Conjecture on the unblown extended period map for I-surfaces
Conjecture on the unblown extended period map for I-surfaces
Let be the moduli space of I-surfaces, let be the natural blowup of the moduli space along the boundary, let be the period map, and let be the partial compactification in the extended period map conjecture. Let be the elliptic ruled stratum with multiplicities . Unblown extended period map conjecture. In addition to the extended period map conjecture, (iii) extends to a holomorphic map
so no additional blowups are necessary; (iv) is an orbifold near , and is there an orbifold normal-crossings divisor locally consisting of three orbifold smooth divisors; and (v) the normal derivative of to is injective at a general smooth point, hence its derivative is injective at a general point of . 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- 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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