Talk:Virtual fundamental class

Todo

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

• 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 ${\overline {M}}_{0,0}(X,d)$ for a generic quintic threefold
• Mention how the VFC has the "right" dimension in the chow ring, it's virtual dimension

Constructions

BF

• 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

• List out the many properties VFC's have

Examples

• Show the BF VFC generalizes the fundamental class. Consider the commutative square

${\begin{matrix}X&\to &\mathbb {P} ^{n}\\\downarrow &&\downarrow \\X&\to &\mathbb {P} ^{n}\end{matrix}}$ Then the VFC is $i^{!}(X)=X\cdot [\mathbb {P} ^{n}]=[X]$, hence it can be considered as a generalization of the fundamental class. (Although P^n could be replaced by a smooth ambient)

• If $X$ is defined by a section of a vector bundle ${\mathcal {E}}$ then there is a square

${\begin{matrix}{\mathcal {E}}|_{X}&\to &{\mathcal {E}}\\\downarrow &&\downarrow \\X&\to &\mathbb {P} ^{n}\end{matrix}}$ giving another VFC construction. Show this applied to mapping stacks

Other Constructions

List out references to constructions of VFC

• https://arxiv.org/pdf/math/0509076.pdf
• https://web.math.princeton.edu/~jpardon/papers/09_implicitatlas.pdf
• https://arxiv.org/pdf/1701.07821.pdf
• https://users.math.msu.edu/users/parker/NaturalVFC.pdf
• https://projecteuclid.org/euclid.jdg/1442364653

13/2

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 $c(E/E')$. 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)