On the non-vanishing conjecture and existence of log minimal models

  title={On the non-vanishing conjecture and existence of log minimal models},
  author={Kenta Hashizume},
  journal={arXiv: Algebraic Geometry},
  • K. Hashizume
  • Published 1 September 2016
  • Mathematics
  • arXiv: Algebraic Geometry
We prove that the non-vanishing conjecture and the log minimal model conjecture for projective log canonical pairs can be reduced to the non-vanishing conjecture for smooth projective varieties such that the boundary divisor is zero. 
On semipositivity, injectivity and vanishing theorems
This is a survey article on the recent developments of semipositivity, injectivity, and vanishing theorems for higher-dimensional complex projective varieties.
On Nonvanishing for uniruled log canonical pairs
We prove the Nonvanishing conjecture for uniruled log canonical pairs of dimension $n$, assuming the Nonvanishing conjecture for smooth projective varieties in dimension $n-1$. We also show that theExpand
The isomorphism problem of projective schemes and related algorithmic problems
We discuss the isomorphism problem of projective schemes; given two projective schemes, can we algorithmically decide whether they are isomorphic? We give affirmative answers in the case ofExpand
On minimal models
We show that minimal models of log canonical pairs exist, assuming the existence of minimal models of smooth varieties.


Remarks on the non-vanishing conjecture
We discuss a difference between the rational and the real non-vanishing conjecture for pseudo-effective log canonical divisors of log canonical pairs. We also show the log non-vanishing theorem forExpand
On existence of log minimal models II
Abstract We prove that the existence of log minimal models in dimension d essentially implies the LMMP with scaling in dimension d. As a consequence we prove that a weak nonvanishing conjecture inExpand
Extension theorems, non-vanishing and the existence of good minimal models
We prove an extension theorem for effective purely log-terminal pairs (X, S + B) of non-negative Kodaira dimension $${\kappa (K_X+S+B)\ge 0}$$ . The main new ingredient is a refinement of theExpand
On canonical bundle formulaS and subadjunctions
We consider a canonical bundle formula for generically finite proper surjective morphisms and obtain subadjunction formulae for minimal log canonical centers of log canonical pairs. We also treatExpand
Existence of minimal models for varieties of log general type
Assuming finite generation in dimension n − 1, we prove that pl-flips exist in dimension n.
On existence of log minimal models
  • C. Birkar
  • Mathematics
  • Compositio Mathematica
  • 2010
Abstract In this paper, we prove that the log minimal model program in dimension d−1 implies the existence of log minimal models for effective lc pairs (e.g. of non-negative Kodaira dimension) inExpand
Minimal model theory for relatively trivial log canonical pairs
We study relative log canonical pairs with relatively trivial log canonical divisors. We fix such a pair $(X,\Delta)/Z$ and establish the minimal model theory for the pair $(X,\Delta)$ assuming theExpand
Existence of log canonical flips and a special LMMP
Let (X/Z,B+A) be a Q-factorial dlt pair where B,A≥0 are Q-divisors and KX+B+A∼Q0/Z. We prove that any LMMP/Z on KX+B with scaling of an ample/Z divisor terminates with a good log minimal model or aExpand
On existence of log minimal models and weak Zariski decompositions
We first introduce a weak type of Zariski decomposition in higher dimensions: an $${\mathbb {R}}$$ -Cartier divisor has a weak Zariski decomposition if birationally and in a numerical sense it can beExpand
ACC for log canonical thresholds
We show that log canonical thresholds satisfy the ACC.