I'm reading about virtual fundamental classes, following Behrend and Fantechi's approach in their paper, The Intrinsic Normal Cone.
I understand that these classes enjoy the following desirable properties
- They refine the top chern class (related to the chosen perfect obstruction theory);
- They agree with the usual fundamental class in the unobstructed case;
- They are used to define Gromov-Witten classes, in the sense that these definitions satisfy the GW-axioms.
My question is: Other than the above properties, is there another reason that this is the "correct" substitute for the fundamental class?
For example, does it satisfy some kind of Poincare Duality property? Any further insight would be much appreciated.