Talk:Virtual fundamental class
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Todo edit
There should be a dedicated virtual fundamental class page describing its various constructions, such as from DAG, or as the virtual fundamental cycle in symplectic geometry.
Motivation edit
- Give motivational examples, such as how Kontsevich moduli spaces have boundary components with extra dimensions, or having dimension greater than 0, for the case of for a generic quintic threefold
- Mention how the VFC has the "right" dimension in the chow ring, it's virtual dimension
Constructions edit
BF edit
- Intrinsic Normal Cone https://arxiv.org/pdf/alg-geom/9601010.pdf
- Gromov Witten invariants in algebraic geometry https://arxiv.org/pdf/alg-geom/9601011.pdf
- Behrand-Fantechi give a construction using DM morphisms of Artin stacks
- Using DAG, mention how Konsevich conjecture (https://arxiv.org/abs/hep-th/9405035 page 9) of VFC using characteristic classes of derived schemes is shown to be true on differential graded manifolds, this is proven by Kapranov https://arxiv.org/abs/math/0703214
- Li, Tian - https://arxiv.org/abs/alg-geom/9608032
Properties edit
- List out the many properties VFC's have
Examples edit
- Show the BF VFC generalizes the fundamental class. Consider the commutative square
Then the VFC is , hence it can be considered as a generalization of the fundamental class. (Although P^n could be replaced by a smooth ambient)
- If is defined by a section of a vector bundle then there is a square
giving another VFC construction. Show this applied to mapping stacks
Other Constructions edit
List out references to constructions of VFC
13/2 edit
Note that 13/2 ways of counting curves gives the technical reasoning behind using excess intersections. Check out the appendix and look at the chern class . This is also in pages 9-10 of Thomas' original paper: https://arxiv.org/abs/math/9806111 — Preceding unsigned comment added by 71.196.136.221 (talk) 18:40, 25 August 2022 (UTC)