Conjecture on the unblown extended period map for I-surfaces

Let M\mathcal{M} be the moduli space of I-surfaces, let M~\widetilde{\mathcal{M}} be the natural blowup of the moduli space along the boundary, let Φ ⁣:MΓ\D\Phi\colon\mathcal{M}\to\Gamma\backslash D be the period map, and let Z\overline{Z} be the partial compactification in the extended period map conjecture. Let S1,1,1\mathcal{S}_{1,1,1} be the elliptic ruled stratum with multiplicities (1,1,1)(1,1,1). Unblown extended period map conjecture. In addition to the extended period map conjecture, (iii) Φ\Phi extends to a holomorphic map

Φ~ ⁣:M~Z,\widetilde{\Phi}\colon\widetilde{\mathcal{M}}\to\overline{Z},

so no additional blowups are necessary; (iv) Z\overline{Z} is an orbifold near Φ~(S1,1,1)\widetilde{\Phi}(\mathcal{S}_{1,1,1}), and ZZ\overline{Z}-Z is there an orbifold normal-crossings divisor locally consisting of three orbifold smooth divisors; and (v) the normal derivative of Φ~\widetilde{\Phi} to S1,1,1\mathcal{S}_{1,1,1} is injective at a general smooth point, hence its derivative is injective at a general point of S1,1,1\mathcal{S}_{1,1,1}. This strengthens the preceding conjecture by asserting that the natural blowup already suffices and by describing the local boundary and generic infinitesimal behavior near the distinguished codimension-33 stratum; the source does not state whether it has been proved or disproved.

Sources & referencesView supporting material

Primary source

Robert Friedman and Phillip Griffiths, “Limiting mixed Hodge structures associated to I-surfaces with simple elliptic singularities”, arXiv:2408.02062 (2024).

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