# Existence of minimal models for varieties of log general type

@article{Hacon2006ExistenceOM, title={Existence of minimal models for varieties of log general type}, author={Christopher D. Hacon and James McKernan}, journal={Journal of the American Mathematical Society}, year={2006}, volume={23}, pages={405-468} }

Assuming finite generation in dimension n − 1, we prove that pl-flips exist in dimension n.

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#### References

SHOWING 1-10 OF 117 REFERENCES

On the existence of flips

- Mathematics
- 2005

We prove the existence of flips in dimension n, contingent on the termination of real flips in dimension n-1.

Minimal models and boundedness of stable varieties

- Mathematics
- 1998

We consider a class of stable smoothable n-dimensional varieties, the analogs of stable curves. Assuming the minimal model program in dimension n+1, we prove that this class is bounded. From Kollar's… Expand

Birational Geometry of Algebraic Varieties

- Mathematics
- 1998

1. Rational curves and the canonical class 2. Introduction to minimal model program 3. Cone theorems 4. Surface singularities 5. Singularities of the minimal model program 6. Three dimensional flops… Expand

On the extension problem of pluricanonical forms

- Mathematics
- 1998

We review some recent development on the extension problem of pluricanonical forms from a divisor to the ambient space in [Si], [K5] and [N3] with simplified proofs.

Restrictions of log canonical algebras of general type

- Mathematics
- 2005

We introduce a diophantine property of a log canonical algebra, and use it to describe the restriction of a log canonical algebra of general type to a log canonical center of codimension one.

Pluricanonical systems of projective varieties of general type II

- Mathematics
- 2004

We prove that there exists a positive integer n depending only on n such that for every smooth projective n-fold of general type X defined over complex numbers, m K X gives a birational rational map… Expand

Towards a Mori Theory on Compact Kähler Threefolds, I

- Mathematics
- 1997

In this paper we construct from non-splitting families of rational curves special contractions on compact Kahler threefolds, i.e., morphisms which are analogous to those in the Mori theory in… Expand

A CANONICAL BUNDLE FORMULA

- Mathematics
- 2000

A higher dimensional analogue of Kodaira’s canonical bundle formula is obtained. As applications, we prove that the log-canonical ring of a klt pair with κ ≤ 3is finitely generated, and that there… Expand

Towards a Mori theory on compact Kähler threefolds, II

- Mathematics
- 1998

We prove the existence of a Mori contraction on a compact Kaehler threefold whose canonical bundle is (analytically) not nef if the threefold can be approximated by projective threefolds or if the… Expand