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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2509.01002 |
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| _version_ | 1866912563259768832 |
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| author | Collins, Tristan C. |
| author_facet | Collins, Tristan C. |
| contents | These lecture notes introduce conifold transitions between complex threefolds with trivial canonical bundle from the differential geometric point of view, and with a particular view towards aspects of mathematical physics and string theory. The lecture notes are aimed at beginning graduate students and non-experts, emphasizing explicit calculations and examples. After a brief introduction in Section 1, we recall some basic facts about Calabi-Yau manifolds in Section 2. Section 3 studies the conifold as a Calabi-Yau manifold with singularities, and introduces the local model for a conifold transition. Section 4 discusses global conifold transitions, and recalls the famous result of Friedman concerning the existence of smoothings for nodal Calabi-Yau threefolds. We give a differential geometric proof of the necessity part of Friedman's theorem. Section 5 discusses Reid's fantasy, and the web of Calabi-Yau threefolds. Section 6 discusses metric aspects of the local conifold transition, constructing explicit asymptotically conical Calabi-Yau metrics on the small resolution and the smoothing. Section 7 discusses the metric aspects of global conifold transitions, with a particular emphasis on the heterotic string. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_01002 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An introduction to conifold transitions Collins, Tristan C. Differential Geometry Algebraic Geometry These lecture notes introduce conifold transitions between complex threefolds with trivial canonical bundle from the differential geometric point of view, and with a particular view towards aspects of mathematical physics and string theory. The lecture notes are aimed at beginning graduate students and non-experts, emphasizing explicit calculations and examples. After a brief introduction in Section 1, we recall some basic facts about Calabi-Yau manifolds in Section 2. Section 3 studies the conifold as a Calabi-Yau manifold with singularities, and introduces the local model for a conifold transition. Section 4 discusses global conifold transitions, and recalls the famous result of Friedman concerning the existence of smoothings for nodal Calabi-Yau threefolds. We give a differential geometric proof of the necessity part of Friedman's theorem. Section 5 discusses Reid's fantasy, and the web of Calabi-Yau threefolds. Section 6 discusses metric aspects of the local conifold transition, constructing explicit asymptotically conical Calabi-Yau metrics on the small resolution and the smoothing. Section 7 discusses the metric aspects of global conifold transitions, with a particular emphasis on the heterotic string. |
| title | An introduction to conifold transitions |
| topic | Differential Geometry Algebraic Geometry |
| url | https://arxiv.org/abs/2509.01002 |