van Garens–Ranganathan log-local conjecture for normal crossing divisors

Let XX be a smooth projective variety and let D=D1++DnD=D_1+\cdots+D_n be an effective reduced normal crossing divisor whose components DiD_i are smooth, irreducible, and nef. For a curve class β\beta of XX, let

di:=Diβ>0,d_i:=D_i\cdot\beta>0,

and let FF be the morphism from the moduli space of genus-zero basic stable log maps to (X,D)(X,D) with one maximal-contact relative marking for each DiD_i to the moduli space of stable maps to the total space of i=1nOX(Di)\bigoplus_{i=1}^n\mathcal O_X(-D_i). van Garens–Ranganathan log-local conjecture.

[M0,0(i=1nOX(Di),β)]vir=(i=1n(1)di1di)F[M0,0,(d1),,(dn)(X/D,β)]vir.[\overline{\mathcal M}_{0,0}(\bigoplus_{i=1}^n\mathcal O_X(-D_i),\beta)]^{\operatorname{vir}}=\left(\prod_{i=1}^n\frac{(-1)^{d_i-1}}{d_i}\right)F_*[\overline{\mathcal M}_{0,0,(d_1),\ldots,(d_n)}(X/D,\beta)]^{\operatorname{vir}}.

This identifies genus-zero local invariants with maximal-tangency logarithmic invariants after the displayed normalization. It has been proved in some cases, but remains open in general.

Sources & referencesView supporting material

Primary source

Hsian-Hua Tseng and Fenglong You, “A mirror theorem for multi-root stacks and applications”, arXiv:2006.08991 (2022).

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