Gromov-Witten Invariants in Derived AG
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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.
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Integrating over a fundamental class :
- moduli space of stable maps
- Operads :
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Appearance of Calabi-Yau in Physics
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Mirror symmetry of CYs:
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The major types of “moduli” style invariants
- See quantum invariants
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Why care about coherent sheaves? #todo/questions
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Donaldson-Thomas invariants are supposed to relate to Gromov-Witten invariants :
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Niceness of spaces:
Derived Stacks
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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.
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Pithy explanation of a derived scheme : a space which can be covered by Zariski opens \(Y\cong \operatorname{Spec}A^*\) where \(A\in { \mathsf{c}{\mathsf{dg \mathsf{Alg} } }}_{k}\).
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schemes and stacks can be very singular.
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Derived schemes and derived stacks act a bit like smooth, nonsingular objects.
- Morphisms behave like they are transverse?
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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.