18 research outputs found
Graded Differential Geometry of Graded Matrix Algebras
We study the graded derivation-based noncommutative differential geometry of
the -graded algebra of complex -matrices
with the ``usual block matrix grading'' (for ). Beside the
(infinite-dimensional) algebra of graded forms the graded Cartan calculus,
graded symplectic structure, graded vector bundles, graded connections and
curvature are introduced and investigated. In particular we prove the
universality of the graded derivation-based first-order differential calculus
and show, that is a ``noncommutative graded manifold'' in a
stricter sense: There is a natural body map and the cohomologies of and its body coincide (as in the case of ordinary graded manifolds).Comment: 21 pages, LATE
Semistability vs. nefness for (Higgs) vector bundles
According to Miyaoka, a vector bundle E on a smooth projective curve is
semistable if and only if a certain numerical class in the projectivized bundle
PE is nef. We establish a similar criterion for the semistability of Higgs
bundles: namely, such a bundle is semistable if and only if for every integer s
between 0 and the rank of E, a suitable numerical class in the scheme
parametrizing the rank s locally-free Higgs quotients of E is nef. We also
extend this result to higher-dimensional complex projective varieties by
showing that the nefness of the above mentioned classes is equivalent to the
semistability of the Higgs bundle E together with the vanishing of the
discriminant of E.Comment: Comments: 20 pages, Latex2e, no figures. v2 includes a generalization
to complex projective manifolds of any dimension. To appear in Diff. Geom.
App