The symplectic-cohomology description of the mirror

Let U=YDU=Y\setminus D and let XX be the mirror fiber of the constructed family over s=exp(2πi(B+iω))s=\exp(2\pi i(\mathbf B+i\omega)). Symplectic-cohomology mirror conjecture. There is an isomorphism of C\mathbb C-algebras

SH0(U)H0(X,OX),SH^0(U)\simeq H^0(X,\mathcal O_X),

so that the mirror can be constructed as

X=SpecSH0(U).X=\operatorname{Spec}SH^0(U).

The assertion is motivated by the open–closed string map and homological mirror symmetry; the source gives no resolution status.

Sources & referencesView supporting material

Primary source

Mark Gross, Paul Hacking and Sean Keel, “Mirror symmetry for log Calabi-Yau surfaces I”, arXiv:1106.4977 (2015).

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