metric space (X, d X ).Furthermore, in both cases X is homeomorphic to M 0 and d X is induced by a Ricci-flat Kähler metric on M 0 \S. In the present paper, we shall prove i) and ii) of the above version of Candelas and de la Ossa's conjecture.Let M 0 be a projective normal Cohen-Macaulay n-dimensional variety with singular set S, and let K M 0 be the canonical sheaf of M 0 ([33]). In this paper, all varieties are assumed to be Cohen-Macaulay. We call M 0 Gorenstein if K M 0 is a rank one locally free sheaf. Assume that M 0 has only canonical singularities, i.e., M 0 is Gorenstein, and for any resolution (M,π),where E are effective exceptional divisors. Consider a resolution (M ,π) of M 0 . If α is an ample class in the Picard group of M 0 ,π * α belongs the boundary of the Kähler cone of M . A resolution (M ,π) of M 0 is called a crepant resolution if KM =π * K M 0 and is called a small resolution if the exceptional subvarietyπ −1 (S) satisfies dim Cπ −1 (S) ≤ n − 2. It is obvious that (M ,π) is crepant if it is a small resolution. If M 0 admits a smoothing (M, π) over a unit disc ∆ ⊂ C with an ample line bundle L on M, then there is an embedding M ֒→ CP N × ∆ such that L m = O ∆ (1)| M for some m ≥ 1, π is a proper surjection given by the restriction of the projection from CP N × ∆ to ∆, and the rank of π * is 1 on M\S. This implies that M t , t ∈ ∆\{0}, have the same underlying differential manifoldM . Moreover, if L is a line bundle on M such that the restriction of L on M 0 is ample, then by Proposition 1.41 in [38] L is ample on π −1 (∆ ′ ) where ∆ ′ ⊆ ∆ is a neighborhood of 0.A Calabi-Yau variety is a simply connected projective normal variety M 0 with trivial canonical sheaf K M 0 ∼ = O M 0 and only canonical singularities. If a Calabi-Yau variety M 0

Gromov–Hausdorff collapsing of Calabi–Yau manifolds

“…In the case of non-collapsing limits, analogous results about metric completions have been obtained in [25,26,27], and homeomorphism results have been obtained in [6] (see also [33]). Now we drop the assumption that the dimension of N equals 1, and assume instead that M is irreducible (i.e., simply connected and not the product of two lower-dimensional complex manifolds) and that it admits a holomorphic symplectic form Θ, which is a non-degenerate holomorphic 2-form.…”

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

“…Therefore, near the points representing singular manifolds, roughly speaking, the moduli space of Calabi-Yau threefolds looks like the union of several manifolds with possibly different dimensions. The precise statement was proved by Rong and Zhang [70]. In general the relationship between the birational resolution and the smoothing of a singularity is called an extremel transition.…”

“…The convergence in C ∞ ν -norm follows from the weighted analysis on the cylinder. The pre-compactness on K ∩ {t ≤ T } follows from the C ∞ loc -estimate of ψ τ,s as in Theorem 1.4 of [70].…”

“…Therefore, even though (X, d X ) is non-compact, Cheeger-Colding theory [11,12,13] and Donaldson-Sun theory [30,31] can still be used. As in [70], (X, d X ) is isometric to the completion of the metricω. In particular, when fixing τ i = τ ∞ ∈ T , using the uniqueness of weak solutions, d X is isometric to ω τi,0 .…”

“…However, recall thatω s equals to the pull back of the Fubini-Study metric in this region. Thus, as in the proof of Lemma 3.2 of[70], embed V s into CP N1 and view the composition of the embedding with the identity map as a harmonic map from (V s , ω τ,s ) to (CP N1 , ω F S ). In this case, the Eells-Sampson's Bochner type formula implies that−∆ ωτ,s log tr ωτ,sωs ≥ C(−|Ric ωτ,s | ωτ,s − |Rm ωF S | ωF S tr ωτ,sωs ).The Riemannian curvature of the Fubini-Study metric is indeed bounded.…”