Genus-zero mirror-map conjecture for complete intersections in weighted projective space

Let P(a1,,aNk1,,km)P(a_1,\dots,a_N\mid k_1,\dots,k_m) be a nonsingular complete intersection in weighted projective space. Let tp(x)t^p(x^*) be the mirror maps defined from the genus-zero virtual structure constants, and let xp(t)x^p(t^*) denote their inverse. Write OhaOhb0(t)\langle{\mathcal O}_{h^a}{\mathcal O}_{h^b}\rangle_0(t^*) for the genus-zero two-point Gromov--Witten generating function and w(OhaOhb)0(x)w({\mathcal O}_{h^a}{\mathcal O}_{h^b})_0(x^*) for the corresponding virtual structure-constant generating function. Genus-zero mirror-map conjecture. For all admissible aa and bb,

OhaOhb0(t0,t1,t2,,tNm1)=w(OhaOhb)0(x0(t),x1(t),x2(t),,xNm1(t)).\langle{\mathcal O}_{h^a} {\mathcal O}_{h^b}\rangle_{0}\bigl(t^0,t^1,t^2,\dots,t^{N-m-1}\bigr)=w({\mathcal O}_{h^a} {\mathcal O}_{h^b})_{0}\bigl(x^0(t^*),x^1(t^*),x^2(t^*),\dots,x^{N-m-1}(t^*)\bigr).

This change-of-variables assertion is used to compute genus-zero Gromov--Witten invariants; the source states that it was already proved for projective hypersurfaces, while its status for the stated weighted complete intersections is not specified.

Sources & referencesView supporting material

Primary source

Masao Jinzenji and Ken Kuwata, “Elliptic Virtual Structure Constants and Gromov-Witten Invariants for Complete Intersections in Weighted Projective Space”, arXiv:2504.14273 (2026).

Additional references

2 papers in this index state this conjecture (2006–2025). The statement above is taken from the most recent of them; the others are arXiv:math/0609016.

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