A Twisted bidifferential, second fundamental form of Prym map and projective structures

Abstract

Given a point [(C̃,C)] of the moduli space ℛ_g, where C̃ is an étale double cover of a smooth curve C of genus g ≥ 7, we construct a twisted meromorphic bidifferential η̂_P on C× C with a pole on the diagonal. It is shown that the second fundamental form ρ_P of the Prym map from ℛ_g to 𝒜_g−1 at [(C̃,C)] is given by this η̂_P. This allows us to compare ρ_P with the second fundamental forms of the two Torelli maps, to 𝒜_2g−1 and 𝒜_g, at the points [C̃] and [C] respectively. It is shown that η̂_P induces a projective structure on C that depends on the double cover C̃ ⟶ C; these projective structures together give a 𝒞^∞ section β_P of the bundle of projective structures on the moduli space ℛ_g. Finally, it is proved that the (0, 1)-component of the differential of the section β_P is the pullback of the Kähler form of the Siegel metric on 𝒜_g−1 via the Prym map. This is a key ingredient in the proof that this section β_P is not the pullback of a section of projective structures on ℳ_g and, in addition, that it differs from the known projective structures on ℛ_g.

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BibTeX RIS

BibTeX

@misc{biswas2026twisted,
  title = {{A Twisted bidifferential, second fundamental form of Prym map and projective structures}},
  author = {Indranil Biswas and Elisabetta Colombo and Paola Frediani and Alessandro Ghigi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.28156},
  eprint = {2610.28156},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.28156},
}

RIS

TY  - UNPB
TI  - A Twisted bidifferential, second fundamental form of Prym map and projective structures
AU  - Indranil Biswas
AU  - Elisabetta Colombo
AU  - Paola Frediani
AU  - Alessandro Ghigi
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.28156
UR  - https://arxiv.org/abs/2610.28156
AB  - Given a point [(C̃,C)] of the moduli space ℛ_g, where C̃ is an étale double cover of a smooth curve C of genus g ≥ 7, we construct a twisted meromorphic bidifferential η̂_P on C× C with a pole on the diagonal. It is shown that the second fundamental form ρ_P of the Prym map from ℛ_g to 𝒜_g−1 at [(C̃,C)] is given by this η̂_P. This allows us to compare ρ_P with the second fundamental forms of the two Torelli maps, to 𝒜_2g−1 and 𝒜_g, at the points [C̃] and [C] respectively. It is shown that η̂_P induces a projective structure on C that depends on the double cover C̃ ⟶ C; these projective structures together give a 𝒞^∞ section β_P of the bundle of projective structures on the moduli space ℛ_g. Finally, it is proved that the (0, 1)-component of the differential of the section β_P is the pullback of the Kähler form of the Siegel metric on 𝒜_g−1 via the Prym map. This is a key ingredient in the proof that this section β_P is not the pullback of a section of projective structures on ℳ_g and, in addition, that it differs from the known projective structures on ℛ_g.
ER  -

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