Abstract: The volume of a Fano manifold plays an important role in the
study of Fano manifolds. One interesting question is to find upper bounds
of the volumes for various classes of Fano manifolds. In this talk, I will
consider Fano 3-folds first. By classification results of Fano 3-folds, we
know that the volumes of Fano 3-folds are bounded by 64, which is the
volume of the projective 3-space. I will also show how to compute
explicitly upper bounds of the volumes for some particular classes of Fano
3-folds. This is not going to give any new upper bounds better than 64 for
Fano 3-folds. However, similar questions become more interesting for
higher dimensional Fano manifolds.