set of isomorphism classes of geometric objects
made into a space of some kind(topological space, complex manifold)
Example. The moduli space of complex tori $ℂ/Λ$ (≃moduli space of non-singular cubics up to projective transform.)
$Λ=⟨w_1,w_2⟩_ℤ=$pairs $(w_1,w_2)/GL(2,ℤ)$ for $w_1/w_2∈ℂ∖ℝ$
moduli space of tori $ℂ/Λ$ $≅ℋ=\{z∈ℂ:\Im(z)>0\}$
$SL(2,ℤ)$ acts by Mobius transform
$SL(2,ℤ)=⟨S,T⟩,S=\pmatrix{1&1\\0&1},T=\pmatrix{0&1\\-1&0}$
fundamental domain for
2 special tori: $ℂ/ℤ+iℤ$: $ℤ^4$ symmetry
$ℂ/ℤ+e^{πi/3}ℤ$: $ℤ^6$ symmetry
$C:x^3+y^3+z^3=0$
What $Λ$ is this $ℂ/Λ$ for? $ℂ/ℤ+e^{πi/3}ℤ$: $ℤ^6$ symmetry
$(x,y,z)↦(e^{2πi/3}x,y,z)$ order 3 rotation fixes $(0,1,-1)$
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