The study of moduli space of polarized Calabi-Yau manifolds (CY moduli) is related to seveal branches of mathematics: differential geometry, algerbraic geometry, and mathematical physics, etc. In what follows, we briefly introduce the differential geometry of the CY moduli.

1. Local theory:

We assume that M is the CY moduli of some CY 3 fold, with dimension m. Assume that ω is the Weil-Petersson metric on M. In [1], we defined

ω'=(m+3)ω+Ric(ω),

and we call ω' the Hodge metric. There are many good differential geometric properties of the Hodge metric, which makes it the starting point of the study of moduli space.

2. Semi-global theory:

In 1993, Bershadsky, Cecotti, Ooguri, and Vafa, based on their physics insight of the Mirror Symmetry, proposed the so-called BCOV conjecture. There are two parts of the conjecture: on one side of the mirror is the symplectic geometric part of the conjecture, which was proved by Zinger [2]; on the other side of the mirror is the complex geometric part of the conjecture. This part was proved in Fang-Lu-Yoshikawa [3]. Since in the complex geometric part, we need to study the properties of the moduli space at infinity, we call this kind of results semi-global theory of the CY moduli.

3. Global theory:

Working with physicist M. Douglas, in [4], we proved a Gauss-Bonnet type theorem of the Chern-Weil forms with respec to the Weil-Petersson metric. As a physics application, we proved, under certain conditions, that the number of feasible CY 3 folds (and hence the number of feasible Universes) is finite.

4. Special Kahler manifold:

Special Kahler manfold is dual to CY moduli. In [5], we proved the Freed Conjecture: Any non-trivial special Kahler manifold is incomplete with respect to the special Kahler metric.

For details, see the survey paper here (TBA in 12/07).

References:

[1] Z. Lu, On the Hodge Metric of the Universal Deformation Space of Calabi-Yau Threefolds, J. Geom. Anal., vol 11(1), 2001, pp 103-118.

[2] A. Zinger, The Reduced Genus-One Gromov-Witten Invariants of Calabi-Yau Hypersurfaces, arXiv:0705.2397.

[3] H. Fang, Z. Lu, and K. Yoshikawa, Asymptotic behavior of the BCOV torsion of Calabi-Yau moduli, preprint; math/0601411.

[4] M. Douglas and Z. Lu, in preparation.

[5] Z. Lu, A Note on Special Kahler Manifolds, Math. Ann., vol 313 1999, pp 711-713.