140,867 research outputs found
Calabi-Yau manifolds with isolated conical singularities
Let be a complex projective variety with only canonical singularities and
with trivial canonical bundle. Let be an ample line bundle on . Assume
that the pair is the flat limit of a family of smooth polarized
Calabi-Yau manifolds. Assume that for each singular point there exist
a Kahler-Einstein Fano manifold and a positive integer dividing
such that is very ample and such that the germ is
locally analytically isomorphic to a neighborhood of the vertex of the
blow-down of the zero section of . We prove that up to
biholomorphism, the unique weak Ricci-flat Kahler metric representing on is asymptotic at a polynomial rate near to the natural
Ricci-flat Kahler cone metric on constructed using the Calabi
ansatz. In particular, our result applies if is a nodal
quintic threefold in . This provides the first known examples of
compact Ricci-flat manifolds with non-orbifold isolated conical singularities.Comment: 41 pages, added a short appendix on special Lagrangian vanishing
cycle
The separating semigroup of a real curve
We introduce the separating semigroup of a real algebraic curve of dividing
type. The elements of this semigroup record the possible degrees of the
covering maps obtained by restricting separating morphisms to the real part of
the curve. We also introduce the hyperbolic semigroup which consists of
elements of the separating semigroup arising from morphisms which are
compositions of a linear projection with an embedding of the curve to some
projective space.
We completely determine both semigroups in the case of maximal curves. We
also prove that any embedding of a real curve to projective space of
sufficiently high degree is hyperbolic. Using these semigroups we show that the
hyperbolicity locus of an embedded curve is in general not connected.Comment: 14 pages, 4 figures, published version, comments welcome
- …