A counterexample to a conjecture of complete fan
The quotient of a Griffiths domain \({\mathcal D}\) by a neat subgroup \(\Gamma\subset{\text{Aut}}{\mathcal D}\) is the analogue in logarithmic geometry (in the sense of Kazuya Kato) of a locally symmetric variety, which is the special case where \({\mathcal D}\) is a hermitian symmetric domain. \textit{K. Kato} and \textit{S. Usui} [Classifying spaces of degenerating polarized Hodge structures. Princeton, NJ: Princeton University Press (2009; Zbl 1172.14002)] extended Mumford's theory of toroidal compctification of locally symmetric varieties to the Griffiths domain context. One of the results of Mumford and his collaborators is that toroidal compactifications may be chosen to be projective, by choosing the fans that determine the compactification to be projective (that means satisfying a convexity condition). Kato and Usui gave an analogue in their situation of the notion of projective fan, which they called a complete fan. As evidence that it was the correct generalisation they showed that in the case of a Griffiths domain associated with Hodge numbers \(h^{2,0}=h^{1,1}=h^{0,2}=1\) (and all others zero) a complete fan exists. They conjectured that complete fans always exist. The author of the present paper set out to give more examples and perhaps prove the conjecture, but instead found a counterexample of a rather general kind, with \(h^{2,0}=h^{1,1}=h^{0,2}=2\). As the counterexample is not at all extreme or contrived the correct inference appears to be that ``complete fan'' is not after all quite the right analogue of ``projective fan'', and the article of Kato and Usui [loc. cit.] has been annotated accordingly. The paper, which is notably well-written, first summarises what is needed from Hodge theory and log geometry, which is quite a lot. After that (but only after that) the proof becomes rather clear: in the end the author constructs a point that would have to be in the interior of two different rank~\(2\) cones of a complete fan.
scientific article; zbMATH DE number 5577344