8月19日-报告2-汝敏-K-stability and Nevanlinna-Diophantine theory.pdf

K-stability and Nevanlinna-Diophantine theory Min Ru University of Houston TX, USA Min Ru K-stability and Nevanlinna-Diophantine theory Introduction In the most recent issue (August 2022) of Notice of AMS, Chenyang Xu wrote a very nice survey article entitled K-stablility: The recent interaction between algebraic and complex geometry. Min Ru K-stability and Nevanlinna-Diophantine theory Introduction In the most recent issue (August 2022) of Notice of AMS, Chenyang Xu wrote a very nice survey article entitled K-stablility: The recent interaction between algebraic and complex geometry. Motivated by his article, I will describe part of contents he mentioned, Min Ru K-stability and Nevanlinna-Diophantine theory Introduction In the most recent issue (August 2022) of Notice of AMS, Chenyang Xu wrote a very nice survey article entitled K-stablility: The recent interaction between algebraic and complex geometry. Motivated by his article, I will describe part of contents he mentioned, and explore the still somewhat mysterious connection of its notion with Nevanlinna theory (Diophantine approximation). Min Ru K-stability and Nevanlinna-Diophantine theory Introduction In the most recent issue (August 2022) of Notice of AMS, Chenyang Xu wrote a very nice survey article entitled K-stablility: The recent interaction between algebraic and complex geometry. Motivated by his article, I will describe part of contents he mentioned, and explore the still somewhat mysterious connection of its notion with Nevanlinna theory (Diophantine approximation). This talk is based on the recent paper of Yan He and Min Ru: The stability threshold and Diophantine approximation, Proc. A.M.S., 2022. Min Ru K-stability and Nevanlinna-Diophantine theory Canonical metrics and stability A metric form ω on a complex manifold X is said to be Kähler if ω is a closed 2-form. Min Ru K-stability and Nevanlinna-Diophantine theory Canonical metrics and stability A metric form ω on a complex manifold X is said to be Kähler if ω is a closed 2-form. Its class [ω] ∈ H 2 (X , R) is a topological invariant. Min Ru K-stability and Nevanlinna-Diophantine theory Canonical metrics and stability A metric form ω on a complex manifold X is said to be Kähler if ω is a closed 2-form. Its class [ω] ∈ H 2 (X , R) is a topological invariant. For a Kähler form ω, one can attach the Ricci form Ric(ω). Min Ru K-stability and Nevanlinna-Diophantine theory Canonical metrics and stability A metric form ω on a complex manifold X is said to be Kähler if ω is a closed 2-form. Its class [ω] ∈ H 2 (X , R) is a topological invariant. For a Kähler form ω, one can attach the Ricci form Ric(ω). It is also a closed 2-form and a remarkable fact is that its class [Ric(ω)] is the first Chern class c1 (X ). Min Ru K-stability and Nevanlinna-Diophantine theory Canonical metrics and stability A metric form ω on a complex manifold X is said to be Kähler if ω is a closed 2-form. Its class [ω] ∈ H 2 (X , R) is a topological invariant. For a Kähler form ω, one can attach the Ricci form Ric(ω). It is also a closed 2-form and a remarkable fact is that its class [Ric(ω)] is the first Chern class c1 (X ). Around the 50s, two fundamental questions became central in complex geometry. Min Ru K-stability and Nevanlinna-Diophantine theory Canonical metrics and stability A metric form ω on a complex manifold X is said to be Kähler if ω is a closed 2-form. Its class [ω] ∈ H 2 (X , R) is a topological invariant. For a Kähler form ω, one can attach the Ricci form Ric(ω). It is also a closed 2-form and a remarkable fact is that its class [Ric(ω)] is the first Chern class c1 (X ). Around the 50s, two fundamental questions became central in complex geometry. The first one is Min Ru K-stability and Nevanlinna-Diophantine theory Canonical metrics and stability A metric form ω on a complex manifold X is said to be Kähler if ω is a closed 2-form. Its class [ω] ∈ H 2 (X , R) is a topological invariant. For a Kähler form ω, one can attach the Ricci form Ric(ω). It is also a closed 2-form and a remarkable fact is that its class [Ric(ω)] is the first Chern class c1 (X ). Around the 50s, two fundamental questions became central in complex geometry. The first one is Calabi Conjecture. Given a compact Kähler manifold (X , ω) together with a 2-form R representing c1 (X ), one can always find a Kähler form ω̂ such that [ω] = [ω̂] and Ric (ω̂) = R. Min Ru K-stability and Nevanlinna-Diophantine theory Canonical metrics and stability A metric form ω on a complex manifold X is said to be Kähler if ω is a closed 2-form. Its class [ω] ∈ H 2 (X , R) is a topological invariant. For a Kähler form ω, one can attach the Ricci form Ric(ω). It is also a closed 2-form and a remarkable fact is that its class [Ric(ω)] is the first Chern class c1 (X ). Around the 50s, two fundamental questions became central in complex geometry. The first one is Calabi Conjecture. Given a compact Kähler manifold (X , ω) together with a 2-form R representing c1 (X ), one can always find a Kähler form ω̂ such that [ω] = [ω̂] and Ric (ω̂) = R. The conjecture was proved in Yau’s famous work in the late 70’s. Min Ru K-stability and Nevanlinna-Diophantine theory Canonical metrics and stability A metric form ω on a complex manifold X is said to be Kähler if ω is a closed 2-form. Its class [ω] ∈ H 2 (X , R) is a topological invariant. For a Kähler form ω, one can attach the Ricci form Ric(ω). It is also a closed 2-form and a remarkable fact is that its class [Ric(ω)] is the first Chern class c1 (X ). Around the 50s, two fundamental questions became central in complex geometry. The first one is Calabi Conjecture. Given a compact Kähler manifold (X , ω) together with a 2-form R representing c1 (X ), one can always find a Kähler form ω̂ such that [ω] = [ω̂] and Ric (ω̂) = R. The conjecture was proved in Yau’s famous work in the late 70’s. The second question is Min Ru K-stability and Nevanlinna-Diophantine theory Canonical metrics and stability A metric form ω on a complex manifold X is said to be Kähler if ω is a closed 2-form. Its class [ω] ∈ H 2 (X , R) is a topological invariant. For a Kähler form ω, one can attach the Ricci form Ric(ω). It is also a closed 2-form and a remarkable fact is that its class [Ric(ω)] is the first Chern class c1 (X ). Around the 50s, two fundamental questions became central in complex geometry. The first one is Calabi Conjecture. Given a compact Kähler manifold (X , ω) together with a 2-form R representing c1 (X ), one can always find a Kähler form ω̂ such that [ω] = [ω̂] and Ric (ω̂) = R. The conjecture was proved in Yau’s famous work in the late 70’s. The second question is K-E question. Does there always exists a Kähler from ωKE on X such that Ric (ωKE ) = λωKE ? Min Ru K-stability and Nevanlinna-Diophantine theory Canonical metrics and stability A metric form ω on a complex manifold X is said to be Kähler if ω is a closed 2-form. Its class [ω] ∈ H 2 (X , R) is a topological invariant. For a Kähler form ω, one can attach the Ricci form Ric(ω). It is also a closed 2-form and a remarkable fact is that its class [Ric(ω)] is the first Chern class c1 (X ). Around the 50s, two fundamental questions became central in complex geometry. The first one is Calabi Conjecture. Given a compact Kähler manifold (X , ω) together with a 2-form R representing c1 (X ), one can always find a Kähler form ω̂ such that [ω] = [ω̂] and Ric (ω̂) = R. The conjecture was proved in Yau’s famous work in the late 70’s. The second question is K-E question. Does there always exists a Kähler from ωKE on X such that Ric (ωKE ) = λωKE ? Note, in the class level, c1 (X ) = λ[ω] for λ = 0, 1, −1. Min Ru K-stability and Nevanlinna-Diophantine theory Canonical metrics and stability A metric form ω on a complex manifold X is said to be Kähler if ω is a closed 2-form. Its class [ω] ∈ H 2 (X , R) is a topological invariant. For a Kähler form ω, one can attach the Ricci form Ric(ω). It is also a closed 2-form and a remarkable fact is that its class [Ric(ω)] is the first Chern class c1 (X ). Around the 50s, two fundamental questions became central in complex geometry. The first one is Calabi Conjecture. Given a compact Kähler manifold (X , ω) together with a 2-form R representing c1 (X ), one can always find a Kähler form ω̂ such that [ω] = [ω̂] and Ric (ω̂) = R. The conjecture was proved in Yau’s famous work in the late 70’s. The second question is K-E question. Does there always exists a Kähler from ωKE on X such that Ric (ωKE ) = λωKE ? Note, in the class level, c1 (X ) = λ[ω] for λ = 0, 1, −1. Min Ru K-stability and Nevanlinna-Diophantine theory The case λ = 0, it is true by the solution of Calabi conjecture, when λ = −1, it was proved by Aubin and Yau. Min Ru K-stability and Nevanlinna-Diophantine theory The case λ = 0, it is true by the solution of Calabi conjecture, when λ = −1, it was proved by Aubin and Yau. The case when λ = 1, X is called Fano. Min Ru K-stability and Nevanlinna-Diophantine theory The case λ = 0, it is true by the solution of Calabi conjecture, when λ = −1, it was proved by Aubin and Yau. The case when λ = 1, X is called Fano. In this case, problem becomes more subtle and there is no definite answer. Min Ru K-stability and Nevanlinna-Diophantine theory The case λ = 0, it is true by the solution of Calabi conjecture, when λ = −1, it was proved by Aubin and Yau. The case when λ = 1, X is called Fano. In this case, problem becomes more subtle and there is no definite answer. In late 90’s, Tian introduced the notion of K-stability. Min Ru K-stability and Nevanlinna-Diophantine theory The case λ = 0, it is true by the solution of Calabi conjecture, when λ = −1, it was proved by Aubin and Yau. The case when λ = 1, X is called Fano. In this case, problem becomes more subtle and there is no definite answer. In late 90’s, Tian introduced the notion of K-stability. This was later reformulated in a purely algebro-geometric form by Donaldson. Min Ru K-stability and Nevanlinna-Diophantine theory The case λ = 0, it is true by the solution of Calabi conjecture, when λ = −1, it was proved by Aubin and Yau. The case when λ = 1, X is called Fano. In this case, problem becomes more subtle and there is no definite answer. In late 90’s, Tian introduced the notion of K-stability. This was later reformulated in a purely algebro-geometric form by Donaldson. When the base field is the complex number field, it was recently established (by Xiuxiong Chen, Simon Donaldson, and Song Sun, 2012) that the existence of positive scalar curvature Kähler-Einstein metric is actually equivalent to the K-stability condition. Min Ru K-stability and Nevanlinna-Diophantine theory The case λ = 0, it is true by the solution of Calabi conjecture, when λ = −1, it was proved by Aubin and Yau. The case when λ = 1, X is called Fano. In this case, problem becomes more subtle and there is no definite answer. In late 90’s, Tian introduced the notion of K-stability. This was later reformulated in a purely algebro-geometric form by Donaldson. When the base field is the complex number field, it was recently established (by Xiuxiong Chen, Simon Donaldson, and Song Sun, 2012) that the existence of positive scalar curvature Kähler-Einstein metric is actually equivalent to the K-stability condition. Min Ru K-stability and Nevanlinna-Diophantine theory Valuative criterion of K-stability. The notion of the K-stability of Fano varieties is an algebro-geometric stability condition. Min Ru K-stability and Nevanlinna-Diophantine theory Valuative criterion of K-stability. The notion of the K-stability of Fano varieties is an algebro-geometric stability condition. An important problem in algebraic geometry is to find a simple criterion to test the K -stability of the variety X . Min Ru K-stability and Nevanlinna-Diophantine theory Valuative criterion of K-stability. The notion of the K-stability of Fano varieties is an algebro-geometric stability condition. An important problem in algebraic geometry is to find a simple criterion to test the K -stability of the variety X . One fundamental development is the equivalent description of the notions of K -stability, using the valuation over the function filed K (X ) (ordE f , where E is a irreducible divisor on X ) Min Ru K-stability and Nevanlinna-Diophantine theory The β-constant (in Ru-Vojta’s paper, Amer. J. Math. 2020): Min Ru K-stability and Nevanlinna-Diophantine theory The β-constant (in Ru-Vojta’s paper, Amer. J. Math. 2020): Let L be a holomorphic line bundle and D be an effective divisor on X . Min Ru K-stability and Nevanlinna-Diophantine theory The β-constant (in Ru-Vojta’s paper, Amer. J. Math. 2020): Let L be a holomorphic line bundle and D be an effective divisor on X . Let H 0 (X , mL) be the set of holomorphic sections of L⊗m . Min Ru K-stability and Nevanlinna-Diophantine theory The β-constant (in Ru-Vojta’s paper, Amer. J. Math. 2020): Let L be a holomorphic line bundle and D be an effective divisor on X . Let H 0 (X , mL) be the set of holomorphic sections of L⊗m . Write h0 (L) = dim H 0 (X , L). Min Ru K-stability and Nevanlinna-Diophantine theory The β-constant (in Ru-Vojta’s paper, Amer. J. Math. 2020): Let L be a holomorphic line bundle and D be an effective divisor on X . Let H 0 (X , mL) be the set of holomorphic sections of L⊗m . Write h0 (L) = dim H 0 (X , L). P Define β(L, D) := lim supm→∞ 0 t≥1 h (mL−tD) mh0 (mL) Min Ru . K-stability and Nevanlinna-Diophantine theory The β-constant (in Ru-Vojta’s paper, Amer. J. Math. 2020): Let L be a holomorphic line bundle and D be an effective divisor on X . Let H 0 (X , mL) be the set of holomorphic sections of L⊗m . Write h0 (L) = dim H 0 (X , L). P Define β(L, D) := lim supm→∞ 0 t≥1 h (mL−tD) mh0 (mL) . Regarding H 0 (X , mL − tD) ⊂ H 0 (X , mL) by s 7→ sD⊗t s, Min Ru K-stability and Nevanlinna-Diophantine theory The β-constant (in Ru-Vojta’s paper, Amer. J. Math. 2020): Let L be a holomorphic line bundle and D be an effective divisor on X . Let H 0 (X , mL) be the set of holomorphic sections of L⊗m . Write h0 (L) = dim H 0 (X , L). P Define β(L, D) := lim supm→∞ 0 t≥1 h (mL−tD) mh0 (mL) . Regarding H 0 (X , mL − tD) ⊂ H 0 (X , mL) by s 7→ sD⊗t s, we get a filtration: H 0 (X , mL) ⊇ H 0 (X , mL − D) ⊇ H 0 (X , mL − tD) · · · ⊇ . Min Ru K-stability and Nevanlinna-Diophantine theory The β-constant (in Ru-Vojta’s paper, Amer. J. Math. 2020): Let L be a holomorphic line bundle and D be an effective divisor on X . Let H 0 (X , mL) be the set of holomorphic sections of L⊗m . Write h0 (L) = dim H 0 (X , L). P Define β(L, D) := lim supm→∞ 0 t≥1 h (mL−tD) mh0 (mL) . Regarding H 0 (X , mL − tD) ⊂ H 0 (X , mL) by s 7→ sD⊗t s, we get a filtration: H 0 (X , mL) ⊇ H 0 (X , mL − D) ⊇ H 0 (X , mL −P tD) · · · ⊇ . Assume 1 m D is irreducible, then β(L, D) = limm→∞ Nm N i=1 ordD (si ), where {s1 , . . . , sNm } is a basis of H 0 (X , mL) according to this filtration. Min Ru K-stability and Nevanlinna-Diophantine theory The β-constant (in Ru-Vojta’s paper, Amer. J. Math. 2020): Let L be a holomorphic line bundle and D be an effective divisor on X . Let H 0 (X , mL) be the set of holomorphic sections of L⊗m . Write h0 (L) = dim H 0 (X , L). P Define β(L, D) := lim supm→∞ 0 t≥1 h (mL−tD) mh0 (mL) . Regarding H 0 (X , mL − tD) ⊂ H 0 (X , mL) by s 7→ sD⊗t s, we get a filtration: H 0 (X , mL) ⊇ H 0 (X , mL − D) ⊇ H 0 (X , mL −P tD) · · · ⊇ . Assume 1 m D is irreducible, then β(L, D) = limm→∞ Nm N i=1 ordD (si ), where {s1 , . . . , sNm } is a basis of H 0 (X , mL) according P mto this filtration. Indeed, if we let Sm (D) := sup{s1 ,...,sNm } N1m N i=1 ordD (si ), where sup runs all basis, Min Ru K-stability and Nevanlinna-Diophantine theory The β-constant (in Ru-Vojta’s paper, Amer. J. Math. 2020): Let L be a holomorphic line bundle and D be an effective divisor on X . Let H 0 (X , mL) be the set of holomorphic sections of L⊗m . Write h0 (L) = dim H 0 (X , L). P Define β(L, D) := lim supm→∞ 0 t≥1 h (mL−tD) mh0 (mL) . Regarding H 0 (X , mL − tD) ⊂ H 0 (X , mL) by s 7→ sD⊗t s, we get a filtration: H 0 (X , mL) ⊇ H 0 (X , mL − D) ⊇ H 0 (X , mL −P tD) · · · ⊇ . Assume 1 m D is irreducible, then β(L, D) = limm→∞ Nm N i=1 ordD (si ), where {s1 , . . . , sNm } is a basis of H 0 (X , mL) according P mto this filtration. Indeed, if we let Sm (D) := sup{s1 ,...,sNm } N1m N i=1 ordD (si ), where sup runs all basis, then the sup is achieved by a basis of filtration. Min Ru K-stability and Nevanlinna-Diophantine theory The β-constant (in Ru-Vojta’s paper, Amer. J. Math. 2020): Let L be a holomorphic line bundle and D be an effective divisor on X . Let H 0 (X , mL) be the set of holomorphic sections of L⊗m . Write h0 (L) = dim H 0 (X , L). P Define β(L, D) := lim supm→∞ 0 t≥1 h (mL−tD) mh0 (mL) . Regarding H 0 (X , mL − tD) ⊂ H 0 (X , mL) by s 7→ sD⊗t s, we get a filtration: H 0 (X , mL) ⊇ H 0 (X , mL − D) ⊇ H 0 (X , mL −P tD) · · · ⊇ . Assume 1 m D is irreducible, then β(L, D) = limm→∞ Nm N i=1 ordD (si ), where {s1 , . . . , sNm } is a basis of H 0 (X , mL) according P mto this filtration. Indeed, if we let Sm (D) := sup{s1 ,...,sNm } N1m N i=1 ordD (si ), where sup runs all basis, then the sup is achieved by a basis of filtration. so β(L, D) = limm→∞ Sm (D). Min Ru K-stability and Nevanlinna-Diophantine theory The β-constant (in Ru-Vojta’s paper, Amer. J. Math. 2020): Let L be a holomorphic line bundle and D be an effective divisor on X . Let H 0 (X , mL) be the set of holomorphic sections of L⊗m . Write h0 (L) = dim H 0 (X , L). P Define β(L, D) := lim supm→∞ 0 t≥1 h (mL−tD) mh0 (mL) . Regarding H 0 (X , mL − tD) ⊂ H 0 (X , mL) by s 7→ sD⊗t s, we get a filtration: H 0 (X , mL) ⊇ H 0 (X , mL − D) ⊇ H 0 (X , mL −P tD) · · · ⊇ . Assume 1 m D is irreducible, then β(L, D) = limm→∞ Nm N i=1 ordD (si ), where {s1 , . . . , sNm } is a basis of H 0 (X , mL) according P mto this filtration. Indeed, if we let Sm (D) := sup{s1 ,...,sNm } N1m N i=1 ordD (si ), where sup runs all basis, then the sup is achieved by a basis of filtration. so β(L,RD) = limm→∞ Sm (D). We also have ∞ 1 β(L, D) = Vol(L) 0 Vol(L − tD)dt, where 0 (X ,mL) Vol(L) = lim supm→∞ dim H mn /n! Min Ru K-stability and Nevanlinna-Diophantine theory The β-constant (in Ru-Vojta’s paper, Amer. J. Math. 2020): Let L be a holomorphic line bundle and D be an effective divisor on X . Let H 0 (X , mL) be the set of holomorphic sections of L⊗m . Write h0 (L) = dim H 0 (X , L). P Define β(L, D) := lim supm→∞ 0 t≥1 h (mL−tD) mh0 (mL) . Regarding H 0 (X , mL − tD) ⊂ H 0 (X , mL) by s 7→ sD⊗t s, we get a filtration: H 0 (X , mL) ⊇ H 0 (X , mL − D) ⊇ H 0 (X , mL −P tD) · · · ⊇ . Assume 1 m D is irreducible, then β(L, D) = limm→∞ Nm N i=1 ordD (si ), where {s1 , . . . , sNm } is a basis of H 0 (X , mL) according P mto this filtration. Indeed, if we let Sm (D) := sup{s1 ,...,sNm } N1m N i=1 ordD (si ), where sup runs all basis, then the sup is achieved by a basis of filtration. so β(L,RD) = limm→∞ Sm (D). We also have ∞ 1 β(L, D) = Vol(L) 0 Vol(L − tD)dt, where 0 (X ,mL) Vol(L) = lim supm→∞ dim H (Note: Vol(kL) = k n Vol(L) so mn /n! the volume function can be extended to Q-divisors. Min Ru K-stability and Nevanlinna-Diophantine theory The β-constant (in Ru-Vojta’s paper, Amer. J. Math. 2020): Let L be a holomorphic line bundle and D be an effective divisor on X . Let H 0 (X , mL) be the set of holomorphic sections of L⊗m . Write h0 (L) = dim H 0 (X , L). P Define β(L, D) := lim supm→∞ 0 t≥1 h (mL−tD) mh0 (mL) . Regarding H 0 (X , mL − tD) ⊂ H 0 (X , mL) by s 7→ sD⊗t s, we get a filtration: H 0 (X , mL) ⊇ H 0 (X , mL − D) ⊇ H 0 (X , mL −P tD) · · · ⊇ . Assume 1 m D is irreducible, then β(L, D) = limm→∞ Nm N i=1 ordD (si ), where {s1 , . . . , sNm } is a basis of H 0 (X , mL) according P mto this filtration. Indeed, if we let Sm (D) := sup{s1 ,...,sNm } N1m N i=1 ordD (si ), where sup runs all basis, then the sup is achieved by a basis of filtration. so β(L,RD) = limm→∞ Sm (D). We also have ∞ 1 β(L, D) = Vol(L) 0 Vol(L − tD)dt, where 0 (X ,mL) Vol(L) = lim supm→∞ dim H (Note: Vol(kL) = k n Vol(L) so mn /n! the volume function can be extended to Q-divisors. Also note that Vol( ) depends only on the numerical class of L, so it is defined on NS(X ) := Div (X )/Num(X ) and extends uniquely to a continuous function on NS(X )R ). Min Ru K-stability and Nevanlinna-Diophantine theory Valuative criterion: Min Ru K-stability and Nevanlinna-Diophantine theory Valuative criterion: In 2015, Fujita showed that if (Fano) X is K -(semi) stable, then β(−KX , D) < 1 (resp. β(−KX , D) ≤ 1) for any nonzero effective divisor D on X . Min Ru K-stability and Nevanlinna-Diophantine theory Valuative criterion: In 2015, Fujita showed that if (Fano) X is K -(semi) stable, then β(−KX , D) < 1 (resp. β(−KX , D) ≤ 1) for any nonzero effective divisor D on X . Fujita and C. Li (with a technical assumption, which were removed by Blum-Xu) independently proved that it is indeed an equivalence condition if one goes to the birational model, Min Ru K-stability and Nevanlinna-Diophantine theory Valuative criterion: In 2015, Fujita showed that if (Fano) X is K -(semi) stable, then β(−KX , D) < 1 (resp. β(−KX , D) ≤ 1) for any nonzero effective divisor D on X . Fujita and C. Li (with a technical assumption, which were removed by Blum-Xu) independently proved that it is indeed an equivalence condition if one goes to the birational model, i.e. the Q-fano variety X is K AX (E ) > 1 for any prime divisors E over X stable if and only if β(−K X ,E ) (i.e. E is a prime divisor on a birational model π : X̃ → X ), Min Ru K-stability and Nevanlinna-Diophantine theory Valuative criterion: In 2015, Fujita showed that if (Fano) X is K -(semi) stable, then β(−KX , D) < 1 (resp. β(−KX , D) ≤ 1) for any nonzero effective divisor D on X . Fujita and C. Li (with a technical assumption, which were removed by Blum-Xu) independently proved that it is indeed an equivalence condition if one goes to the birational model, i.e. the Q-fano variety X is K AX (E ) > 1 for any prime divisors E over X stable if and only if β(−K X ,E ) (i.e. E is a prime divisor on a birational model π : X̃ → X ), where AX (E ) := 1 + ordE (KY /X ) and is called the log discrepancy. Min Ru K-stability and Nevanlinna-Diophantine theory Valuative criterion: In 2015, Fujita showed that if (Fano) X is K -(semi) stable, then β(−KX , D) < 1 (resp. β(−KX , D) ≤ 1) for any nonzero effective divisor D on X . Fujita and C. Li (with a technical assumption, which were removed by Blum-Xu) independently proved that it is indeed an equivalence condition if one goes to the birational model, i.e. the Q-fano variety X is K AX (E ) > 1 for any prime divisors E over X stable if and only if β(−K X ,E ) (i.e. E is a prime divisor on a birational model π : X̃ → X ), where AX (E ) := 1 + ordE (KY /X ) and is called the log discrepancy. X is said to have klt singularities if AX (E ) > 0 for all prime divisors over X . Min Ru K-stability and Nevanlinna-Diophantine theory Valuative criterion: In 2015, Fujita showed that if (Fano) X is K -(semi) stable, then β(−KX , D) < 1 (resp. β(−KX , D) ≤ 1) for any nonzero effective divisor D on X . Fujita and C. Li (with a technical assumption, which were removed by Blum-Xu) independently proved that it is indeed an equivalence condition if one goes to the birational model, i.e. the Q-fano variety X is K AX (E ) > 1 for any prime divisors E over X stable if and only if β(−K X ,E ) (i.e. E is a prime divisor on a birational model π : X̃ → X ), where AX (E ) := 1 + ordE (KY /X ) and is called the log discrepancy. X is said to have klt singularities if AX (E ) > 0 for all prime divisors AX (E ) over X . We call δ(L) = inf E β(L,E ) the stability threshold. Min Ru K-stability and Nevanlinna-Diophantine theory Valuative criterion: In 2015, Fujita showed that if (Fano) X is K -(semi) stable, then β(−KX , D) < 1 (resp. β(−KX , D) ≤ 1) for any nonzero effective divisor D on X . Fujita and C. Li (with a technical assumption, which were removed by Blum-Xu) independently proved that it is indeed an equivalence condition if one goes to the birational model, i.e. the Q-fano variety X is K AX (E ) > 1 for any prime divisors E over X stable if and only if β(−K X ,E ) (i.e. E is a prime divisor on a birational model π : X̃ → X ), where AX (E ) := 1 + ordE (KY /X ) and is called the log discrepancy. X is said to have klt singularities if AX (E ) > 0 for all prime divisors AX (E ) over X . We call δ(L) = inf E β(L,E ) the stability threshold. Valuative criterion of K -stability. 1. X is uniformly K -stable (resp. semi-satble) if and only if δ(−KX ) > 1 (resp. ≥ 1) (Fuji-Li). Min Ru K-stability and Nevanlinna-Diophantine theory Valuative criterion: In 2015, Fujita showed that if (Fano) X is K -(semi) stable, then β(−KX , D) < 1 (resp. β(−KX , D) ≤ 1) for any nonzero effective divisor D on X . Fujita and C. Li (with a technical assumption, which were removed by Blum-Xu) independently proved that it is indeed an equivalence condition if one goes to the birational model, i.e. the Q-fano variety X is K AX (E ) > 1 for any prime divisors E over X stable if and only if β(−K X ,E ) (i.e. E is a prime divisor on a birational model π : X̃ → X ), where AX (E ) := 1 + ordE (KY /X ) and is called the log discrepancy. X is said to have klt singularities if AX (E ) > 0 for all prime divisors AX (E ) over X . We call δ(L) = inf E β(L,E ) the stability threshold. Valuative criterion of K -stability. 1. X is uniformly K -stable (resp. semi-satble) if and only if δ(−KX ) > 1 (resp. ≥ 1) (Fuji-Li). 2. X is K -stable if and only if AX (E ) > β(−KX , E ) for any E (Blum-Xu). Min Ru K-stability and Nevanlinna-Diophantine theory K -stability through the base type divisor Blum-Jonsson (Blum-Jonsson, Advances in Math., 2020) used m-basis type to describe the stability threshold AX (E ) δ(L) = inf E β(L,E ). Min Ru K-stability and Nevanlinna-Diophantine theory K -stability through the base type divisor Blum-Jonsson (Blum-Jonsson, Advances in Math., 2020) used m-basis type to describe the stability threshold AX (E ) δ(L) = inf E β(L,E ) . For m sufficient large, we say D is a m-basis 1 ((s1 ) + · · · + (sNm )) where {s1 , . . . , sNm } type divisor if D = mN m 0 forms a basis of H (X , mL). Min Ru K-stability and Nevanlinna-Diophantine theory K -stability through the base type divisor Blum-Jonsson (Blum-Jonsson, Advances in Math., 2020) used m-basis type to describe the stability threshold AX (E ) δ(L) = inf E β(L,E ) . For m sufficient large, we say D is a m-basis 1 ((s1 ) + · · · + (sNm )) where {s1 , . . . , sNm } type divisor if D = mN m 0 forms a basis of H (X , mL). Recall the algebraic geometry AX (E ) definition of “log canonical threshold”: lct(D) = minE ord . E (D) Min Ru K-stability and Nevanlinna-Diophantine theory K -stability through the base type divisor Blum-Jonsson (Blum-Jonsson, Advances in Math., 2020) used m-basis type to describe the stability threshold AX (E ) δ(L) = inf E β(L,E ) . For m sufficient large, we say D is a m-basis 1 ((s1 ) + · · · + (sNm )) where {s1 , . . . , sNm } type divisor if D = mN m 0 forms a basis of H (X , mL). Recall the algebraic geometry AX (E ) definition of “log canonical threshold”: lct(D) = minE ord . E (D) AX (E ) (E ) Let δm (L) := inf D lct(D) = inf D minE ord = inf E ASmX (E ) , where E (D) D ∼Q L runs through over all m-basis type divisors. Min Ru K-stability and Nevanlinna-Diophantine theory K -stability through the base type divisor Blum-Jonsson (Blum-Jonsson, Advances in Math., 2020) used m-basis type to describe the stability threshold AX (E ) δ(L) = inf E β(L,E ) . For m sufficient large, we say D is a m-basis 1 ((s1 ) + · · · + (sNm )) where {s1 , . . . , sNm } type divisor if D = mN m 0 forms a basis of H (X , mL). Recall the algebraic geometry AX (E ) definition of “log canonical threshold”: lct(D) = minE ord . E (D) AX (E ) (E ) Let δm (L) := inf D lct(D) = inf D minE ord = inf E ASmX (E ) , where E (D) D ∼Q L runs through over all m-basis type divisors. Since limm→∞ Sm (E ) = β(L, E ) (as we described earlier), Blum-Jonsson proved that limm→ δm (L) = δ(L). Min Ru K-stability and Nevanlinna-Diophantine theory K -stability through the base type divisor Blum-Jonsson (Blum-Jonsson, Advances in Math., 2020) used m-basis type to describe the stability threshold AX (E ) δ(L) = inf E β(L,E ) . For m sufficient large, we say D is a m-basis 1 ((s1 ) + · · · + (sNm )) where {s1 , . . . , sNm } type divisor if D = mN m 0 forms a basis of H (X , mL). Recall the algebraic geometry AX (E ) definition of “log canonical threshold”: lct(D) = minE ord . E (D) AX (E ) (E ) Let δm (L) := inf D lct(D) = inf D minE ord = inf E ASmX (E ) , where E (D) D ∼Q L runs through over all m-basis type divisors. Since limm→∞ Sm (E ) = β(L, E ) (as we described earlier), Blum-Jonsson proved that limm→ δm (L) = δ(L). We note that this gives us a way to verify K -stability for explicit Fano varieties, by estimating δm (−KX ). Min Ru K-stability and Nevanlinna-Diophantine theory The log canonical threshold through singular metric Tian in 1987 introduced α(L) the log canonical threshold of L as follows: Min Ru K-stability and Nevanlinna-Diophantine theory The log canonical threshold through singular metric Tian in 1987 introduced α(L) the log canonical threshold of L as follows: √ Let h = e −ϕ be a singular metric with ΘL,h ≥ 0, where ΘL,h = π−1 ∂ ∂¯ log ϕ. Min Ru K-stability and Nevanlinna-Diophantine theory The log canonical threshold through singular metric Tian in 1987 introduced α(L) the log canonical threshold of L as follows: √ Let h = e −ϕ be a singular metric with ΘL,h ≥ 0, where ΘL,h = π−1 ∂ ∂¯ log ϕ. Define cp (h) = sup{c | e −2cϕ is locally integrable at p}. Min Ru K-stability and Nevanlinna-Diophantine theory The log canonical threshold through singular metric Tian in 1987 introduced α(L) the log canonical threshold of L as follows: √ Let h = e −ϕ be a singular metric with ΘL,h ≥ 0, where ΘL,h = π−1 ∂ ∂¯ log ϕ. Define cp (h) = sup{c | e −2cϕ is locally integrable at p}. Define, for p ∈ X , αp (L) = inf h:ΘL,h ≥0 cp (h) Min Ru K-stability and Nevanlinna-Diophantine theory The log canonical threshold through singular metric Tian in 1987 introduced α(L) the log canonical threshold of L as follows: √ Let h = e −ϕ be a singular metric with ΘL,h ≥ 0, where ΘL,h = π−1 ∂ ∂¯ log ϕ. Define cp (h) = sup{c | e −2cϕ is locally integrable at p}. Define, for p ∈ X , αp (L) = inf h:ΘL,h ≥0 cp (h) and α(L) = inf p∈X αp (L). Min Ru K-stability and Nevanlinna-Diophantine theory The log canonical threshold through singular metric Tian in 1987 introduced α(L) the log canonical threshold of L as follows: √ Let h = e −ϕ be a singular metric with ΘL,h ≥ 0, where ΘL,h = π−1 ∂ ∂¯ log ϕ. Define cp (h) = sup{c | e −2cϕ is locally integrable at p}. Define, for p ∈ X , αp (L) = inf h:ΘL,h ≥0 cp (h) n , and α(L) = inf p∈X αp (L). Tian proved that if α(−KX ) > n+1 then X is K -stable. Min Ru K-stability and Nevanlinna-Diophantine theory The log canonical threshold through singular metric Tian in 1987 introduced α(L) the log canonical threshold of L as follows: √ Let h = e −ϕ be a singular metric with ΘL,h ≥ 0, where ΘL,h = π−1 ∂ ∂¯ log ϕ. Define cp (h) = sup{c | e −2cϕ is locally integrable at p}. Define, for p ∈ X , αp (L) = inf h:ΘL,h ≥0 cp (h) n , and α(L) = inf p∈X αp (L). Tian proved that if α(−KX ) > n+1 then X is K -stable. Let D be an effective Cartier divisor, Min Ru K-stability and Nevanlinna-Diophantine theory The log canonical threshold through singular metric Tian in 1987 introduced α(L) the log canonical threshold of L as follows: √ Let h = e −ϕ be a singular metric with ΘL,h ≥ 0, where ΘL,h = π−1 ∂ ∂¯ log ϕ. Define cp (h) = sup{c | e −2cϕ is locally integrable at p}. Define, for p ∈ X , αp (L) = inf h:ΘL,h ≥0 cp (h) n , and α(L) = inf p∈X αp (L). Tian proved that if α(−KX ) > n+1 then X is K -stable. Let D be an effective Cartier divisor, then the canonical section sD of [D] gives a singular metric on [D] with ϕ := log |sD |. Min Ru K-stability and Nevanlinna-Diophantine theory The log canonical threshold through singular metric Tian in 1987 introduced α(L) the log canonical threshold of L as follows: √ Let h = e −ϕ be a singular metric with ΘL,h ≥ 0, where ΘL,h = π−1 ∂ ∂¯ log ϕ. Define cp (h) = sup{c | e −2cϕ is locally integrable at p}. Define, for p ∈ X , αp (L) = inf h:ΘL,h ≥0 cp (h) n , and α(L) = inf p∈X αp (L). Tian proved that if α(−KX ) > n+1 then X is K -stable. Let D be an effective Cartier divisor, then the canonical section sD of [D] gives a singular metric on [D] with ϕ := log |sD |. We denote lctp (D) := cp (h) and lct(D) := inf p∈X lctp (D) with such metric. Min Ru K-stability and Nevanlinna-Diophantine theory The log canonical threshold through singular metric Tian in 1987 introduced α(L) the log canonical threshold of L as follows: √ Let h = e −ϕ be a singular metric with ΘL,h ≥ 0, where ΘL,h = π−1 ∂ ∂¯ log ϕ. Define cp (h) = sup{c | e −2cϕ is locally integrable at p}. Define, for p ∈ X , αp (L) = inf h:ΘL,h ≥0 cp (h) n , and α(L) = inf p∈X αp (L). Tian proved that if α(−KX ) > n+1 then X is K -stable. Let D be an effective Cartier divisor, then the canonical section sD of [D] gives a singular metric on [D] with ϕ := log |sD |. We denote lctp (D) := cp (h) and lct(D) := inf p∈X lctp (D) with such metric. Use the fact that, for ϕ = log |f |, e −2cϕ = |f 1|2c , Min Ru K-stability and Nevanlinna-Diophantine theory The log canonical threshold through singular metric Tian in 1987 introduced α(L) the log canonical threshold of L as follows: √ Let h = e −ϕ be a singular metric with ΘL,h ≥ 0, where ΘL,h = π−1 ∂ ∂¯ log ϕ. Define cp (h) = sup{c | e −2cϕ is locally integrable at p}. Define, for p ∈ X , αp (L) = inf h:ΘL,h ≥0 cp (h) n , and α(L) = inf p∈X αp (L). Tian proved that if α(−KX ) > n+1 then X is K -stable. Let D be an effective Cartier divisor, then the canonical section sD of [D] gives a singular metric on [D] with ϕ := log |sD |. We denote lctp (D) := cp (h) and lct(D) := inf p∈X lctp (D) with such metric. Use R the fact that, for ϕ = log |f |, e −2cϕ = |f 1|2c , and the fact that |z|1a2λ < ∞ iff λa − 1 < 0, i.e. λ < 1a , Min Ru K-stability and Nevanlinna-Diophantine theory The log canonical threshold through singular metric Tian in 1987 introduced α(L) the log canonical threshold of L as follows: √ Let h = e −ϕ be a singular metric with ΘL,h ≥ 0, where ΘL,h = π−1 ∂ ∂¯ log ϕ. Define cp (h) = sup{c | e −2cϕ is locally integrable at p}. Define, for p ∈ X , αp (L) = inf h:ΘL,h ≥0 cp (h) n , and α(L) = inf p∈X αp (L). Tian proved that if α(−KX ) > n+1 then X is K -stable. Let D be an effective Cartier divisor, then the canonical section sD of [D] gives a singular metric on [D] with ϕ := log |sD |. We denote lctp (D) := cp (h) and lct(D) := inf p∈X lctp (D) with such metric. Use R the fact that, for ϕ = log |f |, e −2cϕ = |f 1|2c , and the fact that |z|1a2λ < ∞ iff λa − 1 < 0, i.e. λ < 1a , this links with the (algebraic geometry) definition for lct(D). Min Ru K-stability and Nevanlinna-Diophantine theory The log canonical threshold through singular metric Tian in 1987 introduced α(L) the log canonical threshold of L as follows: √ Let h = e −ϕ be a singular metric with ΘL,h ≥ 0, where ΘL,h = π−1 ∂ ∂¯ log ϕ. Define cp (h) = sup{c | e −2cϕ is locally integrable at p}. Define, for p ∈ X , αp (L) = inf h:ΘL,h ≥0 cp (h) n , and α(L) = inf p∈X αp (L). Tian proved that if α(−KX ) > n+1 then X is K -stable. Let D be an effective Cartier divisor, then the canonical section sD of [D] gives a singular metric on [D] with ϕ := log |sD |. We denote lctp (D) := cp (h) and lct(D) := inf p∈X lctp (D) with such metric. Use R the fact that, for ϕ = log |f |, e −2cϕ = |f 1|2c , and the fact that |z|1a2λ < ∞ iff λa − 1 < 0, i.e. λ < 1a , this links with the (algebraic geometry) definition for lct(D). According to Demailly, α(L) = inf{lct(D) | D is effective, D ∼Q L}. Min Ru K-stability and Nevanlinna-Diophantine theory The log canonical threshold through singular metric Tian in 1987 introduced α(L) the log canonical threshold of L as follows: √ Let h = e −ϕ be a singular metric with ΘL,h ≥ 0, where ΘL,h = π−1 ∂ ∂¯ log ϕ. Define cp (h) = sup{c | e −2cϕ is locally integrable at p}. Define, for p ∈ X , αp (L) = inf h:ΘL,h ≥0 cp (h) n , and α(L) = inf p∈X αp (L). Tian proved that if α(−KX ) > n+1 then X is K -stable. Let D be an effective Cartier divisor, then the canonical section sD of [D] gives a singular metric on [D] with ϕ := log |sD |. We denote lctp (D) := cp (h) and lct(D) := inf p∈X lctp (D) with such metric. Use R the fact that, for ϕ = log |f |, e −2cϕ = |f 1|2c , and the fact that |z|1a2λ < ∞ iff λa − 1 < 0, i.e. λ < 1a , this links with the (algebraic geometry) definition for lct(D). According to Demailly, α(L) = inf{lct(D) | D is effective, D ∼Q L}. This allows purely algebro-geometric proofs of Käher-Einstein metrics. Min Ru K-stability and Nevanlinna-Diophantine theory Nevanlinna theory Min Ru K-stability and Nevanlinna-Diophantine theory Nevanlinna theory The Second Main Theorem(Nevanlinna, 1929). Min Ru K-stability and Nevanlinna-Diophantine theory Nevanlinna theory The Second Main Theorem(Nevanlinna, 1929). Let f be meromorphic (non-constant) on C and a1 , ..., aq ∈ C ∪ {∞} distinct. Min Ru K-stability and Nevanlinna-Diophantine theory Nevanlinna theory The Second Main Theorem(Nevanlinna, 1929). Let f be meromorphic (non-constant) on C and a1 , ..., aq ∈ C ∪ {∞} distinct. Then, for any Pϵq > 0, (q − 2 − ϵ)Tf (r ) ≤exc j=1 Nf (r , aj ), Min Ru K-stability and Nevanlinna-Diophantine theory Nevanlinna theory The Second Main Theorem(Nevanlinna, 1929). Let f be meromorphic (non-constant) on C and a1 , ..., aq ∈ C ∪ {∞} distinct. Then, for any Pϵq > 0, (q − 2 − ϵ)Tf (r ) ≤exc j=1 Nf (r , aj ), or equivalently q X mf (r , aj ) ≤exc (2 + ϵ)Tf (r ) , j=1 Min Ru K-stability and Nevanlinna-Diophantine theory Nevanlinna theory The Second Main Theorem(Nevanlinna, 1929). Let f be meromorphic (non-constant) on C and a1 , ..., aq ∈ C ∪ {∞} distinct. Then, for any Pϵq > 0, (q − 2 − ϵ)Tf (r ) ≤exc j=1 Nf (r , aj ), or equivalently q X mf (r , aj ) ≤exc (2 + ϵ)Tf (r ) , j=1 where ≤exc means that the inequality holds for r ∈ [0, +∞) outside a set E with finite measure. Min Ru K-stability and Nevanlinna-Diophantine theory Nevanlinna theory The Second Main Theorem(Nevanlinna, 1929). Let f be meromorphic (non-constant) on C and a1 , ..., aq ∈ C ∪ {∞} distinct. Then, for any Pϵq > 0, (q − 2 − ϵ)Tf (r ) ≤exc j=1 Nf (r , aj ), or equivalently q X mf (r , aj ) ≤exc (2 + ϵ)Tf (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 K-stability and Nevanlinna-Diophantine theory Nevanlinna theory The Second Main Theorem(Nevanlinna, 1929). Let f be meromorphic (non-constant) on C and a1 , ..., aq ∈ C ∪ {∞} distinct. Then, for any Pϵq > 0, (q − 2 − ϵ)Tf (r ) ≤exc j=1 Nf (r , aj ), or equivalently q X mf (r , aj ) ≤exc (2 + ϵ)Tf (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 K-stability and Nevanlinna-Diophantine theory Cartan’s Theorem (1933). Min Ru K-stability and Nevanlinna-Diophantine theory Cartan’s Theorem (1933). Let f : C → Pn (C) be a linearly non-degenerate holomorphic map. Min Ru K-stability and Nevanlinna-Diophantine theory Cartan’s Theorem (1933). Let f : C → Pn (C) be a linearly non-degenerate holomorphic map. Let H1 , . . . , Hq be the hyperplanes in general position on Pn (C). Min Ru K-stability and Nevanlinna-Diophantine theory Cartan’s Theorem (1933). Let f : C → Pn (C) be a linearly non-degenerate holomorphic map. Let H1 , . . . , Hq be the hyperplanes in general position on Pn (C). Then, for any ϵ > 0, P q j=1 mf (r , Hj ) ≤exc (n + 1 + ϵ)Tf (r ). Min Ru K-stability and Nevanlinna-Diophantine theory Cartan’s Theorem (1933). Let f : C → Pn (C) be a linearly non-degenerate holomorphic map. Let H1 , . . . , Hq be the hyperplanes in general position on Pn (C). Then, for any ϵ > 0, P q j=1 mf (r , Hj ) ≤exc (n + 1 + ϵ)Tf (r ). In 2004, Ru extended the above result to hypersurfaces for fP: C → Pn (C) with Zariski dense image. q 1 j=1 dj mf (r , Dj ) ≤exc (n + 1 + ϵ)Tf (r ). Min Ru K-stability and Nevanlinna-Diophantine theory Cartan’s Theorem (1933). Let f : C → Pn (C) be a linearly non-degenerate holomorphic map. Let H1 , . . . , Hq be the hyperplanes in general position on Pn (C). Then, for any ϵ > 0, P q j=1 mf (r , Hj ) ≤exc (n + 1 + ϵ)Tf (r ). In 2004, Ru extended the above result to hypersurfaces for fP: C → Pn (C) with Zariski dense image. q 1 j=1 dj mf (r , Dj ) ≤exc (n + 1 + ϵ)Tf (r ). Theorem (Ru, 2009). Let f : C → X be holo and Zariski dense, D1 , . . . , Dq be divisors in general position in X . Min Ru K-stability and Nevanlinna-Diophantine theory Cartan’s Theorem (1933). Let f : C → Pn (C) be a linearly non-degenerate holomorphic map. Let H1 , . . . , Hq be the hyperplanes in general position on Pn (C). Then, for any ϵ > 0, P q j=1 mf (r , Hj ) ≤exc (n + 1 + ϵ)Tf (r ). In 2004, Ru extended the above result to hypersurfaces for fP: C → Pn (C) with Zariski dense image. q 1 j=1 dj mf (r , Dj ) ≤exc (n + 1 + ϵ)Tf (r ). Theorem (Ru, 2009). Let f : C → X be holo and Zariski dense, D1 , . . . , Dq be divisors in general position in X . Assume that Dj ∼ dj A (A being ample). Then, for ∀ ϵ > 0, q X 1 mf (r , Dj ) ≤exc (dim X + 1 + ϵ)Tf ,A (r ). dj j=1 Min Ru K-stability and Nevanlinna-Diophantine theory Notations: λD (x) = − log ∥sD (x)∥ =− log distance from x to D (Weil function for D), Min Ru K-stability and Nevanlinna-Diophantine theory Notations: λD (x) = − log ∥sD (x)∥ =− log distance from x to D R 2π dθ (Proximity (Weil function for D), mf (r , D) = 0 λD (f (re iθ )) 2π function). Min Ru K-stability and Nevanlinna-Diophantine theory Notations: λD (x) = − log ∥sD (x)∥ =− log distance from x to D R 2π dθ (Proximity (Weil function for D), mf (r , D) = 0 λD (f (re iθ )) 2π R r dt R ∗ function). Tf ,L (r ) := 1 t |z| 0, q X βj (L , Dj )mf (r , Dj ) ≤exc (1 + ϵ)Tf ,L (r ) j=1 Min Ru K-stability and Nevanlinna-Diophantine theory Notations: λD (x) = − log ∥sD (x)∥ =− log distance from x to D R 2π dθ (Proximity (Weil function for D), mf (r , D) = 0 λD (f (re iθ )) 2π R r dt R ∗ function). Tf ,L (r ) := 1 t |z| 0, q X βj (L , Dj )mf (r , Dj ) ≤exc (1 + ϵ)Tf ,L (r ) j=1 where P β(L , D) = lim sup m≥1 dim H 0 (X , L N (−mD)) N dim H 0 (X , L N ) N→+∞ Min Ru . K-stability and Nevanlinna-Diophantine theory Notations: λD (x) = − log ∥sD (x)∥ =− log distance from x to D R 2π dθ (Proximity (Weil function for D), mf (r , D) = 0 λD (f (re iθ )) 2π R r dt R ∗ function). Tf ,L (r ) := 1 t |z| 0, q X βj (L , Dj )mf (r , Dj ) ≤exc (1 + ϵ)Tf ,L (r ) j=1 where P β(L , D) = lim sup m≥1 dim H 0 (X , L N (−mD)) N dim H 0 (X , L N ) N→+∞ . q In the case when Dj ∼ A, then β(D, Dj ) = n+1 , where D = D1 + · · · + Dq . Min Ru K-stability and Nevanlinna-Diophantine theory Diophantine approximation Theorem (Ru-Vojta, 2020) [Arithmetic Part] Let X be a projective variety over a number field k, and D1 , . . . , Dq be effective Cartier divisors intersecting properly on X . Let L be a line bundle on X with h0 (LN ) ≥ 1 for N big enough. Let S ⊂ Mk be a finite set of places. Then, for every ϵ > 0, the inequality q X β(L, Dj )mS (x, Dj ) ≤ (1 + ϵ)hL (x) i=1 holds for all k-rational points outside a proper Zariski-closed subset of X . Min Ru K-stability and Nevanlinna-Diophantine theory The proof is based on the following basic theorem, which is basically a reformulation of Cartan’s theorem above: Min Ru K-stability and Nevanlinna-Diophantine theory The proof is based on the following basic theorem, which is basically a reformulation of Cartan’s theorem above: The Basic Theorem (m-base estimate). Let X be a complex projective variety and let L be a line sheaf on X with dim H 0 (X , L) ≥ 1. Min Ru K-stability and Nevanlinna-Diophantine theory The proof is based on the following basic theorem, which is basically a reformulation of Cartan’s theorem above: The Basic Theorem (m-base estimate). Let X be a complex projective variety and let L be a line sheaf on X with dim H 0 (X , L) ≥ 1. Let s1 , . . . , sq ∈ H 0 (X , L). Min Ru K-stability and Nevanlinna-Diophantine theory The proof is based on the following basic theorem, which is basically a reformulation of Cartan’s theorem above: The Basic Theorem (m-base estimate). Let X be a complex projective variety and let L be a line sheaf on X with dim H 0 (X , L) ≥ 1. Let s1 , . . . , sq ∈ H 0 (X , L). Let f : C → X be a holomorphic map with Zariski-dense image. Min Ru K-stability and Nevanlinna-Diophantine theory The proof is based on the following basic theorem, which is basically a reformulation of Cartan’s theorem above: The Basic Theorem (m-base estimate). Let X be a complex projective variety and let L be a line sheaf on X with dim H 0 (X , L) ≥ 1. Let s1 , . . . , sq ∈ H 0 (X , L). Let f : C → X be a holomorphic map with Zariski-dense image. Then, for any ϵ > 0, Z 2π X dθ ≤exc (dim H 0 (X , L) + ϵ)Tf ,L (r ) λsj (f (re iθ )) max J 2π 0 j∈J where the set J ranges over all subsets of {1, . . . , q} such that the sections (sj )j∈J are linearly independent. Min Ru K-stability and Nevanlinna-Diophantine theory The proof is based on the following basic theorem, which is basically a reformulation of Cartan’s theorem above: The Basic Theorem (m-base estimate). Let X be a complex projective variety and let L be a line sheaf on X with dim H 0 (X , L) ≥ 1. Let s1 , . . . , sq ∈ H 0 (X , L). Let f : C → X be a holomorphic map with Zariski-dense image. Then, for any ϵ > 0, Z 2π X dθ ≤exc (dim H 0 (X , L) + ϵ)Tf ,L (r ) λsj (f (re iθ )) max J 2π 0 j∈J where the set J ranges over all subsets of {1, . . . , q} such that the sections (sj )j∈J are linearlyPindependent. Note: The D ∼Q L is of 1 m-basis type if D := mN s∈B (s), where B is a basis of m 0 ⊗m H (X , L ), where Nm = dim H 0 (X , L⊗m ). Min Ru K-stability and Nevanlinna-Diophantine theory Theorem (Weak version of Ru-Vojta). Let X be a complex projective variety and let D1 , . . . , Dq be effective Cartier divisors such that at most ℓ of such divisors meet at any point of X . Min Ru K-stability and Nevanlinna-Diophantine theory Theorem (Weak version of Ru-Vojta). Let X be a complex projective variety and let D1 , . . . , Dq be effective Cartier divisors such that at most ℓ of such divisors meet at any point of X . Let L be a line sheaf on X with h0 (LN ) ≥ 1 for N big enough. Let f : C → X be a holomorphic map with Zariski-dense image. Min Ru K-stability and Nevanlinna-Diophantine theory Theorem (Weak version of Ru-Vojta). Let X be a complex projective variety and let D1 , . . . , Dq be effective Cartier divisors such that at most ℓ of such divisors meet at any point of X . Let L be a line sheaf on X with h0 (LN ) ≥ 1 for N big enough. Let f : C → X be a holomorphic map with Zariski-dense image. Then, Pq for every ϵ > 0, j=1 β(L, Dj )mf (r , Dj ) ≤exc ℓ (1 + ϵ) Tf ,L (r ). Min Ru K-stability and Nevanlinna-Diophantine theory Theorem (Weak version of Ru-Vojta). Let X be a complex projective variety and let D1 , . . . , Dq be effective Cartier divisors such that at most ℓ of such divisors meet at any point of X . Let L be a line sheaf on X with h0 (LN ) ≥ 1 for N big enough. Let f : C → X be a holomorphic map with Zariski-dense image. Then, Pq for every ϵ > 0, j=1 β(L, Dj )mf (r , Dj ) ≤exc ℓ (1 + ϵ) Tf ,L (r ). The proof is using the Basic Theorem (m-base estimate) by choosing a a suitable m-basis of H 0 (X , mL) through the filtration t = H 0 (X , mL − tE ), t ≥ 0 of H 0 (X , mL) filtration Fm Min Ru K-stability and Nevanlinna-Diophantine theory . Min Ru K-stability and Nevanlinna-Diophantine theory Outline of the proof: Min Ru K-stability and Nevanlinna-Diophantine theory Outline of the proof: For each f (z) = x ∈ X , from the condition that at most ℓ of Dj , 1 ≤ j ≤ q, meet at x, Min Ru K-stability and Nevanlinna-Diophantine theory Outline of the proof: For each f (z) = x ∈ X , from the condition that Pq at most ℓ of Dj , 1 ≤ j ≤ q, meet at x, we have j=1 βj λDj (x) ≤ ℓβi0 λDi0 (x) + O(1). Min Ru K-stability and Nevanlinna-Diophantine theory Outline of the proof: For each f (z) = x ∈ X , from the condition that Pq at most ℓ of Dj , 1 ≤ j ≤ q, meet at x, we have j=1 βj λDj (x) ≤ ℓβi0 λDi0 (x) + O(1). t = H 0 (X , mL − tD ), t ≥ 0, of H 0 (X , mL) and Consider the Fm 0 choose a basis s1 , · · · , sNm ∈ H 0 (X , mL) according to this filtration. Min Ru K-stability and Nevanlinna-Diophantine theory Outline of the proof: For each f (z) = x ∈ X , from the condition that Pq at most ℓ of Dj , 1 ≤ j ≤ q, meet at x, we have j=1 βj λDj (x) ≤ ℓβi0 λDi0 (x) + O(1). t = H 0 (X , mL − tD ), t ≥ 0, of H 0 (X , mL) and Consider the Fm 0 choose a basis s1 , · · · , sNm ∈ H 0 (X , mL) according to this filtration. Notice that for s ∈ H 0 (X , mL − tDi0 )), we have (s) ≥ tDi0 , Min Ru K-stability and Nevanlinna-Diophantine theory Outline of the proof: For each f (z) = x ∈ X , from the condition that Pq at most ℓ of Dj , 1 ≤ j ≤ q, meet at x, we have j=1 βj λDj (x) ≤ ℓβi0 λDi0 (x) + O(1). t = H 0 (X , mL − tD ), t ≥ 0, of H 0 (X , mL) and Consider the Fm 0 choose a basis s1 , · · · , sNm ∈ H 0 (X , mL) according to this filtration. Notice that for s ∈ H 0 (X , mL − tDi0 )), we have (s) ≥ tDi0 , so N m 1 X (sj ) ≥ mNm P∞ j=1 Min Ru 0 t=1 h (mL − tDi0 ) mNm Di0 . K-stability and Nevanlinna-Diophantine theory Outline of the proof: For each f (z) = x ∈ X , from the condition that Pq at most ℓ of Dj , 1 ≤ j ≤ q, meet at x, we have j=1 βj λDj (x) ≤ ℓβi0 λDi0 (x) + O(1). t = H 0 (X , mL − tD ), t ≥ 0, of H 0 (X , mL) and Consider the Fm 0 choose a basis s1 , · · · , sNm ∈ H 0 (X , mL) according to this filtration. Notice that for s ∈ H 0 (X , mL − tDi0 )), we have (s) ≥ tDi0 , so N m 1 X (sj ) ≥ mNm P∞ j=1 0 t=1 h (mL − tDi0 ) mNm Di0 . It then follows from the Basic Theorem. Min Ru K-stability and Nevanlinna-Diophantine theory Outline of the proof: For each f (z) = x ∈ X , from the condition that Pq at most ℓ of Dj , 1 ≤ j ≤ q, meet at x, we have j=1 βj λDj (x) ≤ ℓβi0 λDi0 (x) + O(1). t = H 0 (X , mL − tD ), t ≥ 0, of H 0 (X , mL) and Consider the Fm 0 choose a basis s1 , · · · , sNm ∈ H 0 (X , mL) according to this filtration. Notice that for s ∈ H 0 (X , mL − tDi0 )), we have (s) ≥ tDi0 , so N m 1 X (sj ) ≥ mNm P∞ j=1 0 t=1 h (mL − tDi0 ) mNm Di0 . It then follows from the Basic Theorem. Min Ru K-stability and Nevanlinna-Diophantine theory With the filtration (multi-parameter filtration) in Ru-Vojta, we can prove that Theorem (He-Ru, Proc. A.M.S., 2022). δ(L) ≤ 1 max1≤i≤q β(Di , L) lct(D), for any divisor D = D1 + · · · + Dq with D1 , . . . , Dq are in general position on X . Min Ru K-stability and Nevanlinna-Diophantine theory With the filtration (multi-parameter filtration) in Ru-Vojta, we can prove that Theorem (He-Ru, Proc. A.M.S., 2022). δ(L) ≤ 1 max1≤i≤q β(Di , L) lct(D), for any divisor D = D1 + · · · + Dq with D1 , . . . , Dq are in general position on X . Ru-Vojta theorem is just above result plus the Basic Theorem. Min Ru K-stability and Nevanlinna-Diophantine theory Three interesting constants Let L be ample, we define Min Ru K-stability and Nevanlinna-Diophantine theory Three interesting constants Let L be ample, we define Seshadri constant ϵ(L, D): Min Ru K-stability and Nevanlinna-Diophantine theory Three interesting constants Let L be ample, we define Seshadri constant ϵ(L, D): ϵ(L, D) = sup{γ ∈ Q : L − γD is nef}. T (L, D) = sup{γ ∈ Q : L−γD is effective or pseudo-effective}. Min Ru K-stability and Nevanlinna-Diophantine theory Three interesting constants Let L be ample, we define Seshadri constant ϵ(L, D): ϵ(L, D) = sup{γ ∈ Q : L − γD is nef}. T (L, D) = sup{γ ∈ Q : L−γD is effective or pseudo-effective}. Then we have (Blum-Jonsson) ϵ(L, D) ≤ T (L, D) Min Ru K-stability and Nevanlinna-Diophantine theory Three interesting constants Let L be ample, we define Seshadri constant ϵ(L, D): ϵ(L, D) = sup{γ ∈ Q : L − γD is nef}. T (L, D) = sup{γ ∈ Q : L−γD is effective or pseudo-effective}. Then we have (Blum-Jonsson) ϵ(L, D) ≤ T (L, D) and 1 n+1 T (L, D) ≤ β(L, D) ≤ T (L, D). Min Ru K-stability and Nevanlinna-Diophantine theory Three interesting constants Let L be ample, we define Seshadri constant ϵ(L, D): ϵ(L, D) = sup{γ ∈ Q : L − γD is nef}. T (L, D) = sup{γ ∈ Q : L−γD is effective or pseudo-effective}. Then we have (Blum-Jonsson) ϵ(L, D) ≤ T (L, D) and 1 n+1 T (L, D) ≤ β(L, D) ≤ T (L, D). ) A(E ) Furthermore, α(L) = inf E TA(E (L,E ) (while δ(L) = inf E β(L,E ) ), Min Ru K-stability and Nevanlinna-Diophantine theory Three interesting constants Let L be ample, we define Seshadri constant ϵ(L, D): ϵ(L, D) = sup{γ ∈ Q : L − γD is nef}. T (L, D) = sup{γ ∈ Q : L−γD is effective or pseudo-effective}. Then we have (Blum-Jonsson) ϵ(L, D) ≤ T (L, D) and 1 n+1 T (L, D) ≤ β(L, D) ≤ T (L, D). ) A(E ) Furthermore, α(L) = inf E TA(E (L,E ) (while δ(L) = inf E β(L,E ) ), and α(L) ≤ δ(L) ≤ (n + 1)α(L). Min Ru K-stability and Nevanlinna-Diophantine theory