Birational Calabi–Yau threefolds and BPS state counting

This paper contains some applications of Bridgeland–Douglas stability conditions on triangulated categories, and Joyce’s work on counting invariants of semistable objects, to the study of birational geometry. We introduce the notion of motivic Gopakumar–Vafa invariants as counting invariants of D2-branes, and show that they are invariant under birational transformations between Calabi–Yau threefolds. The result is similar to the fact that birational Calabi–Yau threefolds have the same betti numbers or Hodge numbers.

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Published 1 January 2008