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Talk:Virtual fundamental class

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Todo

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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

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  • 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

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BF

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Properties

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  • List out the many properties VFC's have

Examples

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  • 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

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List out references to constructions of VFC

13/2

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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)[reply]