1,720,967 research outputs found
Some families of special Lagrangian tori
No full textWe give an explicit proof of the local version of Bryant's result [ 11, stating that any 3-dimensional real-analytic Riemannian manifold can be isometrically embedded as a special Lagrangian submanifold in a Calabi-Yau manifold. We then refine the theorem proving that a certain class of real-analytic one-parameter families of metrics on a 3-torus can be isometrically embedded in a Calabi-Yau manifold as a one-parameter family of special Lagrangian submanifolds. Two applications of these results show how the geometry of the moduli space of 3-dimesional special Lagrangian submanifolds differs considerably from the 2-dimensional one. First of all, applying Bryant's theorem and a construction due to Calabi we show that nearby elements of the local moduli space of a special Lagrangian 3-torus can intersect themselves. Secondly, we use our examples of one-parameter families to show that in dimension three (and higher) the moduli space of special Lagrangian tori is not, in general, special Lagrangian in the sense of Hitchin [13]
Isometric Embeddings of Families of Special Lagrangian Submanifolds
No full textWe 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
Semi-global invariants of piecewise smooth Lagrangian fibrations
No full textWe study certain types of piecewise smooth Lagrangian fibrations of smooth symplectic manifolds, which we call stitched Lagrangian fibrations. We extend the classical theory of action-angle co-ordinates to these fibrations by defining certain invariants which give a semi-global classification of germs of stitched fibrations. We then describe stitched fibrations with monodromy in terms of these invariants
The fixed point set of antisymplectic involutions of Lagrangian fibrations
Get PDFWe discuss some results and ideas on the topology of Lagrangian submanifolds obtained as the fixed point locus of certain anti-symplectic involutions preserving the fibres of a Lagrangian fibration f: X -→ B. Here X is a symplectic manifold diffeomorphic to a Calabi-Yau manifold
Conifold transitions via affine geometry and mirror symmetry
Get PDFMirror symmetry of Calabi-Yau manifolds can be understood via a Legendre duality between a pair of certain affine manifolds with singularities called tropical manifolds. In this article, we study conifold transitions from the point of view of Gross and Siebert. We introduce the notions of tropical nodal singularity, tropical conifolds, tropical resolutions and smoothings. We interpret known global obstructions to the complex smoothing and symplectic small resolution of compact nodal Calabi-Yau manifolds in terms of certain tropical 2-cycles containing the nodes in their associated tropical conifolds. We prove that the existence of such cycles implies the simultaneous vanishing of the obstruction to smoothing the original Calabi-Yau and to resolving its mirror. We formulate a conjecture suggesting that the existence of these cycles should imply that the tropical conifold can be resolved and its mirror can be smoothed, thus showing that the mirror of the resolution is a smoothing. We partially prove the conjecture for certain configurations of nodes and for some interesting examples
On homological mirror symmetry of toric Calabi-Yau three-folds
Get PDFWe use Lagrangian torus fibrations on the mirror of a toric Calabi-Yau
threefold to construct Lagrangian sections and various Lagrangian
spheres on . We then propose an explicit correspondence between the sections
and line bundles on and between spheres and sheaves supported on the
toric divisors of . We conjecture that these correspondences induce
an embedding of the relevant derived Fukaya category of inside the derived
category of coherent sheaves on .Comment: 79 pages, 22 Figures. Accepted manuscript to appear in Journal of
Symplectic Geometry Vol. 16, no.
Some piece-wise smooth Lagrangian fibrations
No full textThis paper was motivated by the Strominger-Yau-Zaslow [A. Strominger, S.-T. Yau and E. Zaslow, Nuclear Phys. B 479 (1996), no. 1-2, 243–259; MR1429831 (97j:32022)] conjecture and work surrounding it. This conjecture predicts that mirror symmetry can be explained in terms of dualizing special Lagrangian fibrations on Calabi-Yau manifolds. The paper under review deals with the question of constructing Lagrangian, rather than special Lagrangian, fibrations.
One way to construct a smooth Lagrangian torus bundle is to start with an affine manifold, i.e., a real manifold with transition maps in , whose transition maps in fact have integral linear part. Then there is a local system contained in the cotangent bundle of , generated locally by where are local affine coordinates. Because of the restriction on transition maps, is well-defined, and inherits the canonical symplectic form on and is a Lagrangian torus bundle over .
Now the basic problem is that interesting Lagrangian fibrations will have singular fibres. One considers affine manifolds with singularities, i.e., topological manifolds with a dense open set which has an affine structure. Ideally, should have codimension two in . One then seeks compactifications as symplectic manifolds. Of course, one's ability to do this will depend on the nature of the affine structure around . It is not difficult to carry this out in two dimensions for some standard types of singularities [see, for example, M. Symington, in Topology and geometry of manifolds (Athens, GA, 2001), 153–208, Proc. Sympos. Pure Math., 71, Amer. Math. Soc., Providence, RI, 2003; MR2024634 (2005b:53142)]. The paper under review is concerned with aspects which only arise in higher dimensions. In particular, it appears that in higher dimensions there are some naturally occurring singularities which can only be compactified using piecewise smooth fibrations. This phenomenon was first seen in work of W.-D. Ruan [in Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999), 297–332, Amer. Math. Soc., Providence, RI, 2001; MR1876075 (2002m:32041)] and was demonstrated by D. D. Joyce [see, for example, Comm. Anal. Geom. 11 (2003), no. 5, 859–907; MR2032503 (2004m:53094)] to be crucial in understanding the Strominger-Yau-Zaslow conjecture.
The paper under review is partly expository and partly an introduction to some new ideas of the authors. It begins with a nice exposition of the basic problems and examples that arise in this context, and then proceeds to give some general constructions for producing piecewise smooth Lagrangian fibrations, including ones which have the correct topology for compactifying the "negative vertex'', one of the two basic singularities which occur in three-dimensional Calabi-Yaus. This is the hard case; local models for the "positive vertex'' have been known for a long time. This example has the feature that the discriminant locus is not codimension two, but is a codimension one fattening of a trivalent graph. This appears to be a necessary feature of such examples. The authors then consider periods of such piecewise smooth fibrations, and give some hints at upcoming work on more powerful methods of constructing piecewise smooth fibrations
Lagrangian 3-torus fibrations
Get PDFWe prove that Mark Gross' topological Calabi-Yau compactifications can be made into symplectic compactifications. To prove this we develop a method to construct singular Lagrangian 3-torus fibrations over certain a priori given integral affine manifolds with singularities, which we call simple. This produces pairs of compact symplectic 6-manifolds homeomorphic to mirror pairs of Calabi-Yau 3-folds together with Lagrangian fibrations whose underlying integral affine structures are dual
Lagrangian submanifolds from tropical hypersurfaces
Get PDFWe prove that a smooth tropical hypersurface in can be lifted to a smooth embedded Lagrangian submanifold in . The idea of the proof is to use Lagrangian pairs of pants, which are the lifts of tropical hyperplanes introduced by the author in an earlier paper, as the main building blocks
- …
