296 research outputs found
Functions dividing their Hessian determinants and affine spheres
The nonzero level sets of a homogeneous, logarithmically homogeneous, or
translationally homogeneous function are affine spheres if and only if the
Hessian determinant of the function is a multiple of a power or an exponential
of the function. In particular, the nonzero level sets of a homogeneous
polynomial are proper affine spheres if some power of it equals a nonzero
multiple of its Hessian determinant. The relative invariants of real forms of
regular irreducible prehomogeneous vector spaces yield many such polynomials
which are moreover irreducible. For example, the nonzero level sets of the
Cayley hyperdeterminant are affine spheres.Comment: v4 is greatly shortened with respect to v3. Some of the omitted
material will be posted in a different articl
Rohlin's invariant and gauge theory III. Homology 4--tori
This is the third in our series of papers relating gauge theoretic invariants
of certain 4-manifolds with invariants of 3-manifolds derived from Rohlin's
theorem. Such relations are well-known in dimension three, starting with
Casson's integral lift of the Rohlin invariant of a homology sphere. We
consider two invariants of a spin 4-manifold that has the integral homology of
a 4-torus. The first is a degree zero Donaldson invariant, counting flat
connections on a certain SO(3)-bundle. The second, which depends on the choice
of a 1-dimensional cohomology class, is a combination of Rohlin invariants of a
3-manifold carrying the dual homology class. We prove that these invariants,
suitably normalized, agree modulo 2, by showing that they coincide with the
quadruple cup product of 1-dimensional cohomology classes.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol9/paper47.abs.htm
Projective BGG equations, algebraic sets, and compactifications of Einstein geometries
For curved projective manifolds we introduce a notion of a normal tractor
frame field, based around any point. This leads to canonical systems of
(redundant) coordinates that generalise the usual homogeneous coordinates on
projective space. These give preferred local maps to the model projective space
that encode geometric contact with the model to a level that is optimal, in a
suitable sense. In terms of the trivialisations arising from the special
frames, normal solutions of classes of natural linear PDE (so-called first BGG
equations) are shown to be necessarily polynomial in the generalised
homogeneous coordinates; the polynomial system is the pull back of a polynomial
system that solves the corresponding problem on the model. Thus questions
concerning the zero locus of solutions, as well as related finer geometric and
smooth data, are reduced to a study of the corresponding polynomial systems and
algebraic sets. We show that a normal solution determines a canonical manifold
stratification that reflects an orbit decomposition of the model. Applications
include the construction of structures that are analogues of Poincare-Einstein
manifolds.Comment: 22 page
The infinitesimal projective rigidity under Dehn filling
To a hyperbolic manifold one can associate a canonical projective structure
and ask whether it can be deformed or not. In a cusped manifold, one can ask
about the existence of deformations that are trivial on the boundary. We prove
that if the canonical projective structure of a cusped manifold is
infinitesimally projectively rigid relative to the boundary, then infinitely
many Dehn fillings are projectively rigid. We analyze in more detail the figure
eight knot and the Withehead link exteriors, for which we can give explicit
infinite families of slopes with projectively rigid Dehn fillings.Comment: Accepted for publication at G
Complete spelling rules for the Monster tower over three-space
The Monster tower, also known as the Semple tower, is a sequence of manifolds
with distributions of interest to both differential and algebraic geometers.
Each manifold is a projective bundle over the previous. Moreover, each level is
a fiber compactified jet bundle equipped with an action of finite jets of the
diffeomorphism group. There is a correspondence between points in the tower and
curves in the base manifold. These points admit a stratification which can be
encoded by a word called the RVT code. Here, we derive the spelling rules for
these words in the case of a three dimensional base. That is, we determine
precisely which words are realized by points in the tower. To this end, we
study the incidence relations between certain subtowers, called Baby Monsters,
and present a general method for determining the level at which each Baby
Monster is born. Here, we focus on the case where the base manifold is three
dimensional, but all the methods presented generalize to bases of arbitrary
dimension.Comment: 14 pages, 4 figures; new titl
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