62 research outputs found
Mass in K\"ahler Geometry
We prove a simple, explicit formula for the mass of any asymptotically
locally Euclidean (ALE) K\"ahler manifold, assuming only the sort of weak
fall-off conditions required for the mass to actually be well-defined. For ALE
scalar-flat K\"ahler manifolds, the mass turns out to be a topological
invariant, depending only on the underlying smooth manifold, the first Chern
class of the complex structure, and the K\"ahler class of the metric. When the
metric is actually AE (asymptotically Euclidean), our formula not only implies
a positive mass theorem for K\"ahler metrics, but also yields a Penrose-type
inequality for the mass.Comment: 53 pages, minor corrections and improvements, final versio
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
Asymptotically conical Calabi-Yau manifolds, I
This is the first part in a two-part series on complete Calabi-Yau manifolds
asymptotic to Riemannian cones at infinity. We begin by proving general
existence and uniqueness results. The uniqueness part relaxes the decay
condition needed in earlier work to ,
relying on some new ideas about harmonic functions. We then look at a few
examples: (1) Crepant resolutions of cones. This includes a new class of
Ricci-flat small resolutions associated with flag manifolds. (2) Affine
deformations of cones. One focus here is the question of the precise rate of
decay of the metric to its tangent cone. We prove that the optimal rate for the
Stenzel metric on is .Comment: 27 pages, various corrections, final versio
Asymptotically cylindrical Calabi-Yau manifolds
Let be a complete Ricci-flat Kahler manifold with one end and assume that
this end converges at an exponential rate to for some
compact connected Ricci-flat manifold . We begin by proving general
structure theorems for ; in particular we show that there is no loss of
generality in assuming that is simply-connected and irreducible with
Hol SU, where is the complex dimension of . If we
then show that there exists a projective orbifold and a divisor
in with torsion normal bundle such that is
biholomorphic to , thereby settling a long-standing
question of Yau in the asymptotically cylindrical setting. We give examples
where is not smooth: the existence of such examples appears not to
have been noticed previously. Conversely, for any such pair we give a short and self-contained proof of the existence and
uniqueness of exponentially asymptotically cylindrical Calabi-Yau metrics on
.Comment: 33 pages, various updates and minor corrections, final versio
- …