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10 - Minimal Models

Published online by Cambridge University Press:  29 September 2023

Masayuki Kawakita
Affiliation:
Kyoto University, Japan
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Summary

The final chapter treats minimal threefolds. We explain the abundance for threefolds due to Miyaoka and Kawamata depending on the numerical Kodaira dimension. The initial step is to prove the non-vanishing which means the existence of a global section of some pluricanonical divisor. If the irregularity is not zero, then the Albanese map provides enough geometric information. In the case of irregularity zero, Miyaoka applied the generic semi-positivity via positive characteristic. We derive abundance from non-vanishing after replacing the threefold by a special divisorially log terminal pair. Birational minimal models are connected by flops and have the same Betti and Hodge numbers. In dimension three, they have the same analytic singularities. One can expect the finiteness of minimal models ignoring the marking map. This is a part of Kawamata and Morrison's cone conjecture for Calabi-Yau fibrations. We explain Kawamata's work on the conjecture for threefold fibrations with non-trivial base. In dimension three, there exists a uniform number for l such that the l-th pluricanonical map is birational to the Iitaka fibration. We find this number explicitly in the case of general type.

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Publisher: Cambridge University Press
Print publication year: 2023

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  • Minimal Models
  • Masayuki Kawakita, Kyoto University, Japan
  • Book: Complex Algebraic Threefolds
  • Online publication: 29 September 2023
  • Chapter DOI: https://doi.org/10.1017/9781108933988.011
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  • Minimal Models
  • Masayuki Kawakita, Kyoto University, Japan
  • Book: Complex Algebraic Threefolds
  • Online publication: 29 September 2023
  • Chapter DOI: https://doi.org/10.1017/9781108933988.011
Available formats
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Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Minimal Models
  • Masayuki Kawakita, Kyoto University, Japan
  • Book: Complex Algebraic Threefolds
  • Online publication: 29 September 2023
  • Chapter DOI: https://doi.org/10.1017/9781108933988.011
Available formats
×