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I'm going to redirect Hodge variety here. AFAIK Hodge variety isn't a current term; it may have been the 'working title' of the manifolds K proved projective (cf the Bourbaki seminar backlink). Anyone who knows better, please say. Charles Matthews 10:33, 24 October 2005 (UTC)
Hodge metric is defined by Griffiths & Harris (rational (1,1) form). I'm temporarily confused about whether to speak in terms of rational classes, or proportional integral classes. The Chern class certainly is integral. Charles Matthews 10:39, 24 October 2005 (UTC)