34 research outputs found
Lie-algebra Dolbeault cohomology and small deformations of nilmanifolds
We consider nilmanifolds with left-invariant complex structure and prove that
small deformations of such structures are again left invariant if the
Dolbeault-cohomology of the nilmanifold can be calculated using left-invariant
forms. By a result of Console and Fino this is generically the case. Our main
tool is an analog of Dolbeault-cohomology for Lie-algebras with complex
structure.Comment: 19 pages; updated first part of arXiv:0709.0467v1 containing the
results on small deformations. The results on deformations in the large will
appear in an independent pape
Torsion in equivariant cohomology and Cohen-Macaulay G-actions
We show that the well-known fact that the equivariant cohomology of a torus
action is a torsion-free module if and only if the map induced by the inclusion
of the fixed point set is injective generalises to actions of arbitrary compact
connected Lie groups if one replaces the fixed point set by the set of points
with maximal isotropy rank. This is true essentially because the action on this
set is always equivariantly formal. In case this set is empty we show that the
induced action on the set of points with highest occuring isotropy rank is
Cohen-Macaulay. It turns out that just as equivariant formality of an action is
equivalent to equivariant formality of the action of a maximal torus, the same
holds true for equivariant injectivity and the Cohen-Macaulay property. In
addition, we find a topological criterion for equivariant injectivity in terms
of orbit spaces.Comment: 14 pages, 1 figur
Canonical rings of Gorenstein stable Godeaux surfaces
Extending the description of canonical rings from \cite{reid78} we show that
every Gorenstein stable Godeaux surface with torsion of order at least is
smoothable.Comment: 17 page
Geometry of nilmanifolds with left-invariant complex structure and deformations in the large
The relation between nilmanifolds with left-invariant complex structure and
iterated principal holomorphic torus bundles is clarified and we give criteria
under which deformations in the large are again of such type. As an application
we obtain a fairly complete picture in complex dimension three.Comment: 40 pages, revised version of the second part of arXiv:0709.0467v1. To
appear in Proc. LM
Computing invariants of semi-log-canonical surfaces
We describe some methods to compute fundamental groups, (co)homology, and
irregularity of semi-log-canonical surfaces.
As an application, we show that there are exactly two irregular Gorenstein
stable surfaces with , both of which have and
but different homotopy type.Comment: 17 page