636 research outputs found
Potentials of a Frobenius like structure
This paper proves the existence of potentials of the first and second kind of
a Frobenius like structure in a frame which encompasses families of
arrangements.
The frame uses the notion of matroids. For the proof of the existence of the
potentials, a power series ansatz is made. The proof that it works requires
that certain decompositions of tuples of coordinate vector fields are related
by certain elementary transformations. This is shown with a nontrivial result
on matroid partition.Comment: The paper generalizes arXiv:1608.08423. Theorem 2.1 in the new paper
is a matroid version of the linear algebra result theorem 1.3 in the old
paper. The main changes between old and new paper are in chapter 2. The new
paper has 13 pages. It is accepted for publication in the Glasgow
Mathematical Journa
Regular F-manifolds: initial conditions and Frobenius metrics
A regular F-manifold is an F-manifold (with Euler field) (M, \circ, e, E),
such that the endomorphism {\mathcal U}(X) := E \circ X of TM is regular at any
p\in M. We prove that the germ ((M,p), \circ, e, E) is uniquely determined (up
to isomorphism) by the conjugacy class of {\mathcal U}_{p} : T_{p}M \rightarrow
T_{p}M. We obtain that any regular F-manifold admits a preferred system of
local coordinates and we find conditions, in these coordinates, for a metric to
be Frobenius. We study the Lie algebra of infinitesimal symmetries of regular
F-manifolds. We show that any regular F-manifold is locally isomorphic to the
parameter space of a Malgrange universal connection. We prove an initial
condition theorem for Frobenius metrics on regular F-manifolds.Comment: 35 pages; with respect to the previous version, Section 4 is
reorganised; reference [17] is added; other minor correction
Hermitian metrics on F-manifolds
An -manifold is complex manifold with a multiplication on the holomorphic
tangent bundle with a certain integrability condition. Important examples are
Frobenius manifolds and especially base spaces of universal unfoldings of
isolated hypersurface singularities. This paper reviews the construction of
hermitian metrics on -manifolds from geometry. It clarifies the logic
between several notions. It also introduces a new {\it canonical} hermitian
metric. Near irreducible points it makes the manifold almost hyperbolic. This
holds for the singularity case and will hopefully lead to applications there.Comment: 2nd version 36 pages. Compared to the 1st version (32 pages), the
sections 2.4 and 2.5 have been extende
Nilpotent orbits of a generalization of Hodge structures
We study a generalization of Hodge structures which first appeared in the
work of Cecotti and Vafa. It consists of twistors, that is, holomorphic vector
bundles on P^1, with additional structure, a flat connection on C^*, a real
subbundle and a pairing. We call these objects TERP-structures. We generalize
to TERP-structures a correspondence of Cattani, Kaplan and Schmid between
nilpotent orbits of Hodge structures and polarized mixed Hodge structures. The
proofs use work of Simpson and Mochizuki on variations of twistor structures
and a control of the Stokes structures of the poles at zero and infinity. The
results are applied to TERP-structures which arise via oscillating integrals
from holomorphic functions with isolated singularities.Comment: 43 pages, some very minor modifications, misprints correcte
Semistable bundles on curves and reducible representations of the fundamental group
Bolibruch's examples of representations of pi_1(P^1-finitely many points)
which are not realizable by Fuchsian differential systems are adapted to curves
of higher genus.Comment: 10 pages, LaTe
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