Scientists / Open questions

Caucher Birkar

Algebraic geometry; birational geometry

2018 Fields Medal

17 questions

Boundedness of the index of Calabi-Yau varieties and pairs: uniform pluricanonical non-vanishing when the canonical class is numerically trivial
The Iitaka conjecture $C_{n,m}$ : subadditivity of the Kodaira dimension for algebraic fibre spaces
The Iitaka conjecture $C_{n,m}$ for algebraic fibre spaces beyond dimension six
Termination of terminal log flips in dimensions at least five and the minimal model conjecture for klt pairs
Termination of log flips in dimension four for non-pseudo-effective log canonical pairs
Existence of weak Zariski decompositions for pseudo-effective R-Cartier divisors (Birkar's Question 1.4, from Nakayama's problem)
The weak form of Iitaka's $C_{3,m}$ conjecture for threefold fibrations over algebraically closed fields of positive characteristic
Finite generation of log canonical algebras and modules for log canonical pairs beyond the known minimal model program cases, and the induced existence of log minimal models with semi-ample canonical class
Completing Song–Tian's analytic minimal model program: the Kähler–Ricci flow through birational surgeries and metric completions, canonical measures without finite generation, and analytic abundance
Extension of Birkar's equivalence between good log minimal models and finite generation of log canonical algebras and modules to arbitrary (non-affine) base varieties
Existence of log minimal models or Mori fibre spaces for log canonical pairs in arbitrary dimension (the minimal model conjecture)
The Weak Nonvanishing Conjecture: is every pseudo-effective log canonical divisor numerically equivalent to an effective divisor?
The log minimal model program for threefold pairs in characteristics 2 and 3
ACC and lower semicontinuity for minimal log discrepancies, and termination of log flips: the surviving open core of Birkar-Shokurov's "Mld's vs thresholds and flips"
Termination of the log minimal model program for polarized pairs and existence of minimal models for generalized pairs with an arbitrary nef real part
Iitaka's $C_{n,m}$ conjecture in positive characteristic: the remaining regimes (higher dimensions, small characteristics, and wild geometric generic fibres)
Good log minimal models and finite generation of log canonical rings for log canonical pairs of log general type without base-locus hypotheses

Back to open problems

Boundedness of the index of Calabi-Yau varieties and pairs: uniform pluricanonical non-vanishing when the canonical class is numerically trivial

The Iitaka conjecture Cn,mC_{n,m}Cn,m​: subadditivity of the Kodaira dimension for algebraic fibre spaces

The Iitaka conjecture Cn,mC_{n,m}Cn,m​ for algebraic fibre spaces beyond dimension six

Termination of terminal log flips in dimensions at least five and the minimal model conjecture for klt pairs

Termination of log flips in dimension four for non-pseudo-effective log canonical pairs

Existence of weak Zariski decompositions for pseudo-effective R-Cartier divisors (Birkar's Question 1.4, from Nakayama's problem)

The weak form of Iitaka's C3,mC_{3,m}C3,m​ conjecture for threefold fibrations over algebraically closed fields of positive characteristic

Finite generation of log canonical algebras and modules for log canonical pairs beyond the known minimal model program cases, and the induced existence of log minimal models with semi-ample canonical class

Completing Song–Tian's analytic minimal model program: the Kähler–Ricci flow through birational surgeries and metric completions, canonical measures without finite generation, and analytic abundance

Extension of Birkar's equivalence between good log minimal models and finite generation of log canonical algebras and modules to arbitrary (non-affine) base varieties

Existence of log minimal models or Mori fibre spaces for log canonical pairs in arbitrary dimension (the minimal model conjecture)

The Weak Nonvanishing Conjecture: is every pseudo-effective log canonical divisor numerically equivalent to an effective divisor?

The log minimal model program for threefold pairs in characteristics 2 and 3

ACC and lower semicontinuity for minimal log discrepancies, and termination of log flips: the surviving open core of Birkar-Shokurov's "Mld's vs thresholds and flips"

Termination of the log minimal model program for polarized pairs and existence of minimal models for generalized pairs with an arbitrary nef real part

Iitaka's Cn,mC_{n,m}Cn,m​ conjecture in positive characteristic: the remaining regimes (higher dimensions, small characteristics, and wild geometric generic fibres)

Good log minimal models and finite generation of log canonical rings for log canonical pairs of log general type without base-locus hypotheses

The Iitaka conjecture $C_{n,m}$ : subadditivity of the Kodaira dimension for algebraic fibre spaces

The Iitaka conjecture $C_{n,m}$ for algebraic fibre spaces beyond dimension six

The weak form of Iitaka's $C_{3,m}$ conjecture for threefold fibrations over algebraically closed fields of positive characteristic

Iitaka's $C_{n,m}$ conjecture in positive characteristic: the remaining regimes (higher dimensions, small characteristics, and wild geometric generic fibres)