Application of a Bogomolov-Gieseker type inequality to counting invariants

I will work on a  Calabi-Yau 3-fold X which satisfies the Bogomolov-Gieseker conjecture of Bayer-Macr\`i-Toda for weak stability conditions, such as the quintic threefold. I will explain how wall-crossing with respect to weak stability conditions gives an expression of Joyce’s generalised Donaldson-Thomas invariants counting Gieseker semistable sheaves of any rank greater than or equal to one on X in terms of those counting sheaves of rank 0 and pure dimension 2.  This is joint work with Richard Thomas.

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