Mirror Symmetry via Logarithmic Degeneration Data I

Toric degenerations of toric varieties and tropical curves

“…In connection with the mirror symmetry program with M. Gross [GrSi1], [GrSi2], the second author has long suspected that this counting problem should have a purely combinatorial counterpart in integral affine geometry (cf. also [FuOh], [KoSo], and the work on Gromov-Witten invariants for degenerations [LiARu], [IoPa], [LiJ2], [Si]).…”

Singular Semi-Flat Calabi-Yau Metrics on <i>S</i>²

“…Then there is a unique map f :M → R 3 so that f z (p) = V and f satisfies the evolution equations (11)(12).…”

“…On the other hand, there are combinatorial constructions of integral affine manifolds with singularities due to Haase-Zharkov [15] and GrossSiebert [13,12], which discuss mirror symmetry from combinatorial and algebro-geometric points of view. Haase-Zharkov [16] also construct affine Kähler metrics on their examples, but these do not satisfy the MongeAmpère equation.…”

“…For any y 0, the linear path from (0, y) to (2π, y) corresponds to a loop |z| = e −y around the singularity p j . Also, {dev x , dev y } is a frame of the tangent space to hypersurface H. Therefore, integrating the initial value problem (11)(12) along such path computes the linear part of the holonomy. So the solution to the ODE X x =ÃX computes this part of the holonomy.…”

“…To calculate dev = dev y dy, compute from the structure equations (11)(12) ⎛ As above, letB be the bottom right 2 × 2 submatrix of B, and we have…”