A counterexample to a conjecture of complete fan

Abstract

If a Griffiths domain $D$ is a symmetric Hermitian domain, the toroidal compactification of the quotient space $\Gamma\backslash D$, associated to a projective fan and a discrete subgroup $\Gamma$ of ${\rm Aut}(D)$, was constructed by Mumford et al. Kazuya Kato and Sampei Usui studied extensions of $\Gamma\backslash D$ for a Griffiths domain $D$ in general, and introduced a notion of \lq\lq complete fan\rq\rq$\;$ as a generalization of a notion of projective fan. The existence of complete fans is expected. In this paper, we give an example of $D$ which has no complete fan.