Questions: Johannes Nordstrom

Nolan Fitzpatrick: What is a "semi-fano 3-fold?" 

Johannes Nordstrom: "Semi-Fano" is not very standard terminology. It's a slightly stronger condition than "weak Fano", which usually means a projective variety whose anticanonical bundle is nef and big (although some authors use it in slightly different ways). In some papers with Alessio Corti, Mark Haskins and Tommaso Pacini, we found it useful to focus on the special case of weak Fanos where the morphism defined by the anticanonical bundle is in addition "semi-small". In the case of 3-folds, that means it can contract curves to points and divisors to curves, but must not contract divisors to points (in other words, the fibres have dimension at most 

Nolan Fitzpatrick: What is the significance of Kovalev's 2003 work?

Johannes Nordstrom: Kovalev's construction provided the second class of examples of closed G2-manifolds, after Joyce's flat orbifold resolutions. Like Joyce's examples they come in families that exhibit a kind of degeneration, but of a different kind. Joyce's degenerate back to the original orbifold, while Kovalev's develop a long neck (so if you rescale so that the diameter is constant, then the volume goes to zero). The twisted connected sums also turn out to have more computable topology than examples from the orbifold construction.