8,633 research outputs found
Numerical Ricci-flat metrics on K3
We develop numerical algorithms for solving the Einstein equation on
Calabi-Yau manifolds at arbitrary values of their complex structure and Kahler
parameters. We show that Kahler geometry can be exploited for significant gains
in computational efficiency. As a proof of principle, we apply our methods to a
one-parameter family of K3 surfaces constructed as blow-ups of the T^4/Z_2
orbifold with many discrete symmetries. High-resolution metrics may be obtained
on a time scale of days using a desktop computer. We compute various geometric
and spectral quantities from our numerical metrics. Using similar resources we
expect our methods to practically extend to Calabi-Yau three-folds with a high
degree of discrete symmetry, although we expect the general three-fold to
remain a challenge due to memory requirements.Comment: 38 pages, 10 figures; program code and animations of figures
downloadable from http://schwinger.harvard.edu/~wiseman/K3/ ; v2 minor
corrections, references adde
Cremona transformations and diffeomorphisms of surfaces
We show that the action of Cremona transformations on the real points of
quadrics exhibits the full complexity of the diffeomorphisms of the sphere, the
torus, and of all non-orientable surfaces. The main result says that if X is
rational, then Aut(X), the group of algebraic automorphisms, is dense in
Diff(X), the group of self-diffeomorphisms of X.Comment: 17 pages, 11 figures, shorter proofs and improvement of the result
Curves in Hilbert modular varieties
We prove a boundedness-theorem for families of abelian varieties with real
multiplication. More generally, we study curves in Hilbert modular varieties
from the point of view of the Green Griffiths-Lang conjecture claiming that
entire curves in complex projective varieties of general type should be
contained in a proper subvariety. Using holomorphic foliations theory, we
establish a Second Main Theorem following Nevanlinna theory. Finally, with a
metric approach, we establish the strong Green-Griffiths-Lang conjecture for
Hilbert modular varieties up to finitely many possible exceptions.Comment: Final version, to appear in Asian J. Mat
D-branes at Singularities, Compactification, and Hypercharge
We report on progress towards the construction of SM-like gauge theories on
the world-volume of D-branes at a Calabi-Yau singularity. In particular, we
work out the topological conditions on the embedding of the singularity inside
a compact CY threefold, that select hypercharge as the only light U(1) gauge
factor. We apply this insight to the proposed open string realization of the SM
of hep-th/0508089, based on a D3-brane at a dP_8 singularity, and present a
geometric construction of a compact Calabi-Yau threefold with all the required
topological properties. We comment on the relevance of D-instantons to the
breaking of global U(1) symmetries.Comment: 39 pages, 9 figures. Minor correction
Mean Curvature Flow, Orbits, Moment Maps
Given a compact Riemannian manifold together with a group of isometries, we
discuss MCF of the orbits and some applications: eg, finding minimal orbits. We
then specialize to Lagrangian orbits in Kaehler manifolds. In particular, in
the Kaehler-Einstein case we find a relation between MCF and moment maps which,
for example, proves that the minimal Lagrangian orbits are isolated.Comment: 18 pages; minor change
- …