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Polarized Hodge Structures for Clemens Manifolds

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Abstract

Let X be a Calabi-Yau threefold. A conifold transition first contracts X along disjoint rational curves with normal bundles of type \((-1,-1)\), and then smooth the resulting singular complex space \(\bar{X}\) to a new compact complex manifold Y. Such Y is called a Clemens manifold and can be non-Kähler. We prove that any small smoothing Y of \(\bar{X}\) satisfies \(\partial \bar{\partial }\)-lemma. We also show that the resulting pure Hodge structure of weight three on \(H^3(Y)\) is polarized by the cup product. These results answer some questions of R. Friedman.

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Notes

  1. There is a proof in https://mathoverflow.net/questions/85407/intersection-of-subvector-bundles.

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Acknowledgements

The author is partially supported by NSF (Grant No. DMS-1810867) and an Alfred P. Sloan research fellowship. The author thanks R. Friedman and G. Tian for their interest. He also thanks G. Tian and S. Rao for helpful comments.

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  1. Department of Mathematics, Rutgers University, Piscataway, NJ, 08854-8019, USA

    Chi Li

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Li, C. Polarized Hodge Structures for Clemens Manifolds. Math. Ann. 389, 525–541 (2024). https://doi.org/10.1007/s00208-023-02650-6

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