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February 2, 2026manuscripta mathematica0 citationsOpen Access

Periods of families of curves in threefolds

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HMHossein Movasati

Key Points

  • The research investigates Clemens' conjecture regarding the finiteness of rational curves in quintic threefolds and the implications if it is false.
  • Generalization of Noether's theorem proof
  • Utilization of infinitesimal variation of Hodge structures
  • Reformulation in terms of integrals and Gauss-Manin connection
  • Certain periods of rational curves must vanish if Clemens' conjecture is false
  • The approach connects Hodge structures with periods in threefolds

Abstract

Abstract Clemens’ conjecture states that the number of rational curve in a generic quintic threefold is finite. If it is false we prove that certain periods of rational curves in such a quintic threefold must vanish. Our method is based on a generalization of a proof of Max Noether’s theorem using infinitesimal variation of Hodge structures and its reformulation in terms of integrals and Gauss-Manin connection.

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Cite This Study

Hossein Movasati (2026) studied this question.

synapsesocial.com/papers/69810013c1c9540dea8131a2https://doi.org/10.1007/s00229-026-01686-7
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