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

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
$$ \begin{matrix} X & \to & \mathbb{P}^n \\ \downarrow & & \downarrow \\ X & \to & \mathbb{P}^n \end{matrix} $$ 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) $$ \begin{matrix} \mathcal{E}|_X & \to & \mathcal{E} \\ \downarrow & & \downarrow \\ X & \to & \mathbb{P}^n \end{matrix} $$ giving another VFC construction. Show this applied to mapping stacks
 * Show the BF VFC generalizes the fundamental class. Consider the commutative square
 * If $$X$$ is defined by a section of a vector bundle $$\mathcal{E}$$ then there is a square

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)