12 research outputs found
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 on a 3-dimensional manifold
with volume form independent of and with a real-analytic family of
nowhere vanishing harmonic one forms , then can be
realized as a family of special Lagrangian submanifolds of a Calabi-Yau
manifold . 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 -parameter families of special
Lagrangian tori inside -dimensional Calabi-Yau manifolds with torus
symmetry. We also compute McLean's metric of 3-dimensional special Lagrangian
fibrations with -symmetry.Comment: 27 page
Lagrangian submanifolds from tropical hypersurfaces
We prove that a smooth tropical hypersurface in can be lifted
to a smooth embedded Lagrangian submanifold in . This
completes the proof of the result announced in the article "Lagrangian pairs
pants" arXiv:1802.02993. The idea of the proof is to use Lagrangian pairs of
pants as the main building blocks.Comment: 59 pages, 14 Figures. This article completes the proof of the result
announced in "Lagrangian pairs pants" arXiv:1802.0299
Conifold transitions via affine geometry and mirror symmetry
Mirror 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-Yaus in terms of certain tropical
-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 \emph{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.Comment: 82 pages, 28 figures. Published version. The main conjecture
(Conjecture 8.3) has been reformulated. We added Section 9.5 where we
partially prove the conjecture in an example. Improved expositio
Lagrangian pairs of pants
We construct a Lagrangian submanifold, inside the cotangent bundle of a real
torus, which we call a Lagrangian pair of pants. It is given as the graph of
the differential of a smooth function defined on the real blow up of a
Lagrangian coamoeba. Lagrangian pairs of pants are the main building blocks in
a construction of smooth Lagrangian submanifolds of which
lift tropical subvarieties in . As an example we explain how to
lift tropical curves in to Lagrangian submanifolds of
. We also give several new examples of Lagrangian
submanifolds inside toric varieties, some of which are monotone.Comment: 50 pages, 18 Figure
On homological mirror symmetry of toric Calabi-Yau threefolds
We 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.
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 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.Comment: 45 page
Constructions of Calabi Yau metrics and of special Lagrangian submanifolds
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