Seidel's quintic divisor-complement conjecture

Let M\overline{M} be the quintic three-fold, let DMD\subset\overline{M} be the smooth hyperplane section used in the paper, and set M=M\DM=\overline{M}\backslash D. Write H0(M)\operatorname{H}^0(\overline{M}) and SH0(M)\operatorname{SH}^0(M) for the degree-zero groups in the paper's adapted grading. Let [D][D] be the canonical generator of SH0(M)\operatorname{SH}^0(M) representing the Morse–Bott family of period-one closed Reeb orbits given by the fibres of NDDN_D\to D. Seidel's quintic conjecture. The small quantum product

t:H0(M)H0(M)H0(M)\star_t:\operatorname{H}^0(\overline{M})\otimes\operatorname{H}^0(\overline{M})\to\operatorname{H}^0(\overline{M})

encoding counts of rational curves and their multiple covers agrees with the family of deformed pair-of-pants products

q:SH0(M)SH0(M)SH0(M)\star_q:\operatorname{SH}^0(M)\otimes\operatorname{SH}^0(M)\to\operatorname{SH}^0(M)

when q=t[D]q=t[D]. The conjecture proposes a precise relationship between quantum cohomology of the quintic and symplectic cohomology of its divisor complement, potentially providing an alternative route to mirror symmetry for the quintic. It is stated without a resolution in the paper.

Sources & referencesView supporting material

Primary source

Oliver Fabert, “Higher algebraic structures in Hamiltonian Floer theory”, arXiv:1412.2682 (2019).

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