## Existence of minimal models for varieties of log general type

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Caucher Birkar, Paolo Cascini, Christopher D. Hacon and James McKernan
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## Abstract:

We prove that the canonical ring of a smooth projective variety is finitely generated.## References

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## Additional Information

**Caucher Birkar**- Affiliation: DPMMS, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, United Kingdom
- Email: c.birkar@dpmms.cam.ac.uk
**Paolo Cascini**- Affiliation: Department of Mathematics, University of California at Santa Barbara, Santa Barbara, California 93106 and Imperial College London, 180 Queens Gate, London SW7 2A2, United Kingdom
- MR Author ID: 674262
- Email: cascini@math.ucsb.edu, p.cascini@imperial.ac.uk
**Christopher D. Hacon**- Affiliation: Department of Mathematics, University of Utah, 155 South 1400 East, JWB 233, Salt Lake City, Utah 84112
- MR Author ID: 613883
- Email: hacon@math.utah.edu
**James McKernan**- Affiliation: Department of Mathematics, University of California at Santa Barbara, Santa Barbara, California 93106 and Department of Mathematics, MIT, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
- Email: mckernan@math.ucsb.edu, mckernan@math.mit.edu
- Received by editor(s): August 13, 2008
- Published electronically: November 13, 2009
- Additional Notes: The first author was partially supported by EPSRC grant GR/S92854/02

The second author was partially supported by NSF research grant no: 0801258

The third author was partially supported by NSF research grant no: 0456363 and an AMS Centennial fellowship

The fourth author was partially supported by NSA grant no: H98230-06-1-0059 and NSF grant no: 0701101 and would like to thank Sogang University and Professor Yongnam Lee for their generous hospitality, where some of the work for this paper was completed

All authors would like to thank Dan Abramovich, Valery Alexeev, Florin Ambro, Tommaso de Fernex, Stephane Dreul, Seán Keel, Kalle Karu, János Kollár, Sándor Kovács, Michael McQuillan, Shigefumi Mori, Martin Olsson, Genia Tevelev, Burt Totaro, Angelo Vistoli and Chengyang Xu for answering many of our questions and pointing out some errors in an earlier version of this paper. They would also like to thank the referee for some useful comments. - © Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**23**(2010), 405-468 - MSC (2010): Primary 14E30
- DOI: https://doi.org/10.1090/S0894-0347-09-00649-3
- MathSciNet review: 2601039