The β constant appeared in algebraic and complex geometry Min Ru University of Houston TX, USA Min Ru The β constant appeared in algebraic and complex geometry
Nevanlinna theory: Introduction the notations Let X be a complex projective variety and let D be an effective Cartier divisor. Let s D be the canonical section of [ D ] (i.e. [ s D = 0] = D ) and � � be an hemitian metric, i.e. � s � 2 = | s α | 2 h α . Min Ru The β constant appeared in algebraic and complex geometry
Nevanlinna theory: Introduction the notations Let X be a complex projective variety and let D be an effective Cartier divisor. Let s D be the canonical section of [ D ] (i.e. [ s D = 0] = D ) and � � be an hemitian metric, i.e. � s � 2 = | s α | 2 h α . Let f : C → X be a holomorphic map. By Poincare-Lelong formula, − dd c [log � f ∗ s D � 2 ] = − f ∗ D + f ∗ c 1 ([ D ]) . Min Ru The β constant appeared in algebraic and complex geometry
Nevanlinna theory: Introduction the notations Let X be a complex projective variety and let D be an effective Cartier divisor. Let s D be the canonical section of [ D ] (i.e. [ s D = 0] = D ) and � � be an hemitian metric, i.e. � s � 2 = | s α | 2 h α . Let f : C → X be a holomorphic map. By Poincare-Lelong formula, − dd c [log � f ∗ s D � 2 ] = − f ∗ D + f ∗ c 1 ([ D ]) . Applying � t dt � | z | < t and use Green-Jensen (Stoke’s theorem), we get the 1 t First Main Theorem: m f ( r , D ) + N f ( r , D ) = T f , D ( r ) + O (1) where λ D ( x ) = − log � s D ( x ) � = − log distance from x to D (Weil � 2 π λ D ( f ( re i θ )) d θ function for D ), m f ( r , D ) = 2 π (Approximation 0 � r dt | z | < t f ∗ c 1 ( L ) (Height function). � function). T f , L ( r ) := 1 t Min Ru The β constant appeared in algebraic and complex geometry
Nevanlinna theory: Introduction the notations Let X be a complex projective variety and let D be an effective Cartier divisor. Let s D be the canonical section of [ D ] (i.e. [ s D = 0] = D ) and � � be an hemitian metric, i.e. � s � 2 = | s α | 2 h α . Let f : C → X be a holomorphic map. By Poincare-Lelong formula, − dd c [log � f ∗ s D � 2 ] = − f ∗ D + f ∗ c 1 ([ D ]) . Applying � t dt � | z | < t and use Green-Jensen (Stoke’s theorem), we get the 1 t First Main Theorem: m f ( r , D ) + N f ( r , D ) = T f , D ( r ) + O (1) where λ D ( x ) = − log � s D ( x ) � = − log distance from x to D (Weil � 2 π λ D ( f ( re i θ )) d θ function for D ), m f ( r , D ) = 2 π (Approximation 0 � r dt | z | < t f ∗ c 1 ( L ) (Height function). � function). T f , L ( r ) := 1 t From First Main Theorem, N f ( r , D ) ≤ T f , D ( r ). The Second Main Theorem (in the spirit of Nevanlinna-Cartan) is to control T f , D ( r ) in terms of N f ( r , D ), or equivalently, to control m f ( r , D ) in terms of T f , D ( r ). Min Ru The β constant appeared in algebraic and complex geometry
Nevanlinna’s SMT for meromorphic functions The Second Main Theorem(Nevanlinna, 1929). Let f be meromorphic (non-constant) on C and a 1 , ..., a q ∈ C ∪ {∞} distinct. Then, for any ǫ > 0, � q ( q − 2 − ǫ ) T f ( r ) ≤ exc j =1 N f ( r , a j ) , or equivalently q � m f ( r , a j ) ≤ exc (2 + ǫ ) T f ( r ) , j =1 where ≤ exc means that the inequality holds for r ∈ [0 , + ∞ ) outside a set E with finite measure. Min Ru The β constant appeared in algebraic and complex geometry
Nevanlinna’s SMT for meromorphic functions The Second Main Theorem(Nevanlinna, 1929). Let f be meromorphic (non-constant) on C and a 1 , ..., a q ∈ C ∪ {∞} distinct. Then, for any ǫ > 0, � q ( q − 2 − ǫ ) T f ( r ) ≤ exc j =1 N f ( r , a j ) , or equivalently q � m f ( r , a j ) ≤ exc (2 + ǫ ) T f ( r ) , j =1 where ≤ exc means that the inequality holds for r ∈ [0 , + ∞ ) outside a set E with finite measure. This implies the well-known little Picard theorem: If a meromorphic function f on C omits three points in C ∪ {∞} , then f must be constant. Min Ru The β constant appeared in algebraic and complex geometry
Nevanlinna’s SMT for meromorphic functions The Second Main Theorem(Nevanlinna, 1929). Let f be meromorphic (non-constant) on C and a 1 , ..., a q ∈ C ∪ {∞} distinct. Then, for any ǫ > 0, � q ( q − 2 − ǫ ) T f ( r ) ≤ exc j =1 N f ( r , a j ) , or equivalently q � m f ( r , a j ) ≤ exc (2 + ǫ ) T f ( r ) , j =1 where ≤ exc means that the inequality holds for r ∈ [0 , + ∞ ) outside a set E with finite measure. This implies the well-known little Picard theorem: If a meromorphic function f on C omits three points in C ∪ {∞} , then f must be constant. Min Ru The β constant appeared in algebraic and complex geometry
Cartan’s Theorem (1933). Let f : C → P n ( C ) be a linearly non-degenerate holomorphic map. Let H 1 , . . . , H q be the hyperplanes in general position on P n ( C ). Then, for any ǫ > 0, � q j =1 m f ( r , H j ) ≤ exc ( n + 1 + ǫ ) T f ( r ) . Min Ru The β constant appeared in algebraic and complex geometry
Cartan’s Theorem (1933). Let f : C → P n ( C ) be a linearly non-degenerate holomorphic map. Let H 1 , . . . , H q be the hyperplanes in general position on P n ( C ). Then, for any ǫ > 0, � q j =1 m f ( r , H j ) ≤ exc ( n + 1 + ǫ ) T f ( r ) . In 2004, Ru extended the above result to hypersurfaces for f : C → P n ( C ) with Zariski dense image. � q 1 d j m f ( r , D j ) ≤ exc ( n + 1 + ǫ ) T f ( r ) . j =1 Min Ru The β constant appeared in algebraic and complex geometry
Cartan’s Theorem (1933). Let f : C → P n ( C ) be a linearly non-degenerate holomorphic map. Let H 1 , . . . , H q be the hyperplanes in general position on P n ( C ). Then, for any ǫ > 0, � q j =1 m f ( r , H j ) ≤ exc ( n + 1 + ǫ ) T f ( r ) . In 2004, Ru extended the above result to hypersurfaces for f : C → P n ( C ) with Zariski dense image. � q 1 d j m f ( r , D j ) ≤ exc ( n + 1 + ǫ ) T f ( r ) . j =1 Theorem (Ru, 2009). Let f : C → X be holo and Zariski dense, D 1 , . . . , D q be divisors in general position in X . Assume that D j ∼ d j A ( A being ample). Then, for ∀ ǫ > 0, q 1 � m f ( r , D j ) ≤ exc (dim X + 1 + ǫ ) T f , A ( r ) d j j =1 Min Ru The β constant appeared in algebraic and complex geometry
Theorem (Ru-Vojta, Amer. J. Math., 2020). Let X be a smooth complex projective variety and let D 1 , . . . , D q be effective Cartier divisors in general position. Let D = D 1 + · · · + D q . Let L be a line sheaf on X with h 0 ( L N ) ≥ 1 for N big enough. Let f : C → X be a holomorphic map with Zariski image. Then, for every ǫ > 0, q � β j ( L , D j ) m f ( r , D j ) ≤ exc (1 + ǫ ) T f , L ( r ) j =1 where m ≥ 1 dim H 0 ( X , L N ( − mD )) � β ( L , D ) = lim sup . N dim H 0 ( X , L N ) N → + ∞ Min Ru The β constant appeared in algebraic and complex geometry
Theorem (Ru-Vojta, Amer. J. Math., 2020). Let X be a smooth complex projective variety and let D 1 , . . . , D q be effective Cartier divisors in general position. Let D = D 1 + · · · + D q . Let L be a line sheaf on X with h 0 ( L N ) ≥ 1 for N big enough. Let f : C → X be a holomorphic map with Zariski image. Then, for every ǫ > 0, q � β j ( L , D j ) m f ( r , D j ) ≤ exc (1 + ǫ ) T f , L ( r ) j =1 where m ≥ 1 dim H 0 ( X , L N ( − mD )) � β ( L , D ) = lim sup . N dim H 0 ( X , L N ) N → + ∞ q In the case when D j ∼ A , then β ( D , D j ) = n +1 , where D = D 1 + · · · + D q . Min Ru The β constant appeared in algebraic and complex geometry
The proof is based on the following basic theorem, which is basically a reformulation of Cartan’s theorem above: Min Ru The β constant appeared in algebraic and complex geometry
The proof is based on the following basic theorem, which is basically a reformulation of Cartan’s theorem above: The Basic Theorem. Let X be a complex projective variety and let L be a line sheaf on X with dim H 0 ( X , L ) ≥ 1. Let s 1 , . . . , s q ∈ H 0 ( X , L ). Let f : C → X be a holomorphic map with Zariski-dense image. Then, for any ǫ > 0, � 2 π λ s j ( f ( re i θ )) d θ � 2 π ≤ exc (dim H 0 ( X , L ) + ǫ ) T f , L ( r ) max J 0 j ∈ J where the set J ranges over all subsets of { 1 , . . . , q } such that the sections ( s j ) j ∈ J are linearly independent. Min Ru The β constant appeared in algebraic and complex geometry
The proof is based on the following basic theorem, which is basically a reformulation of Cartan’s theorem above: The Basic Theorem. Let X be a complex projective variety and let L be a line sheaf on X with dim H 0 ( X , L ) ≥ 1. Let s 1 , . . . , s q ∈ H 0 ( X , L ). Let f : C → X be a holomorphic map with Zariski-dense image. Then, for any ǫ > 0, � 2 π λ s j ( f ( re i θ )) d θ � 2 π ≤ exc (dim H 0 ( X , L ) + ǫ ) T f , L ( r ) max J 0 j ∈ J where the set J ranges over all subsets of { 1 , . . . , q } such that the sections ( s j ) j ∈ J are linearly independent. Note: The D ∼ Q L is of 1 m-basis type if D := � s ∈ B ( s ), where B is a basis of mN m H 0 ( X , L ⊗ m ), where N m = dim H 0 ( X , L ⊗ m ). Min Ru The β constant appeared in algebraic and complex geometry
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