Brauer-Manin obstruction Brauer-Manin obstruction : rational points versus zero-cycles Yongqi LIANG Université Paris-Sud 11, Orsay, France RAGE 2011/05/19 Atlanta, U.S. Yongqi LIANG Université Paris-Sud 11, Orsay, France
Brauer-Manin obstruction Rational points vs. Zero-cycles Notations k : number field k v , for v ∈ Ω k . Ω f k , Ω ∞ k , Ω R k , Ω C k X : projective variety (separated scheme of finite type, geometrically integral) over k Br ( X ) := H 2 ét ( X , G m ) the cohomological Brauer group X v = X ⊗ k k v Yongqi LIANG Université Paris-Sud 11, Orsay, France
Brauer-Manin obstruction Rational points vs. Zero-cycles Rational points X ( k ) ⊂ � v ∈ Ω X ( k v ) Brauer-Manin pairing �� � v ∈ Ω X ( k v ) × Br ( X ) → Q / Z � ( { x v } v ∈ Ω , β ) �→ �{ x v } v , β � := inv v ( β ( x v )) v ∈ Ω �� � Br = left kernel of the pairing v ∈ Ω X ( k v ) �� � Br (by class field theory) Fact. X ( k ) ⊆ v ∈ Ω X ( k v ) X ( k ) : closure of X ( k ) in � v X ( k v ) (product topology) If = , Brauer-Manin obstruction is the only obstruction to weak approximation Yongqi LIANG Université Paris-Sud 11, Orsay, France
Brauer-Manin obstruction Rational points vs. Zero-cycles Rational points X ( k ) ⊂ � v ∈ Ω X ( k v ) Brauer-Manin pairing �� � v ∈ Ω X ( k v ) × Br ( X ) → Q / Z � ( { x v } v ∈ Ω , β ) �→ �{ x v } v , β � := inv v ( β ( x v )) v ∈ Ω �� � Br = left kernel of the pairing v ∈ Ω X ( k v ) �� � Br (by class field theory) Fact. X ( k ) ⊆ v ∈ Ω X ( k v ) X ( k ) : closure of X ( k ) in � v X ( k v ) (product topology) If = , Brauer-Manin obstruction is the only obstruction to weak approximation Yongqi LIANG Université Paris-Sud 11, Orsay, France
Brauer-Manin obstruction Rational points vs. Zero-cycles Rational points X ( k ) ⊂ � v ∈ Ω X ( k v ) Brauer-Manin pairing �� � v ∈ Ω X ( k v ) × Br ( X ) → Q / Z � ( { x v } v ∈ Ω , β ) �→ �{ x v } v , β � := inv v ( β ( x v )) v ∈ Ω �� � Br = left kernel of the pairing v ∈ Ω X ( k v ) �� � Br (by class field theory) Fact. X ( k ) ⊆ v ∈ Ω X ( k v ) X ( k ) : closure of X ( k ) in � v X ( k v ) (product topology) If = , Brauer-Manin obstruction is the only obstruction to weak approximation Yongqi LIANG Université Paris-Sud 11, Orsay, France
Brauer-Manin obstruction Rational points vs. Zero-cycles Rational points X ( k ) ⊂ � v ∈ Ω X ( k v ) Brauer-Manin pairing �� � v ∈ Ω X ( k v ) × Br ( X ) → Q / Z � ( { x v } v ∈ Ω , β ) �→ �{ x v } v , β � := inv v ( β ( x v )) v ∈ Ω �� � Br = left kernel of the pairing v ∈ Ω X ( k v ) �� � Br (by class field theory) Fact. X ( k ) ⊆ v ∈ Ω X ( k v ) X ( k ) : closure of X ( k ) in � v X ( k v ) (product topology) If = , Brauer-Manin obstruction is the only obstruction to weak approximation Yongqi LIANG Université Paris-Sud 11, Orsay, France
Brauer-Manin obstruction Rational points vs. Zero-cycles Rational points X ( k ) ⊂ � v ∈ Ω X ( k v ) Brauer-Manin pairing �� � v ∈ Ω X ( k v ) × Br ( X ) → Q / Z � ( { x v } v ∈ Ω , β ) �→ �{ x v } v , β � := inv v ( β ( x v )) v ∈ Ω �� � Br = left kernel of the pairing v ∈ Ω X ( k v ) �� � Br (by class field theory) Fact. X ( k ) ⊆ v ∈ Ω X ( k v ) X ( k ) : closure of X ( k ) in � v X ( k v ) (product topology) If = , Brauer-Manin obstruction is the only obstruction to weak approximation Yongqi LIANG Université Paris-Sud 11, Orsay, France
Brauer-Manin obstruction Rational points vs. Zero-cycles Zero-cycles (Colliot-Thélène) Similarly, Brauer-Manin pairing �� � v ∈ Ω Z 0 ( X v ) × Br ( X ) → Q / Z �� � v ∈ Ω CH 0 ( X v ) × Br ( X ) → Q / Z �� � v ∈ Ω CH ′ 0 ( X v ) × Br ( X ) → Q / Z The modified Chow group: v ∈ Ω f CH 0 ( X v ) , CH ′ v ∈ Ω R 0 ( X v ) = CH 0 ( X v ) / N C | R CH 0 ( X v ) , v ∈ Ω C 0 , complex CH 0 ( X ) → � v ∈ Ω CH ′ 0 ( X v ) → Hom ( Br ( X ) , Q / Z ) Yongqi LIANG Université Paris-Sud 11, Orsay, France
Brauer-Manin obstruction Rational points vs. Zero-cycles Zero-cycles (Colliot-Thélène) Similarly, Brauer-Manin pairing �� � v ∈ Ω Z 0 ( X v ) × Br ( X ) → Q / Z �� � v ∈ Ω CH 0 ( X v ) × Br ( X ) → Q / Z �� � v ∈ Ω CH ′ 0 ( X v ) × Br ( X ) → Q / Z The modified Chow group: v ∈ Ω f CH 0 ( X v ) , CH ′ v ∈ Ω R 0 ( X v ) = CH 0 ( X v ) / N C | R CH 0 ( X v ) , v ∈ Ω C 0 , complex CH 0 ( X ) → � v ∈ Ω CH ′ 0 ( X v ) → Hom ( Br ( X ) , Q / Z ) Yongqi LIANG Université Paris-Sud 11, Orsay, France
Brauer-Manin obstruction Rational points vs. Zero-cycles Zero-cycles (Colliot-Thélène) Similarly, Brauer-Manin pairing �� � v ∈ Ω Z 0 ( X v ) × Br ( X ) → Q / Z �� � v ∈ Ω CH 0 ( X v ) × Br ( X ) → Q / Z �� � v ∈ Ω CH ′ 0 ( X v ) × Br ( X ) → Q / Z The modified Chow group: v ∈ Ω f CH 0 ( X v ) , CH ′ v ∈ Ω R 0 ( X v ) = CH 0 ( X v ) / N C | R CH 0 ( X v ) , v ∈ Ω C 0 , complex CH 0 ( X ) → � v ∈ Ω CH ′ 0 ( X v ) → Hom ( Br ( X ) , Q / Z ) Yongqi LIANG Université Paris-Sud 11, Orsay, France
Brauer-Manin obstruction Rational points vs. Zero-cycles Zero-cycles − n M / nM = M ⊗ � M � := lim Z for any abelian group M ← deg A 0 ( X ) := ker ( CH 0 ( X ) − → Z ) complex ( E ) �� �� v ∈ Ω CH ′ [ CH 0 ( X )] � → 0 ( X v ) − → Hom ( Br ( X ) , Q / Z ) similarly, complex ( E 0 ) �� �� [ A 0 ( X )] � → v ∈ Ω A 0 ( X v ) − → Hom ( Br ( X ) , Q / Z ) Question : Are they exact? Remark (Wittenberg) Exactness of ( E ) = ⇒ - Exactness of ( E 0 ) - ( E 1 ) : Existence of z ∈ CH 0 ( X ) of degree 1 supposing the existence of a family of degree 1 zero-cycles { z v }⊥ Br ( X ) . Yongqi LIANG Université Paris-Sud 11, Orsay, France
Brauer-Manin obstruction Rational points vs. Zero-cycles Zero-cycles − n M / nM = M ⊗ � M � := lim Z for any abelian group M ← deg A 0 ( X ) := ker ( CH 0 ( X ) − → Z ) complex ( E ) �� �� v ∈ Ω CH ′ [ CH 0 ( X )] � → 0 ( X v ) − → Hom ( Br ( X ) , Q / Z ) similarly, complex ( E 0 ) �� �� [ A 0 ( X )] � → v ∈ Ω A 0 ( X v ) − → Hom ( Br ( X ) , Q / Z ) Question : Are they exact? Remark (Wittenberg) Exactness of ( E ) = ⇒ - Exactness of ( E 0 ) - ( E 1 ) : Existence of z ∈ CH 0 ( X ) of degree 1 supposing the existence of a family of degree 1 zero-cycles { z v }⊥ Br ( X ) . Yongqi LIANG Université Paris-Sud 11, Orsay, France
Brauer-Manin obstruction Rational points vs. Zero-cycles Zero-cycles − n M / nM = M ⊗ � M � := lim Z for any abelian group M ← deg A 0 ( X ) := ker ( CH 0 ( X ) − → Z ) complex ( E ) �� �� v ∈ Ω CH ′ [ CH 0 ( X )] � → 0 ( X v ) − → Hom ( Br ( X ) , Q / Z ) similarly, complex ( E 0 ) �� �� [ A 0 ( X )] � → v ∈ Ω A 0 ( X v ) − → Hom ( Br ( X ) , Q / Z ) Question : Are they exact? Remark (Wittenberg) Exactness of ( E ) = ⇒ - Exactness of ( E 0 ) - ( E 1 ) : Existence of z ∈ CH 0 ( X ) of degree 1 supposing the existence of a family of degree 1 zero-cycles { z v }⊥ Br ( X ) . Yongqi LIANG Université Paris-Sud 11, Orsay, France
Brauer-Manin obstruction Rational points vs. Zero-cycles Examples and a conjecture (Cassels-Tate) ( E 0 ) is exact if X = A is an abelian variety (with finiteness of X ( A ) supposed). (Colliot-Thélène) ( E ) is exact if X = C is a smooth curve (with finiteness of X ( Jac ( C )) supposed). Conjecture (Colliot-Thélène/Sansuc, Kato/Saito, Colliot-Thélène) The complex ( E 0 ) is exact for all smooth projective varieties. Yongqi LIANG Université Paris-Sud 11, Orsay, France
Brauer-Manin obstruction Rational points vs. Zero-cycles Examples and a conjecture (Cassels-Tate) ( E 0 ) is exact if X = A is an abelian variety (with finiteness of X ( A ) supposed). (Colliot-Thélène) ( E ) is exact if X = C is a smooth curve (with finiteness of X ( Jac ( C )) supposed). Conjecture (Colliot-Thélène/Sansuc, Kato/Saito, Colliot-Thélène) The complex ( E 0 ) is exact for all smooth projective varieties. Yongqi LIANG Université Paris-Sud 11, Orsay, France
Brauer-Manin obstruction Rational points vs. Zero-cycles Examples and a conjecture (Cassels-Tate) ( E 0 ) is exact if X = A is an abelian variety (with finiteness of X ( A ) supposed). (Colliot-Thélène) ( E ) is exact if X = C is a smooth curve (with finiteness of X ( Jac ( C )) supposed). Conjecture (Colliot-Thélène/Sansuc, Kato/Saito, Colliot-Thélène) The complex ( E 0 ) is exact for all smooth projective varieties. Yongqi LIANG Université Paris-Sud 11, Orsay, France
Brauer-Manin obstruction Rational points vs. Zero-cycles Rationally connectedness Definition X / k is called rationally connected , if for any P , Q ∈ X ( C ) , there exists a C -morphism f : P 1 C → X C such that f ( 0 ) = P and f ( ∞ ) = Q . Counter-examples: - An abelian variety is never rationally connected. - A smooth curve of genus > 0 is never rationally connected. Example: - A homogeneous space of a connected linear algebraic group is rationally connected. Yongqi LIANG Université Paris-Sud 11, Orsay, France
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