On singular fibres of Lagrangian fibrations over holomorphic symplectic manifolds

We classify singular fibres over general points of the discriminant locus of projective Lagrangian fibrations over 4-dimensional holomorphic symplectic manifolds. The singular fibre F is the following either one: F is isomorphic to the product of an elliptic curve and a Kodaira singular fibre up to finite unramified covering or F is a normal crossing variety consisting of several copies of a minimal elliptic ruled surface of which the dual graph is Dynkin diagram of type or . Moreover, we show all types of the above singular fibres actually occur. Received: 10 March 2000 / Revised version: 29 September 2000 / Published online: 24 September 2001     

Families of Lagrangian fibrations on hyperkähler manifolds

A holomorphic Lagrangian fibration on a holomorphically symplectic manifold is a holomorphic map with Lagrangian fibers. It is known (due to Huybrechts) that a given compact manifold admits only finitely many holomorphic symplectic structures, up to deformation. We prove that a given compact, simple hyperkähler manifold with _b2?7$b_2?7$ admits only finitely many deformation types of holomorphic Lagrangian fibrations. We also prove that all known hyperkähler manifolds are never Kobayashi hyperbolic.     

On almost holomorphic Lagrangian fibrations

We prove that the pull back of an ample line bundle by an almost holomorphic Lagrangian fibration is nef. As an application, we show birational semi rigidity of Lagrangian fibrations.     

Symmetries of Lagrangian fibrations

We construct fiber-preserving anti-symplectic involutions for a large class of symplectic manifolds with Lagrangian torus fibrations. In particular, we treat the K3 surface and the six-dimensional examples constructed by Castaño-Bernard and Matessi (2009) [8], which include a six-dimensional symplectic manifold homeomorphic to the quintic threefold. We interpret our results as corroboration of the view that in homological mirror symmetry, an anti-symplectic involution is the mirror of duality. In the same setting, we construct fiber-preserving symplectomorphisms that can be interpreted as the mirror to twisting by a holomorphic line bundle.     

Base manifolds for fibrations of projective irreducible symplectic manifolds

Given a projective irreducible symplectic manifold M of dimension 2n, a projective manifold X and a surjective holomorphic map f:M→X with connected fibers of positive dimension, we prove that X is biholomorphic to the projective space of dimension n. The proof is obtained by exploiting two geometric structures at general points of X: the affine structure arising from the action variables of the Lagrangian fibration f and the structure defined by the variety of minimal rational tangents on the Fano manifold X.     

Singular Lagrangian manifolds and their Lagrangian maps

The paper examines the singularity theory of Lagrangian manifolds and its connection with variational calculus, classification of Coxeter groups, and symplectic topology. We consider the application of the theory to the problem of going past an obstacle, to partial differential equations, and to the analysis of singularities of ray systems.Translated from Itogi Nauki i Tekhniki, Seriya Sovermennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 33, pp. 55–112, 1988.     

Classification of Lagrangian fibrations over a Klein bottle

This paper completes the classification of regular Lagrangian fibrations over compact surfaces. Mishachev (Diff Geom Appl 6:301–320, 1996) classifies regular Lagrangian fibrations over \mathbbT²{\mathbb{T}^2}. The main theorem in Fried et al. (Comment Math Helv 56(4):487–523, 1981) is used to in order to classify integral affine structures on the Klein bottle K ² and, hence, regular Lagrangian fibrations over this space.     

Connected sums of manifolds which induce approximate fibrations

Codimension-2 fibrators are -manifolds which automatically induce approximate fibration, in the following sense: given any proper mapping from an -manifold onto a -manifold such that each point-preimage is a copy of the codimension-2 fibrator, is necessarily an approximate fibration. In this paper, we give some answers to the following question: given an -manifold which is a nontrivial connected sum, when is a codimension-2 fibrator?

     

Pseudotoric Lagrangian fibrations of toric and nontoric fano varieties

We introduce the notion of a pseudotoric structure on a symplectic manifold, generalizing the notion of a toric structure. We show that such a pseudotoric structure can exist on toric and nontoric symplectic manifolds. For the toric manifolds, it describes deformations of the standard toric Lagrangian fibrations; for the nontoric ones, it gives Lagrangian fibrations with singularities that are very close to the toric fibrations. We present examples of toric manifolds with different pseudotoric structures and prove that certain nontoric manifolds (smooth complex quadrics) have such structures. In the future, introducing this new structure can be useful for generalizing the geometric quantization and mirror symmetry methods that work well in the toric case to a broader class of Fano varieties.     

Algebraic topology of special Lagrangian manifolds

In this paper, we prove various results on the topology of the Grassmannian of oriented 3-planes in Euclidean 6-space and compute its cohomology ring. We give self-contained proofs. These spaces come up when studying submanifolds of manifolds with calibrated geometries. We collect these results here for the sake of completeness. As applications of our algebraic topological study we present some results on special Lagrangian-free embeddings of surfaces and 3-manifolds into the Euclidean 4 and 6-space.     

Approximate fibrations and bundle maps on Hilbert cube manifolds

This paper is concerned with parameterized families of approximate fibrations from a compact Hilbert cube manifold M to a compact polyhedron B. The main result shows how to straighten out certain of these families to be nearly like a product. As an application of this technique, it is shown that an approximate fibration p:M → B can be approximated arbitrarily closely by bundle maps if and only if p is homotopic via approximate fibrations to a bundle map. Another result is that the space of bundle maps from M to B is locally n-connected for each n ? 0.     

On second order minimal Lagrangian sub manifolds in complex space forms

In this paper, we determine all second order minimal Lagrangian submanifolds in complex space forms whose cubic forms have the largest non-trivial continuous symmetries. We describe these minimal Lagrangian submanifolds from the viewpoint of Bryant and study their geometric properties.     

Lagrangian embeddings into subcritical Stein manifolds

In this paper we obtain new topological restrictions on Lagrangian embeddings into subcritical Stein manifolds. We also extend previous results of Gromov, Oh, Polterovich and Viterbo on Lagrangian submanifolds of ℂ ⁿ to the more general case of subcritical Stein manifolds. Research partially supported by the US-Israel Binational Science Foundation grant 1999086.     

Rational Lagrangian fibrations on punctual Hilbert schemes of K3 surfaces

A rational Lagrangian fibration f on an irreducible symplectic variety V is a rational map which is birationally equivalent to a regular surjective morphism with Lagrangian fibers. By analogy with K3 surfaces, it is natural to expect that a rational Lagrangian fibration exists if and only if V has a divisor D with Bogomolov–Beauville square 0. This conjecture is proved in the case when V is the Hilbert scheme of d points on a generic K3 surface S of genus g under the hypothesis that its degree 2g−2 is a square times 2d−2. The construction of f uses a twisted Fourier–Mukai transform which induces a birational isomorphism of V with a certain moduli space of twisted sheaves on another K3 surface M, obtained from S as its Fourier–Mukai partner.