11,618 research outputs found
Covariant symplectic structure of the complex Monge-Amp\`ere equation
The complex Monge-Amp\`ere equation admits covariant bi-symplectic structure
for complex dimension 3, or higher. The first symplectic 2-form is obtained
from a new variational formulation of complex Monge- Amp\`ere equation in the
framework of the covariant Witten-Zuckerman approach to symplectic structure.
We base our considerations on a reformulation of the Witten-Zuckerman theory in
terms of holomorphic differential forms. The first closed and conserved
Witten-Zuckerman symplectic 2-form for the complex Monge-Amp\`ere equation is
obtained in arbitrary dimension and for all cases elliptic, hyperbolic and
homogeneous.
The connection of the complex Monge-Amp\`ere equation with Ricci-flat
K\"ahler geometry suggests the use of the Hilbert action. However, we point out
that Hilbert's Lagrangian is a divergence for K\"ahler metrics. Nevertheless,
using the surface terms in the Hilbert Lagrangian we obtain the second
Witten-Zuckerman symplectic 2-form for complex dimension>2.Comment: Physics Letters A 268 (2000) 29
Smooth invariants of focus-focus singularities and obstructions to product decomposition
We study focus-focus singularities (also known as nodal singularities, or
pinched tori) of Lagrangian fibrations on symplectic -manifolds. We show
that, in contrast to elliptic and hyperbolic singularities, there exist
homeomorphic focus-focus singularities which are not diffeomorphic.
Furthermore, we obtain an algebraic description of the moduli space of
focus-focus singularities up to smooth equivalence, and show that for double
pinched tori this space is one-dimensional. Finally, we apply our construction
to disprove Zung's conjecture which says that any non-degenerate singularity
can be smoothly decomposed into an almost direct product of standard
singularities.Comment: Final version accepted to Journal of Symplectic Geometry; 25 pages, 2
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Symplectic Calabi-Yau manifolds, minimal surfaces and the hyperbolic geometry of the conifold
Given an SO(3)-bundle with connection, the associated two-sphere bundle
carries a natural closed 2-form. Asking that this be symplectic gives a
curvature inequality first considered by Reznikov. We study this inequality in
the case when the base has dimension four, with three main aims. Firstly, we
use this approach to construct symplectic six-manifolds with c_1=0 which are
never Kahler; e.g., we produce such manifolds with b_1=0=b_3 and also with
c_2.omega <0, answering questions posed by Smith-Thomas-Yau. Examples come from
Riemannian geometry, via the Levi-Civita connection on Lambda^+. The underlying
six-manifold is then the twistor space and often the symplectic structure tames
the Eells-Salamon twistor almost complex structure. Our second aim is to
exploit this to deduce new results about minimal surfaces: if a certain
curvature inequality holds, it follows that the space of minimal surfaces (with
fixed topological invariants) is compactifiable; the minimal surfaces must also
satisfy an adjunction inequality, unifying and generalising results of
Chen--Tian. One metric satisfying the curvature inequality is hyperbolic
four-space H^4. Our final aim is to show that the corresponding symplectic
manifold is symplectomorphic to the small resolution of the conifold xw-yz=0 in
C^4. We explain how this fits into a hyperbolic description of the conifold
transition, with isometries of H^4 acting symplectomorphically on the
resolution and isometries of H^3 acting biholomorphically on the smoothing.Comment: v1. 52 pages, but not overly technical, so don't let the length put
you off! Comments actively encouraged. v2. Typos corrected, addresses
include
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