20,890 research outputs found
Hodge Theory on Metric Spaces
Hodge theory is a beautiful synthesis of geometry, topology, and analysis,
which has been developed in the setting of Riemannian manifolds. On the other
hand, spaces of images, which are important in the mathematical foundations of
vision and pattern recognition, do not fit this framework. This motivates us to
develop a version of Hodge theory on metric spaces with a probability measure.
We believe that this constitutes a step towards understanding the geometry of
vision.
The appendix by Anthony Baker provides a separable, compact metric space with
infinite dimensional \alpha-scale homology.Comment: appendix by Anthony W. Baker, 48 pages, AMS-LaTeX. v2: final version,
to appear in Foundations of Computational Mathematics. Minor changes and
addition
Hodge theory on Cheeger spaces
We extend the study of the de Rham operator with ideal boundary conditions
from the case of isolated conic singularities, as analyzed by Cheeger, to the
case of arbitrary stratified pseudomanifolds. We introduce a class of ideal
boundary operators and the notion of mezzoperversity, which intermediates
between the standard lower and upper middle perversities in intersection
theory, as interpreted in this de Rham setting, and show that the de Rham
operator with these boundary conditions is Fredholm and has compact resolvent.
We also prove an isomorphism between the resulting Hodge and L2 de Rham
cohomology groups, and that these are independent of the choice of iterated
edge metric. On spaces which admit ideal boundary conditions of this type which
are also self-dual, which we call `Cheeger spaces', we show that these Hodge/de
Rham cohomology groups satisfy Poincare Duality.Comment: v2: Slight changes to improve exposition, v3: Improved discussion of
core domain, to appear in Crelle's journa
Moduli Space Holography and the Finiteness of Flux Vacua
A holographic perspective to study and characterize field spaces that arise
in string compactifications is suggested. A concrete correspondence is
developed by studying two-dimensional moduli spaces in supersymmetric string
compactifications. It is proposed that there exist theories on the boundaries
of each moduli space, whose crucial data are given by a Hilbert space, an
Sl(2,C)-algebra, and two special operators. This boundary data is motivated by
asymptotic Hodge theory and the fact that the physical metric on the moduli
space of Calabi-Yau manifolds asymptotes near any infinite distance boundary to
a Poincare metric with Sl(2,R) isometry. The crucial part of the bulk theory on
the moduli space is a sigma model for group-valued matter fields. It is
discussed how this might be coupled to a two-dimensional gravity theory. The
classical bulk-boundary matching is then given by the proof of the famous Sl(2)
orbit theorem of Hodge theory, which is reformulated in a more physical
language. Applying this correspondence to the flux landscape in Calabi-Yau
fourfold compactifications it is shown that there are no infinite tails of
self-dual flux vacua near any co-dimension one boundary. This finiteness result
is a consequence of the constraints on the near boundary expansion of the bulk
solutions that match to the boundary data. It is also pointed out that there is
a striking connection of the finiteness result for supersymmetric flux vacua
and the Hodge conjecture.Comment: 57 pages, 2 figures, v2: minor clarifications, typos corrected,
references adde
Hodge theory on Cheeger spaces
We extend the study of the de Rham operator with ideal boundary conditions from the case of isolated conic singularities, as analyzed by Cheeger, to the case of arbitrary stratified pseudomanifolds. We introduce a class of ideal boundary operators and the notion of mezzoperversity, which intermediates between the standard lower and upper middle perversities in intersection theory, as interpreted in this de Rham setting, and show that the de Rham operator with these boundary conditions is Fredholm and has compact resolvent. We also prove an isomorphism between the resulting Hodge and L2 de Rham cohomology groups, and that these are independent of the choice of iterated edge metric. On spaces which admit ideal boundary conditions of this type which are also self-dual, which we call ‘Cheeger spaces’, we show that these Hodge/de Rham cohomology groups satisfy Poincare' Duality
Adiabatic Limit and the Fr\"olicher Spectral Sequence
Motivated by our conjecture of an earlier work predicting the degeneration at
the second page of the Fr\"olicher spectral sequence of any compact complex
manifold supporting an SKT metric (i.e. such that
), we prove degeneration at whenever the
manifold admits a Hermitian metric whose torsion operator and its
adjoint vanish on -harmonic forms of positive degrees up to
\mbox{dim}_\C X. Besides the pseudo-differential Laplacian inducing a Hodge
theory for that we constructed in earlier work and Demailly's
Bochner-Kodaira-Nakano formula for Hermitian metrics, a key ingredient is a
general formula for the dimensions of the vector spaces featuring in the
Fr\"olicher spectral sequence in terms of the asymptotics, as a positive
constant decreases to zero, of the small eigenvalues of a rescaled
Laplacian , introduced here in the present form, that we adapt to the
context of a complex structure from the well-known construction of the
adiabatic limit and from the analogous result for Riemannian foliations of
\'Alvarez L\'opez and Kordyukov.Comment: 32 page
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