METRIC PROPERTIES OF CALABI-YAU MANIFOLDS

This talk aims to introduce some general ideas relating properties from algebraic
geometry to concepts from metric geometry, in particular that of Gromov-Hausdorff limits
of metric spaces.
Let $X$ be a Calabi-Yau manifold of dimension $n$, that is a complex projective manifold
which admits a nowhere vanishing holomorphic $n$-form, and no holomorphic $i$-forms for
$0 < i < n$. By a famous theorem of Yau, for each K"ahler class in the real second cohomology,
there exists a unique Ricci
at K"ahler metric on X with K"ahler form in the given
class, the Calabi Yau metric; hence there is a well-defined metric space structure on $X$.
A natural question then arises: if we degenerate either the complex or Kähler structures
on X in the sense of algebraic geometry, obtaining a singular projective variety, what
can be said about the metric limits (in the sense of Gromov-Hausdorff) of the corresponding Ricci
at K"ahler manifolds? We will suggest some answers to this question and explain
their relevance for the geometry of Calabi-Yau manifolds.