Simultaneous desingularizations of Calabi–Yau and special Lagrangian 3-folds with conical singularities. II. Examples

This article is a sequel to Chan (Ann Glob Anal Geom, to appear) on simultaneous desingularizations of Calabi–Yau and special Lagrangian (SL) 3-folds with conical singularities. In Chan (Ann Glob Anal Geom, to appear) we treated the question of starting with a conically singular Calabi–Yau 3-fold and an SL 3-fold with conical singularities at the same points and deforming both together to get a smooth situation. In this article, we survey the major result from Chan (Ann Glob Anal Geom, to appear) and describe some examples from our earlier articles (Chan Q J Math 57:151–181, 2006, Q J Math, to appear) on Calabi–Yau desingularizations. We then provide many explicit examples of Asymptotically Conical (AC) SL submanifolds in two specific AC Calabi–Yau manifolds. Using the result in Chan (Ann Glob Anal Geom, to appear), we construct smooth examples of compact SL 3-folds in compact Calabi–Yau 3-folds by gluing those AC SL 3-folds into some conically singular SL 3-folds at the singular points.     

Examples of Calabi–Yau Threefolds as Covers of Almost-Fano Threefolds

We give a method for producing examples of Calabi–Yau threefolds as covers of degree d ≤ 8 of almost-Fano threefolds, computing explicitely their Euler– Poincaré characteristic. Such a method generalizes the well-knownclassical construction of Calabi–Yau threefolds as double covers of the projective space branched along octic surfaces.     

Simultaneous desingularizations of Calabi–Yau and special Lagrangian 3-folds with conical singularities: I. The gluing construction

In this paper we extend our previous results on resolving conically singular Calabi–Yau 3-folds (Chan, Quart. J. Math. 57:151–181, 2006; Quart. J. Math., to appear) to include the desingularizations of special Lagrangian (SL) 3-folds with conical singularities that occur at the same points of the ambient Calabi–Yau. The gluing construction of the SL 3-folds is achieved by applying Joyce’s analytic result (Joyce, Ann. Global. Anal. Geom. 26: 1–58, 2004, Thm. 5.3) on deforming Lagrangian submanifolds to nearby special Lagrangian submanifolds. Our result will in principle be able to construct more examples of compact SL submanifolds in compact Calabi–Yau manifolds. Various explicit examples and applications illustrating the result in this paper can be found in the sequel (Chan, Ann. Global. Anal. Geom., to appear).     

Isometric Embeddings of Families of Special Lagrangian Submanifolds

We prove that certain Riemannian manifolds can be isometrically embedded inside Calabi–Yau manifolds. For example, we prove that given any real-analytic one parameter family of Riemannian metrics g _t on a three-dimensional manifold Y with volume form independent of t and with a real-analytic family of nowhere vanishing harmonic one forms θ _t , then (Y,g _t ) can be realized as a family of special Lagrangian submanifolds of a Calabi–Yau manifold X. We also prove that certain principal torus bundles can be equivariantly and isometrically embedded inside Calabi-Yau manifolds with torus action. We use this to construct examples of n-parameter families of special Lagrangian tori inside n + k-dimensional Calabi–Yau manifolds with torus symmetry. We also compute McLean's metric of 3-dimensional special Lagrangian fibrations with T ²-symmetry. Mathematics Subject Classification (2000): 53-XX, 53C38.Communicated by N. Hitchin (Oxford)     

Homotopy Gerstenhaber Structures and Vertex Algebras

We provide a simple construction of a G _∞-algebra structure on an important class of vertex algebras V, which lifts the Gerstenhaber algebra structure on BRST cohomology of V introduced by Lian and Zuckerman. We outline two applications to algebraic topology: the construction of a sheaf of G _∞ algebras on a Calabi–Yau manifold M, extending the operations of multiplication and bracket of functions and vector fields on M, and of a Lie_∞ structure related to the bracket of Courant (Trans Amer Math Soc 319:631–661, 1990).     

DESINGULARIZATIONS OF CALABI-YAU 3-FOLDS WITH CONICAL SINGULARITIES. II. THE OBSTRUCTED CASE

This is the second of two papers studying Calabi–Yau 3-foldswith conical singularities and their desingularizations. Inour first paper [Y.-M. Chan, Quart. J. Math. 57 (2006), 151–181]we constructed the desingularization of the conically singularmanifold M₀ by gluing an asymptotically conical (AC) Calabi–Yau3-fold Y into M₀ at the singular point, thus obtaining a 1-parameterfamily of compact, non-singular Calabi–Yau 3-folds M_tfor small t > 0. During the gluing process one may encountera kind of cohomological obstruction to defining a 3-form _t onM_t which interpolates between the 3-form ₀ on M₀ and the scaled3-form t³ _Y on Y if the rate at which the AC Calabi–Yau3-fold Y converges to the Calabi–Yau cone is equal to– 3. The first paper [3] studied the simpler case <–3 where there is no obstruction. This paper extends theresult in the first one by considering a more complicated situtationwhen = –3. Assuming the existence of singular Calabi–Yaumetrics on compact complex 3-folds with ordinary double points,our result in this paper can be applied to repairing such kindsof singularities, which is an analytic version of Friedman'sresult giving necessary and sufficient conditions for smoothingordinary double points.     

Global Smoothing of Calabi–Yau Threefolds II

The moduli spaces of Calabi–Yau threefolds are conjectured to be connected by the combination of birational contraction maps and flat deformations. In this context, it is important to calculate dim Def(X) from dim Def(~X) in terms of certain geometric information of f, when we are given a birational morphism f:~XX from a smooth Calabi–Yau threefold ~X to a singular Calabi–Yau threefold X. A typical case of this problem is a conjecture of Morrison-Seiberg which originally came from physics. In this paper we give a mathematical proof to this conjecture. Moreover, by using output of this conjecture, we prove that certain Calabi–Yau threefolds with nonisolated singularities have flat deformations to smooth Calabi–Yau threefolds. We shall use invariants of singularities closely related to Du Bois's work to calculate dim Def(X) from dim Def(~X).     

Correspondences between modular Calabi–Yau fiber products

We describe two ways to construct finite rational morphisms between fiber products of rational elliptic surfaces with section and some Calabi–Yau varieties. We use them to construct correspondences between such fiber products that admit at most five singular fibers and rigid Calabi–Yau threefolds.     

Non-liftable Calabi–Yau spaces

We construct many new non-liftable three-dimensional Calabi–Yau spaces in positive characteristic. The technique relies on lifting a nodal model to a smooth rigid Calabi–Yau space over some number field as introduced by one of us jointily with D. van Straten.     

Isolated rational curves on K3-fibered Calabi–Yau threefolds

In this paper we study 16 complete intersection K3-fibered Calabi--Yau variety types in biprojective space ℙ ^{n 1}}×ℙ¹. These are all the CICY-types that are K3 fibered by the projection on the second factor. We prove existence of isolated rational curves of bidegree (d,0) for every positive integer d on a general Calabi–Yau variety of these types. The proof depends heavily on existence theorems for curves on K3-surfaces proved by S. Mori and K. Oguiso. Some of these varieties are related to Calabi–Yau varieties in projective space by a determinantal contraction, and we use this to prove existence of rational curves of every degree for a general Calabi–Yau variety in projective space. Received: 14 October 1997 / Revised version: 18 January 1998     

Calabi–Yau coverings over some singular varieties and new Calabi–Yau 3-folds with Picard number one

This note is a report on the observation that some singular varieties admit Calabi–Yau coverings. As an application, we construct 18 new Calabi–Yau 3-folds with Picard number one that have some interesting properties.     

Variations of mixed Hodge structure, Higgs fields, and quantum cohomology

Following C. Simpson, we show that every variation of graded-polarized mixed Hodge structure defined over ℚ carries a natural Higgs bundle structure which is invariant under the ℂ^* action studied in [20]. We then specialize our construction to the context of [6], and show that the resulting Higgs field θ determines (and is determined by) the Gromov–Witten potential of the underlying family of Calabi–Yau threefolds. Received: 14 February 2000     

Convergence of Calabi–Yau manifolds

In this paper, we study the convergence of Calabi–Yau manifolds under Kähler degeneration to orbifold singularities and complex degeneration to canonical singularities (including the conifold singularities), and the collapsing of a family of Calabi–Yau manifolds.     

Quivers with potentials and their representations I: Mutations

We study quivers with relations given by noncommutative analogs of Jacobian ideals in the complete path algebra. This framework allows us to give a representation-theoretic interpretation of quiver mutations at arbitrary vertices. This gives a far-reaching generalization of Bernstein–Gelfand–Ponomarev reflection functors. The motivations for this work come from several sources: superpotentials in physics, Calabi–Yau algebras, cluster algebras.      

Moduli of Polarized Calabi–Yau Pairs

We prove that the irreducible components of the moduli space of polarized Calabi–Yau pairs are projective.     

Variation of the unit root along the Dwork family of Calabi–Yau varieties

We study the variation of the unit roots of members of the Dwork families of Calabi–Yau varieties over a finite field by the method of Dwork–Katz and also from the point of view of formal group laws. A p-adic analytic formula for the unit roots away from the Hasse locus is obtained. This work was supported in part by Professor N. Yui’s Discovery Grant from NSERC, Canada.     

The Spectrum of the Martin-Morales-Nadirashvili Minimal Surfaces Is Discrete

We show that the spectrum of a complete submanifold properly immersed into a ball of a Riemannian manifold is discrete, provided the norm of the mean curvature vector is sufficiently small. In particular, the spectrum of a complete minimal surface properly immersed into a ball of ℝ³ is discrete. This gives a positive answer to a question of Yau (Asian J. Math. 4:235–278, 2000).