Some notions from tropical geometry Recall that an (integral, R -affine) polyhedron ∆ in R r is a set N � � ω ∈ R r � � � � u i , ω � ≥ γ i , u i ∈ Z r , γ i ∈ R . ∆ := i = 1 A polyhedral complex Σ in R r consists of a finite set Σ of polyhedra in R r such that ∆ ∈ Σ ⇒ every face of ∆ is in Σ , ∆ , ∆ ′ ∈ Σ ⇒ ∆ ∩ ∆ ′ is empty or a face of ∆ and ∆ ′ . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 7 / 29
Some notions from tropical geometry Recall that an (integral, R -affine) polyhedron ∆ in R r is a set N � � ω ∈ R r � � � � u i , ω � ≥ γ i , u i ∈ Z r , γ i ∈ R . ∆ := i = 1 A polyhedral complex Σ in R r consists of a finite set Σ of polyhedra in R r such that ∆ ∈ Σ ⇒ every face of ∆ is in Σ , ∆ , ∆ ′ ∈ Σ ⇒ ∆ ∩ ∆ ′ is empty or a face of ∆ and ∆ ′ . A weighted polyhedral complex (Σ , m ) of pure dimension n consists of a polyhedral complex Σ of pure dimension n and a weight function m : Σ n := { ∆ ∈ Σ | dim (∆) = n } → Z . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 7 / 29
Lagerberg’s superforms Following Lagerberg we define for U ⊆ R n open q � A p , q ( U ) := A p ( U ) ⊗ C ∞ ( U ) A q ( U ) = A p ( U ) ⊗ Z Z r � bigraded differential alternating algebra ( A · , · , d ′ , d ′′ ) . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 8 / 29
Lagerberg’s superforms Following Lagerberg we define for U ⊆ R n open q � A p , q ( U ) := A p ( U ) ⊗ C ∞ ( U ) A q ( U ) = A p ( U ) ⊗ Z Z r � bigraded differential alternating algebra ( A · , · , d ′ , d ′′ ) . In local coordinates and with multi-index notation � f IJ d ′ x I ∧ d ′′ x J , α = | I | = p , | J | = q � r � ∂ f IJ d ′ α d ′ x i ∧ d ′ x I ∧ d ′′ x J , = ∂ x i i = 1 I , J r � � ∂ f IJ d ′′ α d ′′ x j ∧ d ′ x I ∧ d ′′ x J . = ∂ x j j = 1 I , J Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 8 / 29
Tropical geometry and superforms Integration of α ∈ A n , n c ( R r ) over n -dimensional polyhedron ∆ is well defined � current δ ∆ ∈ D n , n ( R r ) =: D r − n , r − n ( R r ) Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 9 / 29
Tropical geometry and superforms Integration of α ∈ A n , n c ( R r ) over n -dimensional polyhedron ∆ is well defined � current δ ∆ ∈ D n , n ( R r ) =: D r − n , r − n ( R r ) For a weighted polyhedral complex (Σ , m ) of pure dimension n � current δ (Σ , m ) = � ∆ ∈ Σ n m ∆ δ ∆ ∈ D n , n ( R r ) Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 9 / 29
Tropical geometry and superforms Integration of α ∈ A n , n c ( R r ) over n -dimensional polyhedron ∆ is well defined � current δ ∆ ∈ D n , n ( R r ) =: D r − n , r − n ( R r ) For a weighted polyhedral complex (Σ , m ) of pure dimension n � current δ (Σ , m ) = � ∆ ∈ Σ n m ∆ δ ∆ ∈ D n , n ( R r ) A weighted polyhedral complex (Σ , m ) is called a tropical cycle if and only if d ′ ( δ (Σ , m ) ) = 0 = d ′′ ( δ (Σ , m ) ) . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 9 / 29
Tropical geometry and superforms Integration of α ∈ A n , n c ( R r ) over n -dimensional polyhedron ∆ is well defined � current δ ∆ ∈ D n , n ( R r ) =: D r − n , r − n ( R r ) For a weighted polyhedral complex (Σ , m ) of pure dimension n � current δ (Σ , m ) = � ∆ ∈ Σ n m ∆ δ ∆ ∈ D n , n ( R r ) A weighted polyhedral complex (Σ , m ) is called a tropical cycle if and only if d ′ ( δ (Σ , m ) ) = 0 = d ′′ ( δ (Σ , m ) ) . (Mikhalkin, Allermann, Rau, ...) There is a well-defined intersection product for tropical cycles on R r (no equivalence relation is needed). Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 9 / 29
δ -preforms Definition A current in D p , q ( R r ) is a δ -preform of type ( p , q ) if and only if it is of the form N � α i ∧ δ C i i = 1 with α i ∈ A p i , q i ( R r ) and C i = (Σ i , m i ) tropical cycles of codimension k i with ( p , q ) = ( p i + k i , q i + k i ) Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 10 / 29
δ -preforms Definition A current in D p , q ( R r ) is a δ -preform of type ( p , q ) if and only if it is of the form N � α i ∧ δ C i i = 1 with α i ∈ A p i , q i ( R r ) and C i = (Σ i , m i ) tropical cycles of codimension k i with ( p , q ) = ( p i + k i , q i + k i ) Get bigraded differential algebra ( P • , • ( R r ) , d ′ , d ′′ ) , where ( α i ∧ δ C i ) ∧ ( α ′ j ) := ( α i ∧ α ′ j ∧ δ C ′ j ) ∧ δ C i · C ′ j using the tropical intersection product. Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 10 / 29
Non-archimedean analytification From now on, K denotes an algebraically closed field complete with respect to a non-trivial non-archimedean absolute value | | : K → [ 0 , ∞ ) with valuation ring K ◦ and residue class field ˜ K . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 11 / 29
Non-archimedean analytification From now on, K denotes an algebraically closed field complete with respect to a non-trivial non-archimedean absolute value | | : K → [ 0 , ∞ ) with valuation ring K ◦ and residue class field ˜ K . Let X be a projective variety over K and X an the associated Berkovich analytic space. Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 11 / 29
Non-archimedean analytification From now on, K denotes an algebraically closed field complete with respect to a non-trivial non-archimedean absolute value | | : K → [ 0 , ∞ ) with valuation ring K ◦ and residue class field ˜ K . Let X be a projective variety over K and X an the associated Berkovich analytic space. For closed subvariety U of a torus T = G r m over K , consider trop : T an − → R r , p �− → ( − log p ( t 1 ) , . . . , − log p ( t r )) and put Trop ( U ) := trop ( U an ) . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 11 / 29
Non-archimedean analytification From now on, K denotes an algebraically closed field complete with respect to a non-trivial non-archimedean absolute value | | : K → [ 0 , ∞ ) with valuation ring K ◦ and residue class field ˜ K . Let X be a projective variety over K and X an the associated Berkovich analytic space. For closed subvariety U of a torus T = G r m over K , consider trop : T an − → R r , p �− → ( − log p ( t 1 ) , . . . , − log p ( t r )) and put Trop ( U ) := trop ( U an ) . Theorem (Bieri, Groves, Speyer, Sturmfels) Trop ( U ) has a canonical structure of a tropical cycle. Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 11 / 29
Very affine open subsets Let U be an open subset of X . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 12 / 29
Very affine open subsets Let U be an open subset of X . If U is affine then M U := O ( U ) × / K × is a free Z -module of finite rank r (Samuel 1966). Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 12 / 29
Very affine open subsets Let U be an open subset of X . If U is affine then M U := O ( U ) × / K × is a free Z -module of finite rank r (Samuel 1966). Put N U = Hom Z ( M U , Z ) and N U , R = N U ⊗ Z R ∼ = R r . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 12 / 29
Very affine open subsets Let U be an open subset of X . If U is affine then M U := O ( U ) × / K × is a free Z -module of finite rank r (Samuel 1966). Put N U = Hom Z ( M U , Z ) and N U , R = N U ⊗ Z R ∼ = R r . Choosing a Z -basis f 1 , . . . , f r of M U , we obtain (up to translation in T ) a canonical map ϕ U = ( f 1 , . . . , f r ) : U → G r m = T U = Spec K [ M U ] . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 12 / 29
Very affine open subsets Let U be an open subset of X . If U is affine then M U := O ( U ) × / K × is a free Z -module of finite rank r (Samuel 1966). Put N U = Hom Z ( M U , Z ) and N U , R = N U ⊗ Z R ∼ = R r . Choosing a Z -basis f 1 , . . . , f r of M U , we obtain (up to translation in T ) a canonical map ϕ U = ( f 1 , . . . , f r ) : U → G r m = T U = Spec K [ M U ] . Call U very affine , if it satisfies the equivalent conditions U admits closed embedding into a torus, O ( U ) is generated as a K -algebra by O ( U ) × , ϕ U is a closed embedding. Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 12 / 29
Tropical charts Very affine open subsets form basis for Zariski toplogy on X . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 13 / 29
Tropical charts Very affine open subsets form basis for Zariski toplogy on X . Definition A tropical chart ( V , ϕ U ) on X consists of a very affine open subset U of X → T U = G r an associated closed immersion ϕ U : U ֒ m , an open subset Ω of Trop ( U ) such that V is the open subset U (Ω) in U an where trop − 1 ϕ an → T an = ( G r trop → N U , R = R r . trop U : U an U m ) an − − Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 13 / 29
Tropical charts Very affine open subsets form basis for Zariski toplogy on X . Definition A tropical chart ( V , ϕ U ) on X consists of a very affine open subset U of X → T U = G r an associated closed immersion ϕ U : U ֒ m , an open subset Ω of Trop ( U ) such that V is the open subset U (Ω) in U an where trop − 1 ϕ an → T an = ( G r trop → N U , R = R r . trop U : U an U m ) an − − Tropical charts form a basis for the topology on X an . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 13 / 29
Tropical charts Very affine open subsets form basis for Zariski toplogy on X . Definition A tropical chart ( V , ϕ U ) on X consists of a very affine open subset U of X → T U = G r an associated closed immersion ϕ U : U ֒ m , an open subset Ω of Trop ( U ) such that V is the open subset U (Ω) in U an where trop − 1 ϕ an → T an = ( G r trop → N U , R = R r . trop U : U an U m ) an − − Tropical charts form a basis for the topology on X an . Be careful: A tropical chart ( V , ϕ U ) will not give complete local ’information’ about V ! Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 13 / 29
δ -forms and δ -currents Definition A δ -form α on X an is given by tropical charts ( V i , ϕ U i ) i ∈ I covering X an and a family α = ( α ) i ∈ I where α i ∈ P • , • ( N U i , R ) such that � � α = α ′ ⇔ α i � � j = α ′ in a tropical sense (see [GK]) . j V i ∩ V ′ V i ∩ V ′ j Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 14 / 29
δ -forms and δ -currents Definition A δ -form α on X an is given by tropical charts ( V i , ϕ U i ) i ∈ I covering X an and a family α = ( α ) i ∈ I where α i ∈ P • , • ( N U i , R ) such that � � α = α ′ ⇔ α i � � j = α ′ in a tropical sense (see [GK]) . j V i ∩ V ′ V i ∩ V ′ j get bigraded differential algebra ( B • , • ( X an ) , d ′ , d ′′ ) . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 14 / 29
δ -forms and δ -currents Definition A δ -form α on X an is given by tropical charts ( V i , ϕ U i ) i ∈ I covering X an and a family α = ( α ) i ∈ I where α i ∈ P • , • ( N U i , R ) such that � � α = α ′ ⇔ α i � � j = α ′ in a tropical sense (see [GK]) . j V i ∩ V ′ V i ∩ V ′ j get bigraded differential algebra ( B • , • ( X an ) , d ′ , d ′′ ) . Topological dual B • , • c ( X an ) is space of δ -currents E • , • ( X an ) . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 14 / 29
δ -forms and δ -currents Definition A δ -form α on X an is given by tropical charts ( V i , ϕ U i ) i ∈ I covering X an and a family α = ( α ) i ∈ I where α i ∈ P • , • ( N U i , R ) such that � � α = α ′ ⇔ α i � � j = α ′ in a tropical sense (see [GK]) . j V i ∩ V ′ V i ∩ V ′ j get bigraded differential algebra ( B • , • ( X an ) , d ′ , d ′′ ) . Topological dual B • , • c ( X an ) is space of δ -currents E • , • ( X an ) . Observe that no smoothness assumption on X is required. Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 14 / 29
δ -forms and δ -currents Definition A δ -form α on X an is given by tropical charts ( V i , ϕ U i ) i ∈ I covering X an and a family α = ( α ) i ∈ I where α i ∈ P • , • ( N U i , R ) such that � � α = α ′ ⇔ α i � � j = α ′ in a tropical sense (see [GK]) . j V i ∩ V ′ V i ∩ V ′ j get bigraded differential algebra ( B • , • ( X an ) , d ′ , d ′′ ) . Topological dual B • , • c ( X an ) is space of δ -currents E • , • ( X an ) . Observe that no smoothness assumption on X is required. Consider δ -forms as analogs of complex differential forms with logarithmic singularities. Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 14 / 29
Smooth forms after Chambert-Loir, Ducros Using Lagerberg’s superforms and skipping tropical cycles similarly smooth ( p , q ) -forms of Chambert-Loir and Ducros � (see [CD], [Gu]) leading to subalgebra A • , • ( X an ) of B • , • ( X an ) . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 15 / 29
Smooth forms after Chambert-Loir, Ducros Using Lagerberg’s superforms and skipping tropical cycles similarly smooth ( p , q ) -forms of Chambert-Loir and Ducros � (see [CD], [Gu]) leading to subalgebra A • , • ( X an ) of B • , • ( X an ) . Chambert-Loir, Ducros work more generally on analytic spaces with boundary (no balancing condition is known at the boundary). Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 15 / 29
Smooth forms after Chambert-Loir, Ducros Using Lagerberg’s superforms and skipping tropical cycles similarly smooth ( p , q ) -forms of Chambert-Loir and Ducros � (see [CD], [Gu]) leading to subalgebra A • , • ( X an ) of B • , • ( X an ) . Chambert-Loir, Ducros work more generally on analytic spaces with boundary (no balancing condition is known at the boundary). Given a tropical chart ( V , ϕ U ) , f 1 , . . . , f m ∈ O ( U ) × and g : R m → R smooth, the function g ( − log | f 1 | , . . . , − log | f m | ) : V → R is smooth. Not all functions in A 0 ( V ) are of this form! Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 15 / 29
Smooth forms after Chambert-Loir, Ducros Using Lagerberg’s superforms and skipping tropical cycles similarly smooth ( p , q ) -forms of Chambert-Loir and Ducros � (see [CD], [Gu]) leading to subalgebra A • , • ( X an ) of B • , • ( X an ) . Chambert-Loir, Ducros work more generally on analytic spaces with boundary (no balancing condition is known at the boundary). Given a tropical chart ( V , ϕ U ) , f 1 , . . . , f m ∈ O ( U ) × and g : R m → R smooth, the function g ( − log | f 1 | , . . . , − log | f m | ) : V → R is smooth. Not all functions in A 0 ( V ) are of this form! Stone-Weierstraß Theorem (Chambert-Loir, Ducros): A 0 c ( W ) is dense in C 0 c ( W ) if W is open in X an . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 15 / 29
Smooth forms after Chambert-Loir, Ducros Using Lagerberg’s superforms and skipping tropical cycles similarly smooth ( p , q ) -forms of Chambert-Loir and Ducros � (see [CD], [Gu]) leading to subalgebra A • , • ( X an ) of B • , • ( X an ) . Chambert-Loir, Ducros work more generally on analytic spaces with boundary (no balancing condition is known at the boundary). Given a tropical chart ( V , ϕ U ) , f 1 , . . . , f m ∈ O ( U ) × and g : R m → R smooth, the function g ( − log | f 1 | , . . . , − log | f m | ) : V → R is smooth. Not all functions in A 0 ( V ) are of this form! Stone-Weierstraß Theorem (Chambert-Loir, Ducros): A 0 c ( W ) is dense in C 0 c ( W ) if W is open in X an . Cohomology of ( A 0 , · ( X an ) , d ′′ ) computes singular cohomology of X an (Philipp Jell, arXiv:1409.0676 ). Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 15 / 29
Integration Let W be an open subset of X an . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 16 / 29
Integration Let W be an open subset of X an . � W : B n , n There is a well-defined integral c ( W ) → R . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 16 / 29
Integration Let W be an open subset of X an . � W : B n , n There is a well-defined integral c ( W ) → R . Every α ∈ B n , n ( W ) induces continuous map � A 0 c ( W ) − → R , f �→ f · α W which extends to continuous map C 0 c ( X ) → R and induces a signed Borel measure on W (by the Riesz representation Theorem). Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 16 / 29
Integration Let W be an open subset of X an . � W : B n , n There is a well-defined integral c ( W ) → R . Every α ∈ B n , n ( W ) induces continuous map � A 0 c ( W ) − → R , f �→ f · α W which extends to continuous map C 0 c ( X ) → R and induces a signed Borel measure on W (by the Riesz representation Theorem). Get C 0 c ( W ) → E 0 , 0 ( W ) , f �→ [ f ] . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 16 / 29
Metrics on line bundles Let L be a line bundle on X . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 17 / 29
Metrics on line bundles Let L be a line bundle on X . Fix an open covering ( U i ) i ∈ I of X , a family ( s i ) i ∈ I of frames s i ∈ Γ( U i , L ) , and the 1-cocyle ( h ij ) ij determined by s j = h ij s i . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 17 / 29
Metrics on line bundles Let L be a line bundle on X . Fix an open covering ( U i ) i ∈ I of X , a family ( s i ) i ∈ I of frames s i ∈ Γ( U i , L ) , and the 1-cocyle ( h ij ) ij determined by s j = h ij s i . A (continuous) metric � � on L is given by a family ( ρ i ) i of continuous functions ρ i : U an → R such that ρ j = | h ij | · ρ i on i U an ∩ U an for all i , j ∈ I . i j Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 17 / 29
Metrics on line bundles Let L be a line bundle on X . Fix an open covering ( U i ) i ∈ I of X , a family ( s i ) i ∈ I of frames s i ∈ Γ( U i , L ) , and the 1-cocyle ( h ij ) ij determined by s j = h ij s i . A (continuous) metric � � on L is given by a family ( ρ i ) i of continuous functions ρ i : U an → R such that ρ j = | h ij | · ρ i on i U an ∩ U an for all i , j ∈ I . i j Let W ⊆ X an be open. Given sections s ∈ Γ( W , L an ) , the metric � � determines continuous functions � s � : X an − → R such that � s i � = ρ i . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 17 / 29
First Chern delta-current Let L be line bundle on X and � � continuous metric on L an . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 18 / 29
First Chern delta-current Let L be line bundle on X and � � continuous metric on L an . For a local frame s of L over an open subset U of X , [ c 1 ( L , � � )] := d ′ d ′′ [ − log � s � ] ∈ E 1 , 1 ( U an ) is a δ -current independent of the choice s . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 18 / 29
First Chern delta-current Let L be line bundle on X and � � continuous metric on L an . For a local frame s of L over an open subset U of X , [ c 1 ( L , � � )] := d ′ d ′′ [ − log � s � ] ∈ E 1 , 1 ( U an ) is a δ -current independent of the choice s . By partition of unity argument, we get a well-defined δ -current [ c 1 ( L , � � )] called the first Chern δ -current . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 18 / 29
First Chern delta-current Let L be line bundle on X and � � continuous metric on L an . For a local frame s of L over an open subset U of X , [ c 1 ( L , � � )] := d ′ d ′′ [ − log � s � ] ∈ E 1 , 1 ( U an ) is a δ -current independent of the choice s . By partition of unity argument, we get a well-defined δ -current [ c 1 ( L , � � )] called the first Chern δ -current . Similarly as in [CD], we get Poincaré-Lelong formula d ′ d ′′ [ − log � s � ] + δ div ( s ) = [ c 1 ( L , � in E 1 , 1 ( X an ) � )] for any meromorphic section s of L over X . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 18 / 29
Smooth and algebraic metrics A metric � � on L is called smooth if and only if − log � s � ∈ A 0 , 0 ( U ) for any local frame s ∈ Γ( U , L ) . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 19 / 29
Smooth and algebraic metrics A metric � � on L is called smooth if and only if − log � s � ∈ A 0 , 0 ( U ) for any local frame s ∈ Γ( U , L ) . Let K ◦ = valuation ring of K . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 19 / 29
Smooth and algebraic metrics A metric � � on L is called smooth if and only if − log � s � ∈ A 0 , 0 ( U ) for any local frame s ∈ Γ( U , L ) . Let K ◦ = valuation ring of K . Let X be a proper flat model of X over Spec K ◦ . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 19 / 29
Smooth and algebraic metrics A metric � � on L is called smooth if and only if − log � s � ∈ A 0 , 0 ( U ) for any local frame s ∈ Γ( U , L ) . Let K ◦ = valuation ring of K . Let X be a proper flat model of X over Spec K ◦ . Let L be a line bundle on X with isomorphism L| X = L . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 19 / 29
Smooth and algebraic metrics A metric � � on L is called smooth if and only if − log � s � ∈ A 0 , 0 ( U ) for any local frame s ∈ Γ( U , L ) . Let K ◦ = valuation ring of K . Let X be a proper flat model of X over Spec K ◦ . Let L be a line bundle on X with isomorphism L| X = L . Get a unique continuous metric � � L on L such that local frames s of L over X satisfy � s � L = 1. Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 19 / 29
Smooth and algebraic metrics A metric � � on L is called smooth if and only if − log � s � ∈ A 0 , 0 ( U ) for any local frame s ∈ Γ( U , L ) . Let K ◦ = valuation ring of K . Let X be a proper flat model of X over Spec K ◦ . Let L be a line bundle on X with isomorphism L| X = L . Get a unique continuous metric � � L on L such that local frames s of L over X satisfy � s � L = 1. Such a metric is called an algebraic metric . In a similar way get formal metrics from formal models. Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 19 / 29
Smooth and algebraic metrics A metric � � on L is called smooth if and only if − log � s � ∈ A 0 , 0 ( U ) for any local frame s ∈ Γ( U , L ) . Let K ◦ = valuation ring of K . Let X be a proper flat model of X over Spec K ◦ . Let L be a line bundle on X with isomorphism L| X = L . Get a unique continuous metric � � L on L such that local frames s of L over X satisfy � s � L = 1. Such a metric is called an algebraic metric . In a similar way get formal metrics from formal models. Problem : Algebraic and formal metrics are not always smooth! Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 19 / 29
Delta-metrics Definition A δ -metric on L is a piecewise smooth metric � � such that the δ -current [ c 1 ( L , � � )] is represented in an functorial way by a δ -form c 1 ( L , � � ) . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 20 / 29
Delta-metrics Definition A δ -metric on L is a piecewise smooth metric � � such that the δ -current [ c 1 ( L , � � )] is represented in an functorial way by a δ -form c 1 ( L , � � ) . δ -metrics are stable under tensor products and pullback. Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 20 / 29
Delta-metrics Definition A δ -metric on L is a piecewise smooth metric � � such that the δ -current [ c 1 ( L , � � )] is represented in an functorial way by a δ -form c 1 ( L , � � ) . δ -metrics are stable under tensor products and pullback. Smooth metrics are δ -metrics. Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 20 / 29
Delta-metrics Definition A δ -metric on L is a piecewise smooth metric � � such that the δ -current [ c 1 ( L , � � )] is represented in an functorial way by a δ -form c 1 ( L , � � ) . δ -metrics are stable under tensor products and pullback. Smooth metrics are δ -metrics. Algebraic metrics are δ -metrics. Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 20 / 29
Delta-metrics Definition A δ -metric on L is a piecewise smooth metric � � such that the δ -current [ c 1 ( L , � � )] is represented in an functorial way by a δ -form c 1 ( L , � � ) . δ -metrics are stable under tensor products and pullback. Smooth metrics are δ -metrics. Algebraic metrics are δ -metrics. Canonical metrics on line bundles over abelian varieties are δ -metrics (use Mumford’s non-archimedean uniformization of abelian varieties). Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 20 / 29
Delta-metrics Definition A δ -metric on L is a piecewise smooth metric � � such that the δ -current [ c 1 ( L , � � )] is represented in an functorial way by a δ -form c 1 ( L , � � ) . δ -metrics are stable under tensor products and pullback. Smooth metrics are δ -metrics. Algebraic metrics are δ -metrics. Canonical metrics on line bundles over abelian varieties are δ -metrics (use Mumford’s non-archimedean uniformization of abelian varieties). Canonical metrics on line bundles algebraically equivalent to zero are δ -metrics. Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 20 / 29
Measures associated with metrized line bundles Chambert-Loir measure: Let X be projective and � � L an algebraic metric on L an given by ( X , L ) such that the special fibre X s is reduced. Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 21 / 29
Measures associated with metrized line bundles Chambert-Loir measure: Let X be projective and � � L an algebraic metric on L an given by ( X , L ) such that the special fibre X s is reduced. There is a unique (discrete) measure µ on X an such that the projection formula holds and � µ = deg L ( Y ) δ ξ Y Y where Y ranges over the irreducible components of X s and ξ Y is the unique point of X an whose reduction is the generic point of Y . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 21 / 29
Measures associated with metrized line bundles Chambert-Loir measure: Let X be projective and � � L an algebraic metric on L an given by ( X , L ) such that the special fibre X s is reduced. There is a unique (discrete) measure µ on X an such that the projection formula holds and � µ = deg L ( Y ) δ ξ Y Y where Y ranges over the irreducible components of X s and ξ Y is the unique point of X an whose reduction is the generic point of Y . Monge-Ampère measure: Let � � be a δ -metric on L an , i.e. c 1 ( L , � � ) is a δ -form. Then c 1 ( L , � � ) ∧ n ∈ B n , n ( X an ) and it induces a measure on X an . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 21 / 29
Chambert-Loir measure = Monge-Ampère measure Let ( L , � � ) be an algebraic metric. Theorem (GK) Chambert-Loir measure = Monge-Ampère measure c 1 ( L , � � ) ∧ n . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 22 / 29
Chambert-Loir measure = Monge-Ampère measure Let ( L , � � ) be an algebraic metric. Theorem (GK) Chambert-Loir measure = Monge-Ampère measure c 1 ( L , � � ) ∧ n . A variant of this following Theorem was proved before by Chambert-Loir and Ducros. Difference to [CD]: Chambert-Loir and Ducros use an approximation process by smooth metrics as in Bedford-Taylor theory to define [ c 1 ( L , � � )] ∧ n as a wedge product of currents. We obtain c 1 ( L , � � ) ∧ n directly as a wedge product of δ -forms. This simplifies the proof. Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 22 / 29
Non-archimedean Arakelov theory Let X be (as before) a projective variety over K of dimension n . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 23 / 29
Non-archimedean Arakelov theory Let X be (as before) a projective variety over K of dimension n . Definition A δ -current g Z ∈ E p − 1 , p − 1 ( X an ) is called a Green current for a cycle Z ∈ Z p ( X ) : ⇔ d ′ d ′′ g Z + δ Z = [ ω Z ] holds for some δ -form ω ( Z , g Z ) := ω Z ∈ B p , p ( X an ) . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 23 / 29
Non-archimedean Arakelov theory Let X be (as before) a projective variety over K of dimension n . Definition A δ -current g Z ∈ E p − 1 , p − 1 ( X an ) is called a Green current for a cycle Z ∈ Z p ( X ) : ⇔ d ′ d ′′ g Z + δ Z = [ ω Z ] holds for some δ -form ω ( Z , g Z ) := ω Z ∈ B p , p ( X an ) . Let ( L , � � ) be a line bundle with a δ -metric. Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 23 / 29
Non-archimedean Arakelov theory Let X be (as before) a projective variety over K of dimension n . Definition A δ -current g Z ∈ E p − 1 , p − 1 ( X an ) is called a Green current for a cycle Z ∈ Z p ( X ) : ⇔ d ′ d ′′ g Z + δ Z = [ ω Z ] holds for some δ -form ω ( Z , g Z ) := ω Z ∈ B p , p ( X an ) . Let ( L , � � ) be a line bundle with a δ -metric. Let s be a meromorphic section of L over X with Weil divisor D = div ( s ) . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 23 / 29
Non-archimedean Arakelov theory Let X be (as before) a projective variety over K of dimension n . Definition A δ -current g Z ∈ E p − 1 , p − 1 ( X an ) is called a Green current for a cycle Z ∈ Z p ( X ) : ⇔ d ′ d ′′ g Z + δ Z = [ ω Z ] holds for some δ -form ω ( Z , g Z ) := ω Z ∈ B p , p ( X an ) . Let ( L , � � ) be a line bundle with a δ -metric. Let s be a meromorphic section of L over X with Weil divisor D = div ( s ) . Poincaré-Lelong equation ⇒ g D = [ − log � s � ] ∈ E 1 , 1 ( X an ) is a Green current for D with ω D = c 1 ( L , � � ) . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 23 / 29
The star product of Green currents Let Z ∈ Z p ( X ) be a prime cycle with a Green current g Z . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 24 / 29
The star product of Green currents Let Z ∈ Z p ( X ) be a prime cycle with a Green current g Z . Let s be meromorphic section of line bundle ( L , � � ) with δ -metric such that D = div ( s ) intersects Z properly. Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 24 / 29
The star product of Green currents Let Z ∈ Z p ( X ) be a prime cycle with a Green current g Z . Let s be meromorphic section of line bundle ( L , � � ) with δ -metric such that D = div ( s ) intersects Z properly. Another application of the Poincaré-Lelong formula gives: Proposition [GK] If D intersects Z property, then g D ∗ g Z := g D ∧ δ Z + c 1 ( L , � � ) ∧ g Z is a Green current for the intersection D · Z ∈ Z p + 1 ( X ) with ω ( D · Z , g D ∗ g Z ) = ω ( D , g D ) ∧ ω ( Z , g Z ) . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 24 / 29
The star product of Green currents Let Z ∈ Z p ( X ) be a prime cycle with a Green current g Z . Let s be meromorphic section of line bundle ( L , � � ) with δ -metric such that D = div ( s ) intersects Z properly. Another application of the Poincaré-Lelong formula gives: Proposition [GK] If D intersects Z property, then g D ∗ g Z := g D ∧ δ Z + c 1 ( L , � � ) ∧ g Z is a Green current for the intersection D · Z ∈ Z p + 1 ( X ) with ω ( D · Z , g D ∗ g Z ) = ω ( D , g D ) ∧ ω ( Z , g Z ) . The ∗ -product is commutative modulo Im ( d ′ ) + Im ( d ′′ ) . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 24 / 29
Local heights Definition: Let D 0 , . . . , D n be Cartier divisors intersecting properly on X . Let O ( D 0 ) , . . . , O ( D n ) be equipped with δ -metrics � � i and associated Green currents g D i = [ − log � s D i � i ] then λ ( X ) := � g D 0 ∗ · · · ∗ g D n , 1 � is called the local height of X . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 25 / 29
Local heights Definition: Let D 0 , . . . , D n be Cartier divisors intersecting properly on X . Let O ( D 0 ) , . . . , O ( D n ) be equipped with δ -metrics � � i and associated Green currents g D i = [ − log � s D i � i ] then λ ( X ) := � g D 0 ∗ · · · ∗ g D n , 1 � is called the local height of X . Theorem (GK) If we use algebraic metrics on O ( D 0 ) , . . . , O ( D n ) , then λ ( X ) is the usual local height of X in arithmetic geometry given as the intersection number of the Cartier divisors on a corresponding model. Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 25 / 29
Positive forms and plurisubharmonic metrics Let W be open in X an and � � a continuous metric on L an . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 26 / 29
Positive forms and plurisubharmonic metrics Let W be open in X an and � � a continuous metric on L an . Chambert-Loir, Ducros define cones of positive elements in A p , p ( W ) and D p , p ( W ) . Klaus Künnemann (Regensburg) Metrics and delta-forms Papeete 2015 26 / 29
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