2021-04-22

Gromov-Witten Invariants in Derived AG

  • My main question: does introducing derived stacks somehow make some computation easier? #todo/questions

    • I haven’t found any explicit computations of these, but sources alluded to actual counts (numbers) conjecturally coming from physics, where a few have been verified.
  • Integrating over a fundamental class :

attachments/image_2021-04-22-11-57-59.png

attachments/image_2021-04-22-11-59-01.png

  • moduli space of stable maps

attachments/image_2021-04-22-12-00-48.png

attachments/image_2021-04-22-12-02-20.png

  • Operads :

attachments/image_2021-04-22-12-05-38.png

attachments/image_2021-04-22-12-14-47.png

  • Donaldson-Thomas invariants are supposed to relate to Gromov-Witten invariants :

    attachments/image_2021-04-22-12-17-02.png

  • Niceness of spaces:

    attachments/image_2021-04-22-12-17-44.png

Derived Stacks

  • We can’t prove the Tate conjecture? I guess this is an arithmetic analog of the Hodge conjecture. Serre’s book calls some isomorphism the Tate conjecture and says it’s proved though.

  • Pithy explanation of a derived scheme : a space which can be covered by Zariski opens \(Y\cong \operatorname{Spec}A^*\) where \(A\in { \mathsf{c}{\mathsf{dg{\mathsf{Alg}}} }}_{k}\).

    • schemes and stacks can be very singular.

    • Derived schemes and derived stacks act a bit like smooth, nonsingular objects.

      • Morphisms behave like they are transverse?
  • Derived modular stacks of quasicoherent sheaves over \(X\) remember the entire deformation theory of sheaves on \(X\).

    • The homology of its “tangent space” at a point \([E]\) is \(\operatorname{Ext} ^*(E, E)\), which only holds in restricted degrees if you only use a non-derived moduli scheme or stack.
#web/quick-notes #todo/questions