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Let X be the total space of the canonical bundle of P^2. We study thegeneralized Donaldson-Thomas invariants, defined in the work of Joyce-Song, ofthe moduli spaces of the 2-dimensional Gieseker semistable sheaves on X withfirst Chern class equal to k times the class of the zero section of X. Whenk=1, 2 or 3, and semistability implies stability, we express the invariants interms of known modular forms. We prove a combinatorial formula for theinvariants when k=2 in the presence of the strictly semistable sheaves, andverify the BPS integrality conjecture of Joyce-Song in some cases.

Generalized Donaldson-Thomas Invariants of 2-Dimensional sheaves on local P^2