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Introduction Main results Strategy of Thm B Details of proof Related topics On projective manifolds with semi-positive holomorphic sectional curvature Shin-ichi Matsumura ( ) Tohoku University (mshinichi-math@tohoku.ac.jp,

  1. Introduction Main results Strategy of Thm B Details of proof Related topics Theorem (Frankel conjecture prove by Siu-Yau’80, Hartshorne conjecture proved by Mori’ 79) (1) If BSC g is positive ( that is, BSC g ( v , w ) > 0 for any non-zero vectors v , w ∈ T X at any point of X ) , then X ∼ = P n . (2) If T X is ample, then the same conclusion holds. Ampleness of T X is an algebraic counterpart of BSC g > 0. Theorem (Igusa ’55) TFAE. (1) ∃ g: BSC g ≡ 0 (2) ∃ g: R g ≡ 0 (3) X is a torus up to finite ´ etale cover, namely, there exists a torus T s.t. finite ´ etale T − − − − − − → X 4 / 21

  2. Introduction Main results Strategy of Thm B Details of proof Related topics Theorem (Frankel conjecture prove by Siu-Yau’80, Hartshorne conjecture proved by Mori’ 79) (1) If BSC g is positive ( that is, BSC g ( v , w ) > 0 for any non-zero vectors v , w ∈ T X at any point of X ) , then X ∼ = P n . (2) If T X is ample, then the same conclusion holds. Ampleness of T X is an algebraic counterpart of BSC g > 0. Theorem (Igusa ’55) TFAE. (1) ∃ g: BSC g ≡ 0 (2) ∃ g: R g ≡ 0 (3) X is a torus up to finite ´ etale cover, namely, there exists a torus T s.t. finite ´ etale T − − − − − − → X 4 / 21

  3. Introduction Main results Strategy of Thm B Details of proof Related topics Theorem (Frankel conjecture prove by Siu-Yau’80, Hartshorne conjecture proved by Mori’ 79) (1) If BSC g is positive ( that is, BSC g ( v , w ) > 0 for any non-zero vectors v , w ∈ T X at any point of X ) , then X ∼ = P n . (2) If T X is ample, then the same conclusion holds. Ampleness of T X is an algebraic counterpart of BSC g > 0. Theorem (Igusa ’55) TFAE. (1) ∃ g: BSC g ≡ 0 (2) ∃ g: R g ≡ 0 (3) X is a torus up to finite ´ etale cover, namely, there exists a torus T s.t. finite ´ etale T − − − − − − → X 4 / 21

  4. Introduction Main results Strategy of Thm B Details of proof Related topics Theorem (Frankel conjecture prove by Siu-Yau’80, Hartshorne conjecture proved by Mori’ 79) (1) If BSC g is positive ( that is, BSC g ( v , w ) > 0 for any non-zero vectors v , w ∈ T X at any point of X ) , then X ∼ = P n . (2) If T X is ample, then the same conclusion holds. Ampleness of T X is an algebraic counterpart of BSC g > 0. Theorem (Igusa ’55) TFAE. (1) ∃ g: BSC g ≡ 0 (2) ∃ g: R g ≡ 0 (3) X is a torus up to finite ´ etale cover, namely, there exists a torus T s.t. finite ´ etale T − − − − − − → X 4 / 21

  5. Introduction Main results Strategy of Thm B Details of proof Related topics Theorem (Frankel conjecture prove by Siu-Yau’80, Hartshorne conjecture proved by Mori’ 79) (1) If BSC g is positive ( that is, BSC g ( v , w ) > 0 for any non-zero vectors v , w ∈ T X at any point of X ) , then X ∼ = P n . (2) If T X is ample, then the same conclusion holds. Ampleness of T X is an algebraic counterpart of BSC g > 0. Theorem (Igusa ’55) TFAE. (1) ∃ g: BSC g ≡ 0 (2) ∃ g: R g ≡ 0 (3) X is a torus up to finite ´ etale cover, namely, there exists a torus T s.t. finite ´ etale T − − − − − − → X 4 / 21

  6. Introduction Main results Strategy of Thm B Details of proof Related topics Theorem (Frankel conjecture prove by Siu-Yau’80, Hartshorne conjecture proved by Mori’ 79) (1) If BSC g is positive ( that is, BSC g ( v , w ) > 0 for any non-zero vectors v , w ∈ T X at any point of X ) , then X ∼ = P n . (2) If T X is ample, then the same conclusion holds. Ampleness of T X is an algebraic counterpart of BSC g > 0. Theorem (Igusa ’55) TFAE. (1) ∃ g: BSC g ≡ 0 (2) ∃ g: R g ≡ 0 (3) X is a torus up to finite ´ etale cover, namely, there exists a torus T s.t. finite ´ etale T − − − − − − → X 4 / 21

  7. Introduction Main results Strategy of Thm B Details of proof Related topics Theorem (Frankel conjecture prove by Siu-Yau’80, Hartshorne conjecture proved by Mori’ 79) (1) If BSC g is positive ( that is, BSC g ( v , w ) > 0 for any non-zero vectors v , w ∈ T X at any point of X ) , then X ∼ = P n . (2) If T X is ample, then the same conclusion holds. Ampleness of T X is an algebraic counterpart of BSC g > 0. Theorem (Igusa ’55) TFAE. (1) ∃ g: BSC g ≡ 0 (2) ∃ g: R g ≡ 0 (3) X is a torus up to finite ´ etale cover, namely, there exists a torus T s.t. finite ´ etale T − − − − − − → X 4 / 21

  8. Introduction Main results Strategy of Thm B Details of proof Related topics Theorem (Howard-Smyth-Wu’81) If BSC g is non-negative, then ∃ φ : X → Y satisfying that: (1) φ is locally trivial ( all the fibers F are non-singular and biholo ) . (2) ∃ K¨ ahler metric g Y on Y : R g Y ≡ 0 . (3) ∃ K¨ ahler metric g F on F s.t. BSC g F ≥ 0 and Ric g F > q 0 . Moreover, we obtain X univ ∼ = F × C m ( biholo and isometric ) Theorem (Mok’88) A manifold F ( which corresponds to the fiber of φ : X → Y ) satisfying that BSC g F ≥ 0 Ric g F > q 0 and is always a Hermitian symmetric space. 5 / 21

  9. Introduction Main results Strategy of Thm B Details of proof Related topics Theorem (Howard-Smyth-Wu’81) If BSC g is non-negative, then ∃ φ : X → Y satisfying that: (1) φ is locally trivial ( all the fibers F are non-singular and biholo ) . (2) ∃ K¨ ahler metric g Y on Y : R g Y ≡ 0 . (3) ∃ K¨ ahler metric g F on F s.t. BSC g F ≥ 0 and Ric g F > q 0 . Moreover, we obtain X univ ∼ = F × C m ( biholo and isometric ) Theorem (Mok’88) A manifold F ( which corresponds to the fiber of φ : X → Y ) satisfying that BSC g F ≥ 0 Ric g F > q 0 and is always a Hermitian symmetric space. 5 / 21

  10. Introduction Main results Strategy of Thm B Details of proof Related topics Theorem (Howard-Smyth-Wu’81) If BSC g is non-negative, then ∃ φ : X → Y satisfying that: (1) φ is locally trivial ( all the fibers F are non-singular and biholo ) . (2) ∃ K¨ ahler metric g Y on Y : R g Y ≡ 0 . (3) ∃ K¨ ahler metric g F on F s.t. BSC g F ≥ 0 and Ric g F > q 0 . Moreover, we obtain X univ ∼ = F × C m ( biholo and isometric ) Theorem (Mok’88) A manifold F ( which corresponds to the fiber of φ : X → Y ) satisfying that BSC g F ≥ 0 Ric g F > q 0 and is always a Hermitian symmetric space. 5 / 21

  11. Introduction Main results Strategy of Thm B Details of proof Related topics Theorem (Howard-Smyth-Wu’81) If BSC g is non-negative, then ∃ φ : X → Y satisfying that: (1) φ is locally trivial ( all the fibers F are non-singular and biholo ) . (2) ∃ K¨ ahler metric g Y on Y : R g Y ≡ 0 . (3) ∃ K¨ ahler metric g F on F s.t. BSC g F ≥ 0 and Ric g F > q 0 . Moreover, we obtain X univ ∼ = F × C m ( biholo and isometric ) Theorem (Mok’88) A manifold F ( which corresponds to the fiber of φ : X → Y ) satisfying that BSC g F ≥ 0 Ric g F > q 0 and is always a Hermitian symmetric space. 5 / 21

  12. Introduction Main results Strategy of Thm B Details of proof Related topics Theorem (Howard-Smyth-Wu’81) If BSC g is non-negative, then ∃ φ : X → Y satisfying that: (1) φ is locally trivial ( all the fibers F are non-singular and biholo ) . (2) ∃ K¨ ahler metric g Y on Y : R g Y ≡ 0 . (3) ∃ K¨ ahler metric g F on F s.t. BSC g F ≥ 0 and Ric g F > q 0 . Moreover, we obtain X univ ∼ = F × C m ( biholo and isometric ) Theorem (Mok’88) A manifold F ( which corresponds to the fiber of φ : X → Y ) satisfying that BSC g F ≥ 0 Ric g F > q 0 and is always a Hermitian symmetric space. 5 / 21

  13. Introduction Main results Strategy of Thm B Details of proof Related topics Theorem (Howard-Smyth-Wu’81) If BSC g is non-negative, then ∃ φ : X → Y satisfying that: (1) φ is locally trivial ( all the fibers F are non-singular and biholo ) . (2) ∃ K¨ ahler metric g Y on Y : R g Y ≡ 0 . (3) ∃ K¨ ahler metric g F on F s.t. BSC g F ≥ 0 and Ric g F > q 0 . Moreover, we obtain X univ ∼ = F × C m ( biholo and isometric ) Theorem (Mok’88) A manifold F ( which corresponds to the fiber of φ : X → Y ) satisfying that BSC g F ≥ 0 Ric g F > q 0 and is always a Hermitian symmetric space. 5 / 21

  14. Introduction Main results Strategy of Thm B Details of proof Related topics Theorem (Howard-Smyth-Wu’81) If BSC g is non-negative, then ∃ φ : X → Y satisfying that: (1) φ is locally trivial ( all the fibers F are non-singular and biholo ) . (2) ∃ K¨ ahler metric g Y on Y : R g Y ≡ 0 . (3) ∃ K¨ ahler metric g F on F s.t. BSC g F ≥ 0 and Ric g F > q 0 . Moreover, we obtain X univ ∼ = F × C m ( biholo and isometric ) Theorem (Mok’88) A manifold F ( which corresponds to the fiber of φ : X → Y ) satisfying that BSC g F ≥ 0 Ric g F > q 0 and is always a Hermitian symmetric space. 5 / 21

  15. Introduction Main results Strategy of Thm B Details of proof Related topics Theorem (Howard-Smyth-Wu’81) If BSC g is non-negative, then ∃ φ : X → Y satisfying that: (1) φ is locally trivial ( all the fibers F are non-singular and biholo ) . (2) ∃ K¨ ahler metric g Y on Y : R g Y ≡ 0 . (3) ∃ K¨ ahler metric g F on F s.t. BSC g F ≥ 0 and Ric g F > q 0 . Moreover, we obtain X univ ∼ = F × C m ( biholo and isometric ) Theorem (Mok’88) A manifold F ( which corresponds to the fiber of φ : X → Y ) satisfying that BSC g F ≥ 0 Ric g F > q 0 and is always a Hermitian symmetric space. 5 / 21

  16. Introduction Main results Strategy of Thm B Details of proof Related topics Problem (Main problem in this talk) (1) If HSC g > 0 , then what can we say for X?? ( HSC version of Mok’s result ) . (2) If HSC g ≥ 0 , then can we obtain a structure thm?? ( HSC version of Howard-Smyth-Wu’s result ) . Motivation : • HSC g determines R g (in particular BSC g ), namely, if HSC g = HSC h , then R g = R h . • In this sense, HSC g is a more primitive object. • However, a relation between positivity of BSC and HSC is unknown and still mysterious. • HSC g is closely related to the geometry of rational curves . • For example, by Schwarz lemma generalized by Yau, if HSC g < 0, then there is no rational curves in X . 6 / 21

  17. Introduction Main results Strategy of Thm B Details of proof Related topics Problem (Main problem in this talk) (1) If HSC g > 0 , then what can we say for X?? ( HSC version of Mok’s result ) . (2) If HSC g ≥ 0 , then can we obtain a structure thm?? ( HSC version of Howard-Smyth-Wu’s result ) . Motivation : • HSC g determines R g (in particular BSC g ), namely, if HSC g = HSC h , then R g = R h . • In this sense, HSC g is a more primitive object. • However, a relation between positivity of BSC and HSC is unknown and still mysterious. • HSC g is closely related to the geometry of rational curves . • For example, by Schwarz lemma generalized by Yau, if HSC g < 0, then there is no rational curves in X . 6 / 21

  18. Introduction Main results Strategy of Thm B Details of proof Related topics Problem (Main problem in this talk) (1) If HSC g > 0 , then what can we say for X?? ( HSC version of Mok’s result ) . (2) If HSC g ≥ 0 , then can we obtain a structure thm?? ( HSC version of Howard-Smyth-Wu’s result ) . Motivation : • HSC g determines R g (in particular BSC g ), namely, if HSC g = HSC h , then R g = R h . • In this sense, HSC g is a more primitive object. • However, a relation between positivity of BSC and HSC is unknown and still mysterious. • HSC g is closely related to the geometry of rational curves . • For example, by Schwarz lemma generalized by Yau, if HSC g < 0, then there is no rational curves in X . 6 / 21

  19. Introduction Main results Strategy of Thm B Details of proof Related topics Problem (Main problem in this talk) (1) If HSC g > 0 , then what can we say for X?? ( HSC version of Mok’s result ) . (2) If HSC g ≥ 0 , then can we obtain a structure thm?? ( HSC version of Howard-Smyth-Wu’s result ) . Motivation : • HSC g determines R g (in particular BSC g ), namely, if HSC g = HSC h , then R g = R h . • In this sense, HSC g is a more primitive object. • However, a relation between positivity of BSC and HSC is unknown and still mysterious. • HSC g is closely related to the geometry of rational curves . • For example, by Schwarz lemma generalized by Yau, if HSC g < 0, then there is no rational curves in X . 6 / 21

  20. Introduction Main results Strategy of Thm B Details of proof Related topics Problem (Main problem in this talk) (1) If HSC g > 0 , then what can we say for X?? ( HSC version of Mok’s result ) . (2) If HSC g ≥ 0 , then can we obtain a structure thm?? ( HSC version of Howard-Smyth-Wu’s result ) . Motivation : • HSC g determines R g (in particular BSC g ), namely, if HSC g = HSC h , then R g = R h . • In this sense, HSC g is a more primitive object. • However, a relation between positivity of BSC and HSC is unknown and still mysterious. • HSC g is closely related to the geometry of rational curves . • For example, by Schwarz lemma generalized by Yau, if HSC g < 0, then there is no rational curves in X . 6 / 21

  21. Introduction Main results Strategy of Thm B Details of proof Related topics Problem (Main problem in this talk) (1) If HSC g > 0 , then what can we say for X?? ( HSC version of Mok’s result ) . (2) If HSC g ≥ 0 , then can we obtain a structure thm?? ( HSC version of Howard-Smyth-Wu’s result ) . Motivation : • HSC g determines R g (in particular BSC g ), namely, if HSC g = HSC h , then R g = R h . • In this sense, HSC g is a more primitive object. • However, a relation between positivity of BSC and HSC is unknown and still mysterious. • HSC g is closely related to the geometry of rational curves . • For example, by Schwarz lemma generalized by Yau, if HSC g < 0, then there is no rational curves in X . 6 / 21

  22. Introduction Main results Strategy of Thm B Details of proof Related topics Problem (Main problem in this talk) (1) If HSC g > 0 , then what can we say for X?? ( HSC version of Mok’s result ) . (2) If HSC g ≥ 0 , then can we obtain a structure thm?? ( HSC version of Howard-Smyth-Wu’s result ) . Motivation : • HSC g determines R g (in particular BSC g ), namely, if HSC g = HSC h , then R g = R h . • In this sense, HSC g is a more primitive object. • However, a relation between positivity of BSC and HSC is unknown and still mysterious. • HSC g is closely related to the geometry of rational curves . • For example, by Schwarz lemma generalized by Yau, if HSC g < 0, then there is no rational curves in X . 6 / 21

  23. Introduction Main results Strategy of Thm B Details of proof Related topics Problem (Main problem in this talk) (1) If HSC g > 0 , then what can we say for X?? ( HSC version of Mok’s result ) . (2) If HSC g ≥ 0 , then can we obtain a structure thm?? ( HSC version of Howard-Smyth-Wu’s result ) . Motivation : • HSC g determines R g (in particular BSC g ), namely, if HSC g = HSC h , then R g = R h . • In this sense, HSC g is a more primitive object. • However, a relation between positivity of BSC and HSC is unknown and still mysterious. • HSC g is closely related to the geometry of rational curves . • For example, by Schwarz lemma generalized by Yau, if HSC g < 0, then there is no rational curves in X . 6 / 21

  24. Introduction Main results Strategy of Thm B Details of proof Related topics Problem (Main problem in this talk) (1) If HSC g > 0 , then what can we say for X?? ( HSC version of Mok’s result ) . (2) If HSC g ≥ 0 , then can we obtain a structure thm?? ( HSC version of Howard-Smyth-Wu’s result ) . Motivation : • HSC g determines R g (in particular BSC g ), namely, if HSC g = HSC h , then R g = R h . • In this sense, HSC g is a more primitive object. • However, a relation between positivity of BSC and HSC is unknown and still mysterious. • HSC g is closely related to the geometry of rational curves . • For example, by Schwarz lemma generalized by Yau, if HSC g < 0, then there is no rational curves in X . 6 / 21

  25. Introduction Main results Strategy of Thm B Details of proof Related topics Conjecture (Yau’s conjecture) If HSC g is positive, then X is ( projective and ) rationally connected. Definition (1) A curve R ⊂ X is called a rational curve if ∃ f : P 1 → X s.t. R = f ( P 1 ) . (2) X is called rationally connected ( RC ) if ∀ x , y ∈ X , ∃ R ⊂ X s.t. x , y ∈ R. (3) X is called uniruled if ∀ x ∈ X , ∃ R ⊂ X s.t. x ∈ R. 7 / 21

  26. Introduction Main results Strategy of Thm B Details of proof Related topics Conjecture (Yau’s conjecture) If HSC g is positive, then X is ( projective and ) rationally connected. Definition (1) A curve R ⊂ X is called a rational curve if ∃ f : P 1 → X s.t. R = f ( P 1 ) . (2) X is called rationally connected ( RC ) if ∀ x , y ∈ X , ∃ R ⊂ X s.t. x , y ∈ R. (3) X is called uniruled if ∀ x ∈ X , ∃ R ⊂ X s.t. x ∈ R. 7 / 21

  27. Introduction Main results Strategy of Thm B Details of proof Related topics Conjecture (Yau’s conjecture) If HSC g is positive, then X is ( projective and ) rationally connected. Definition (1) A curve R ⊂ X is called a rational curve if ∃ f : P 1 → X s.t. R = f ( P 1 ) . (2) X is called rationally connected ( RC ) if ∀ x , y ∈ X , ∃ R ⊂ X s.t. x , y ∈ R. (3) X is called uniruled if ∀ x ∈ X , ∃ R ⊂ X s.t. x ∈ R. 7 / 21

  28. Introduction Main results Strategy of Thm B Details of proof Related topics Conjecture (Yau’s conjecture) If HSC g is positive, then X is ( projective and ) rationally connected. Definition (1) A curve R ⊂ X is called a rational curve if ∃ f : P 1 → X s.t. R = f ( P 1 ) . (2) X is called rationally connected ( RC ) if ∀ x , y ∈ X , ∃ R ⊂ X s.t. x , y ∈ R. (3) X is called uniruled if ∀ x ∈ X , ∃ R ⊂ X s.t. x ∈ R. 7 / 21

  29. Introduction Main results Strategy of Thm B Details of proof Related topics Conjecture (Yau’s conjecture) If HSC g is positive, then X is ( projective and ) rationally connected. Definition (1) A curve R ⊂ X is called a rational curve if ∃ f : P 1 → X s.t. R = f ( P 1 ) . (2) X is called rationally connected ( RC ) if ∀ x , y ∈ X , ∃ R ⊂ X s.t. x , y ∈ R. (3) X is called uniruled if ∀ x ∈ X , ∃ R ⊂ X s.t. x ∈ R. 7 / 21

  30. Introduction Main results Strategy of Thm B Details of proof Related topics Conjecture (Yau’s conjecture) If HSC g is positive, then X is ( projective and ) rationally connected. Definition (1) A curve R ⊂ X is called a rational curve if ∃ f : P 1 → X s.t. R = f ( P 1 ) . (2) X is called rationally connected ( RC ) if ∀ x , y ∈ X , ∃ R ⊂ X s.t. x , y ∈ R. (3) X is called uniruled if ∀ x ∈ X , ∃ R ⊂ X s.t. x ∈ R. 7 / 21

  31. Introduction Main results Strategy of Thm B Details of proof Related topics Conjecture (Yau’s conjecture) If HSC g is positive, then X is ( projective and ) rationally connected. Definition (1) A curve R ⊂ X is called a rational curve if ∃ f : P 1 → X s.t. R = f ( P 1 ) . (2) X is called rationally connected ( RC ) if ∀ x , y ∈ X , ∃ R ⊂ X s.t. x , y ∈ R. (3) X is called uniruled if ∀ x ∈ X , ∃ R ⊂ X s.t. x ∈ R. 7 / 21

  32. Introduction Main results Strategy of Thm B Details of proof Related topics Definition (Reprint) (1) A curve R ⊂ X is called a rational curve ( R-curve ) if ∃ f : P 1 → X s.t. R = f ( P 1 ) . (2) X is called rationally connected ( RC ) if ∀ x , y ∈ X , ∃ R ⊂ X s.t. x , y ∈ R. (3) X is called uniruled if ∀ x ∈ X , ∃ R ⊂ X s.t. x ∈ R. Example (1) Any Fano manifolds ( including P n ) are RC. (2) Any torus T has no rational curve. (3) X := P ( E ) → Y is always uniruled. (4) X := P ( E ) is not RC when Y is torus. 8 / 21

  33. Introduction Main results Strategy of Thm B Details of proof Related topics Definition (Reprint) (1) A curve R ⊂ X is called a rational curve ( R-curve ) if ∃ f : P 1 → X s.t. R = f ( P 1 ) . (2) X is called rationally connected ( RC ) if ∀ x , y ∈ X , ∃ R ⊂ X s.t. x , y ∈ R. (3) X is called uniruled if ∀ x ∈ X , ∃ R ⊂ X s.t. x ∈ R. Example (1) Any Fano manifolds ( including P n ) are RC. (2) Any torus T has no rational curve. (3) X := P ( E ) → Y is always uniruled. (4) X := P ( E ) is not RC when Y is torus. 8 / 21

  34. Introduction Main results Strategy of Thm B Details of proof Related topics Definition (Reprint) (1) A curve R ⊂ X is called a rational curve ( R-curve ) if ∃ f : P 1 → X s.t. R = f ( P 1 ) . (2) X is called rationally connected ( RC ) if ∀ x , y ∈ X , ∃ R ⊂ X s.t. x , y ∈ R. t . (3) X is called uniruled if ∀ x ∈ X , ∃ R ⊂ X s.t. x ∈ R. R' T → f- Example (1) Any Fano manifolds ( including P n ) are RC. (2) Any torus T has no rational curve. (3) X := P ( E ) → Y is always uniruled. (4) X := P ( E ) is not RC when Y is torus. 8 / 21

  35. Introduction Main results Strategy of Thm B Details of proof Related topics Definition (Reprint) (1) A curve R ⊂ X is called a rational curve ( R-curve ) if ° FIR ∃ f : P 1 → X s.t. R = f ( P 1 ) . h (2) X is called rationally connected ( RC ) if ∀ x , y ∈ X , ∃ R ⊂ X s.t. x , y ∈ R. BE ) (3) X is called uniruled if ⇐ ∀ x ∈ X , ∃ R ⊂ X s.t. x ∈ R. Example (1) Any Fano manifolds ( including P n ) are RC. (2) Any torus T has no rational curve. (3) X := P ( E ) → Y is always uniruled. (4) X := P ( E ) is not RC when Y is torus. 8 / 21

  36. Introduction Main results Strategy of Thm B Details of proof Related topics Definition (Reprint) (1) A curve R ⊂ X is called a rational curve ( R-curve ) if ∃ f : P 1 → X s.t. R = f ( P 1 ) . :¥÷ 7 (2) X is called rationally connected ( RC ) if in ∀ x , y ∈ X , ∃ R ⊂ X s.t. x , y ∈ R. (3) X is called uniruled if ∀ x ∈ X , ∃ R ⊂ X s.t. x ∈ R. Example (1) Any Fano manifolds ( including P n ) are RC. torus (2) Any torus T has no rational curve. (3) X := P ( E ) → Y is always uniruled. (4) X := P ( E ) is not RC when Y is torus. 8 / 21

  37. Introduction Main results Strategy of Thm B Details of proof Related topics Conjecture (Reprint, Yau’s conjecture) If HSC g is positive, then X is projective and RC. Known Reults: • Heier-Wong’15 solved Yau’s conjecture when X is projective. • Yang’18 solved Yau’s conjecture in a general situation. However • Yang’s method is based on the maximal principle and it can not be applied to the quasi-positive case. • It is important to consider the quasi-positive case of Yau’s conjecture from the viewpoint of a structure thm. Theorem (Thm A) If X is projective and satisfies P ( X , g ) = n ( which is satisfied in the quasi-positive case ) , then X is RC. • P ( X , g ) is an invariant defined by an analogue of the numerical Kodaira dimension. • The proof depends on Heier-Wong’s idea and RC positivity (partial positivity) introduced by Yang. 9 / 21

  38. Introduction Main results Strategy of Thm B Details of proof Related topics Conjecture (Reprint, Yau’s conjecture) If HSC g is positive, then X is projective and RC. Known Reults: • Heier-Wong’15 solved Yau’s conjecture when X is projective. • Yang’18 solved Yau’s conjecture in a general situation. However • Yang’s method is based on the maximal principle and it can not be applied to the quasi-positive case. • It is important to consider the quasi-positive case of Yau’s conjecture from the viewpoint of a structure thm. Theorem (Thm A) If X is projective and satisfies P ( X , g ) = n ( which is satisfied in the quasi-positive case ) , then X is RC. • P ( X , g ) is an invariant defined by an analogue of the numerical Kodaira dimension. • The proof depends on Heier-Wong’s idea and RC positivity (partial positivity) introduced by Yang. 9 / 21

  39. Introduction Main results Strategy of Thm B Details of proof Related topics Conjecture (Reprint, Yau’s conjecture) If HSC g is positive, then X is projective and RC. Known Reults: • Heier-Wong’15 solved Yau’s conjecture when X is projective. • Yang’18 solved Yau’s conjecture in a general situation. However • Yang’s method is based on the maximal principle and it can not be applied to the quasi-positive case. • It is important to consider the quasi-positive case of Yau’s conjecture from the viewpoint of a structure thm. Theorem (Thm A) If X is projective and satisfies P ( X , g ) = n ( which is satisfied in the quasi-positive case ) , then X is RC. • P ( X , g ) is an invariant defined by an analogue of the numerical Kodaira dimension. • The proof depends on Heier-Wong’s idea and RC positivity (partial positivity) introduced by Yang. 9 / 21

  40. Introduction Main results Strategy of Thm B Details of proof Related topics Conjecture (Reprint, Yau’s conjecture) If HSC g is positive, then X is projective and RC. Known Reults: • Heier-Wong’15 solved Yau’s conjecture when X is projective. • Yang’18 solved Yau’s conjecture in a general situation. However • Yang’s method is based on the maximal principle and it can not be applied to the quasi-positive case. • It is important to consider the quasi-positive case of Yau’s conjecture from the viewpoint of a structure thm. Theorem (Thm A) If X is projective and satisfies P ( X , g ) = n ( which is satisfied in the quasi-positive case ) , then X is RC. • P ( X , g ) is an invariant defined by an analogue of the numerical Kodaira dimension. • The proof depends on Heier-Wong’s idea and RC positivity (partial positivity) introduced by Yang. 9 / 21

  41. Introduction Main results Strategy of Thm B Details of proof Related topics Conjecture (Reprint, Yau’s conjecture) If HSC g is positive, then X is projective and RC. Known Reults: • Heier-Wong’15 solved Yau’s conjecture when X is projective. • Yang’18 solved Yau’s conjecture in a general situation. However • Yang’s method is based on the maximal principle and it can not be applied to the quasi-positive case. • It is important to consider the quasi-positive case of Yau’s conjecture from the viewpoint of a structure thm. Theorem (Thm A) If X is projective and satisfies P ( X , g ) = n ( which is satisfied in the quasi-positive case ) , then X is RC. • P ( X , g ) is an invariant defined by an analogue of the numerical Kodaira dimension. • The proof depends on Heier-Wong’s idea and RC positivity (partial positivity) introduced by Yang. 9 / 21

  42. Introduction Main results Strategy of Thm B Details of proof Related topics Conjecture (Reprint, Yau’s conjecture) If HSC g is positive, then X is projective and RC. Known Reults: • Heier-Wong’15 solved Yau’s conjecture when X is projective. • Yang’18 solved Yau’s conjecture in a general situation. However • Yang’s method is based on the maximal principle and it can not be applied to the quasi-positive case. • It is important to consider the quasi-positive case of Yau’s conjecture from the viewpoint of a structure thm. Theorem (Thm A) If X is projective and satisfies P ( X , g ) = n ( which is satisfied in the quasi-positive case ) , then X is RC. • P ( X , g ) is an invariant defined by an analogue of the numerical Kodaira dimension. • The proof depends on Heier-Wong’s idea and RC positivity (partial positivity) introduced by Yang. 9 / 21

  43. Introduction Main results Strategy of Thm B Details of proof Related topics Conjecture (Reprint, Yau’s conjecture) If HSC g is positive, then X is projective and RC. Known Reults: • Heier-Wong’15 solved Yau’s conjecture when X is projective. • Yang’18 solved Yau’s conjecture in a general situation. However • Yang’s method is based on the maximal principle and it can not be applied to the quasi-positive case. • It is important to consider the quasi-positive case of Yau’s conjecture from the viewpoint of a structure thm. Theorem (Thm A) If X is projective and satisfies P ( X , g ) = n ( which is satisfied in the quasi-positive case ) , then X is RC. • P ( X , g ) is an invariant defined by an analogue of the numerical Kodaira dimension. • The proof depends on Heier-Wong’s idea and RC positivity (partial positivity) introduced by Yang. 9 / 21

  44. Introduction Main results Strategy of Thm B Details of proof Related topics Conjecture (Reprint, Yau’s conjecture) If HSC g is positive, then X is projective and RC. Known Reults: • Heier-Wong’15 solved Yau’s conjecture when X is projective. • Yang’18 solved Yau’s conjecture in a general situation. However • Yang’s method is based on the maximal principle and it can not be applied to the quasi-positive case. • It is important to consider the quasi-positive case of Yau’s conjecture from the viewpoint of a structure thm. Theorem (Thm A) If X is projective and satisfies P ( X , g ) = n ( which is satisfied in the quasi-positive case ) , then X is RC. • P ( X , g ) is an invariant defined by an analogue of the numerical Kodaira dimension. • The proof depends on Heier-Wong’s idea and RC positivity (partial positivity) introduced by Yang. 9 / 21

  45. Introduction Main results Strategy of Thm B Details of proof Related topics Conjecture (Reprint, Yau’s conjecture) If HSC g is positive, then X is projective and RC. Known Reults: • Heier-Wong’15 solved Yau’s conjecture when X is projective. • Yang’18 solved Yau’s conjecture in a general situation. However • Yang’s method is based on the maximal principle and it can not be applied to the quasi-positive case. • It is important to consider the quasi-positive case of Yau’s conjecture from the viewpoint of a structure thm. Theorem (Thm A) If X is projective and satisfies P ( X , g ) = n ( which is satisfied in the quasi-positive case ) , then X is RC. • P ( X , g ) is an invariant defined by an analogue of the numerical Kodaira dimension. • The proof depends on Heier-Wong’s idea and RC positivity (partial positivity) introduced by Yang. 9 / 21

  46. Introduction Main results Strategy of Thm B Details of proof Related topics Theorem (Thm B) If X is projective and HSC g is non-negative, then ∃ φ : X → Y satisfying that: (1) φ is locally trivial ( all the fibers F are non-singular and biholo ) . (2) ∃ K¨ ahler metric g Y on Y : R g Y ≡ 0 . (3) ∃ K¨ ahler metric g F on F s.t. HSC g F ≥ 0 and F is RC . Moreover, we obtain = F × C m ( biholo and isometric ) X univ ∼ • There are two bad points in Thm B, compared to Howard-Smyth-Wu’s structure theorem. • One is the assumption of X being projective. • The other is the lack of “quasi-positivity” of HSC g F . 10 / 21

  47. Introduction Main results Strategy of Thm B Details of proof Related topics Theorem (Thm B) If X is projective and HSC g is non-negative, then ∃ φ : X → Y satisfying that: (1) φ is locally trivial ( all the fibers F are non-singular and biholo ) . (2) ∃ K¨ ahler metric g Y on Y : R g Y ≡ 0 . (3) ∃ K¨ ahler metric g F on F s.t. HSC g F ≥ 0 and F is RC . Moreover, we obtain = F × C m ( biholo and isometric ) X univ ∼ • There are two bad points in Thm B, compared to Howard-Smyth-Wu’s structure theorem. • One is the assumption of X being projective. • The other is the lack of “quasi-positivity” of HSC g F . 10 / 21

  48. Introduction Main results Strategy of Thm B Details of proof Related topics Theorem (Thm B) If X is projective and HSC g is non-negative, then ∃ φ : X → Y satisfying that: (1) φ is locally trivial ( all the fibers F are non-singular and biholo ) . (2) ∃ K¨ ahler metric g Y on Y : R g Y ≡ 0 . (3) ∃ K¨ ahler metric g F on F s.t. HSC g F ≥ 0 and F is RC . Moreover, we obtain = F × C m ( biholo and isometric ) X univ ∼ • There are two bad points in Thm B, compared to Howard-Smyth-Wu’s structure theorem. • One is the assumption of X being projective. • The other is the lack of “quasi-positivity” of HSC g F . 10 / 21

  49. Introduction Main results Strategy of Thm B Details of proof Related topics Theorem (Thm B) If X is projective and HSC g is non-negative, then ∃ φ : X → Y satisfying that: (1) φ is locally trivial ( all the fibers F are non-singular and biholo ) . (2) ∃ K¨ ahler metric g Y on Y : R g Y ≡ 0 . (3) ∃ K¨ ahler metric g F on F s.t. HSC g F ≥ 0 and F is RC . Moreover, we obtain = F × C m ( biholo and isometric ) X univ ∼ • There are two bad points in Thm B, compared to Howard-Smyth-Wu’s structure theorem. • One is the assumption of X being projective. • The other is the lack of “quasi-positivity” of HSC g F . 10 / 21

  50. Introduction Main results Strategy of Thm B Details of proof Related topics Theorem (Thm B) If X is projective and HSC g is non-negative, then ∃ φ : X → Y satisfying that: (1) φ is locally trivial ( all the fibers F are non-singular and biholo ) . (2) ∃ K¨ ahler metric g Y on Y : R g Y ≡ 0 . (3) ∃ K¨ ahler metric g F on F s.t. HSC g F ≥ 0 and F is RC . Moreover, we obtain = F × C m ( biholo and isometric ) X univ ∼ • There are two bad points in Thm B, compared to Howard-Smyth-Wu’s structure theorem. • One is the assumption of X being projective. • The other is the lack of “quasi-positivity” of HSC g F . 10 / 21

  51. Introduction Main results Strategy of Thm B Details of proof Related topics Theorem (Thm B) If X is projective and HSC g is non-negative, then ∃ φ : X → Y satisfying that: (1) φ is locally trivial ( all the fibers F are non-singular and biholo ) . (2) ∃ K¨ ahler metric g Y on Y : R g Y ≡ 0 . (3) ∃ K¨ ahler metric g F on F s.t. HSC g F ≥ 0 and F is RC . Moreover, we obtain = F × C m ( biholo and isometric ) X univ ∼ • There are two bad points in Thm B, compared to Howard-Smyth-Wu’s structure theorem. • One is the assumption of X being projective. • The other is the lack of “quasi-positivity” of HSC g F . 10 / 21

  52. Introduction Main results Strategy of Thm B Details of proof Related topics Theorem (Thm B) If X is projective and HSC g is non-negative, then ∃ φ : X → Y satisfying that: (1) φ is locally trivial ( all the fibers F are non-singular and biholo ) . (2) ∃ K¨ ahler metric g Y on Y : R g Y ≡ 0 . (3) ∃ K¨ ahler metric g F on F s.t. HSC g F ≥ 0 and F is RC . Moreover, we obtain = F × C m ( biholo and isometric ) X univ ∼ • There are two bad points in Thm B, compared to Howard-Smyth-Wu’s structure theorem. • One is the assumption of X being projective. • The other is the lack of “quasi-positivity” of HSC g F . 10 / 21

  53. Introduction Main results Strategy of Thm B Details of proof Related topics Theorem (Thm B) If X is projective and HSC g is non-negative, then ∃ φ : X → Y satisfying that: (1) φ is locally trivial ( all the fibers F are non-singular and biholo ) . (2) ∃ K¨ ahler metric g Y on Y : R g Y ≡ 0 . (3) ∃ K¨ ahler metric g F on F s.t. HSC g F ≥ 0 and F is RC . Moreover, we obtain = F × C m ( biholo and isometric ) X univ ∼ • There are two bad points in Thm B, compared to Howard-Smyth-Wu’s structure theorem. • One is the assumption of X being projective. • The other is the lack of “quasi-positivity” of HSC g F . 10 / 21

  54. Introduction Main results Strategy of Thm B Details of proof Related topics Theorem (Thm B) If X is projective and HSC g is non-negative, then ∃ φ : X → Y satisfying that: (1) φ is locally trivial ( all the fibers F are non-singular and biholo ) . (2) ∃ K¨ ahler metric g Y on Y : R g Y ≡ 0 . (3) ∃ K¨ ahler metric g F on F s.t. HSC g F ≥ 0 and F is RC . Moreover, we obtain = F × C m ( biholo and isometric ) X univ ∼ • There are two bad points in Thm B, compared to Howard-Smyth-Wu’s structure theorem. • One is the assumption of X being projective. • The other is the lack of “quasi-positivity” of HSC g F . 10 / 21

  55. Introduction Main results Strategy of Thm B Details of proof Related topics Step 1: MRC fibration • ∃ φ : X ��� Y : MRC fibration (Campana, Koll´ ar-Miyaoka-Mori): (1) φ is a rational map whose general fiber is compact. (2) A general fiber F is RC. (3) ∄ horizontal rational curve at “general points”. • For example, X := P ( E ) → Y is always an RC fibration, but it is not necessarily an MRC fibration (consider Y = P m ). • X := P ( E ) → Y is MRC when Y has no rational curve. Remark (1) MRC fibrations are not uniquely determined, but they are unique up to birational models of Y . (2) X is RC i ff dim Y = 0 . (3) X is not uniruled i ff dim Y = dim X. ( 4 ) If MRC is non-trivial ( that is, 0 < dim Y < dim X ) , then K Y is always pseudo-e ff ective ( psef ) , namely, K Y has a singular metric h such that Θ h ( K X ) ≥ 0 ( by the result of Graber-Harris-Starr ) . 11 / 21

  56. Introduction Main results Strategy of Thm B Details of proof Related topics Step 1: MRC fibration • ∃ φ : X ��� Y : MRC fibration (Campana, Koll´ ar-Miyaoka-Mori): (1) φ is a rational map whose general fiber is compact. (2) A general fiber F is RC. (3) ∄ horizontal rational curve at “general points”. ↳ :* • For example, X := P ( E ) → Y is always an RC fibration, but it is not necessarily an MRC fibration (consider Y = P m ). ftp.jiinm • X := P ( E ) → Y is MRC when Y has no rational curve. Remark (1) MRC fibrations are not uniquely determined, but they are unique up to birational models of Y . = . (2) X is RC i ff dim Y = 0 . ¥ (3) X is not uniruled i ff dim Y = dim X. ( 4 ) If MRC is non-trivial ( that is, 0 < dim Y < dim X ) , then K Y is always pseudo-e ff ective ( psef ) , namely, - K Y has a singular metric h such that Θ h ( K X ) ≥ 0 ( by the result of Graber-Harris-Starr ) . 11 / 21

  57. Introduction Main results Strategy of Thm B Details of proof Related topics Step 1: MRC fibration • ∃ φ : X ��� Y : MRC fibration (Campana, Koll´ ar-Miyaoka-Mori): (1) φ is a rational map whose general fiber is compact. (2) A general fiber F is RC. (3) ∄ horizontal rational curve at “general points”. Rf 641 • For example, X := P ( E ) → Y is always an RC fibration, but it is , not necessarily an MRC fibration (consider Y = P m ). EE • X := P ( E ) → Y is MRC when Y has no rational curve. Remark (1) MRC fibrations are not uniquely determined, but they are unique up to birational models of Y . (2) X is RC i ff dim Y = 0 . , (3) X is not uniruled i ff dim Y = dim X. ( 4 ) If MRC is non-trivial ( that is, 0 < dim Y < dim X ) , then K Y is always pseudo-e ff ective ( psef ) , namely, K Y has a singular metric h such that Θ h ( K X ) ≥ 0 ( by the result of Graber-Harris-Starr ) . 11 / 21

  58. Introduction Main results Strategy of Thm B Details of proof Related topics Step 1: MRC fibration • ∃ φ : X ��� Y : MRC fibration (Campana, Koll´ ar-Miyaoka-Mori): (1) φ is a rational map whose general fiber is compact. (2) A general fiber F is RC. (3) ∄ horizontal rational curve at “general points”. • For example, X := P ( E ) → Y is always an RC fibration, but it is not necessarily an MRC fibration (consider Y = P m ). • X := P ( E ) → Y is MRC when Y has no rational curve. Remark (1) MRC fibrations are not uniquely determined, but they are unique up to birational models of Y . (2) X is RC i ff dim Y = 0 . (3) X is not uniruled i ff dim Y = dim X. ( 4 ) If MRC is non-trivial ( that is, 0 < dim Y < dim X ) , then K Y is always pseudo-e ff ective ( psef ) , namely, K Y has a singular metric h such that Θ h ( K X ) ≥ 0 ( by the result of Graber-Harris-Starr ) . 11 / 21

  59. Introduction Main results Strategy of Thm B Details of proof Related topics Step 1: MRC fibration • ∃ φ : X ��� Y : MRC fibration (Campana, Koll´ ar-Miyaoka-Mori): (1) φ is a rational map whose general fiber is compact. (2) A general fiber F is RC. (3) ∄ horizontal rational curve at “general points”. • For example, X := P ( E ) → Y is always an RC fibration, but it is not necessarily an MRC fibration (consider Y = P m ). • X := P ( E ) → Y is MRC when Y has no rational curve. Remark LIE (1) MRC fibrations are not uniquely determined, but they are ho at unique up to birational models of Y . curve (2) X is RC i ff dim Y = 0 . , (3) X is not uniruled i ff dim Y = dim X. I ( 4 ) If MRC is non-trivial ( that is, 0 < dim Y < dim X ) , then K Y is always pseudo-e ff ective ( psef ) , namely, m =D - K Y has a singular metric h such that Θ h ( K X ) ≥ 0 ( by the result of Graber-Harris-Starr ) . 11 / 21

  60. Introduction Main results Strategy of Thm B Details of proof Related topics Step 1: MRC fibration • ∃ φ : X ��� Y : MRC fibration (Campana, Koll´ ar-Miyaoka-Mori): (1) φ is a rational map whose general fiber is compact. (2) A general fiber F is RC. (3) ∄ horizontal rational curve at “general points”. • For example, X := P ( E ) → Y is always an RC fibration, but it is not necessarily an MRC fibration (consider Y = P m ). • X := P ( E ) → Y is MRC when Y has no rational curve. Remark (1) MRC fibrations are not uniquely determined, but they are unique up to birational models of Y . (2) X is RC i ff dim Y = 0 . (3) X is not uniruled i ff dim Y = dim X. ( 4 ) If MRC is non-trivial ( that is, 0 < dim Y < dim X ) , then K Y is always pseudo-e ff ective ( psef ) , namely, K Y has a singular metric h such that Θ h ( K X ) ≥ 0 ( by the result of Graber-Harris-Starr ) . 11 / 21

  61. Introduction Main results Strategy of Thm B Details of proof Related topics Step 1: MRC fibration • ∃ φ : X ��� Y : MRC fibration (Campana, Koll´ ar-Miyaoka-Mori): (1) φ is a rational map whose general fiber is compact. (2) A general fiber F is RC. (3) ∄ horizontal rational curve at “general points”. • For example, X := P ( E ) → Y is always an RC fibration, but it is not necessarily an MRC fibration (consider Y = P m ). • X := P ( E ) → Y is MRC when Y has no rational curve. Remark (1) MRC fibrations are not uniquely determined, but they are unique up to birational models of Y . (2) X is RC i ff dim Y = 0 . (3) X is not uniruled i ff dim Y = dim X. ( 4 ) If MRC is non-trivial ( that is, 0 < dim Y < dim X ) , then K Y is always pseudo-e ff ective ( psef ) , namely, K Y has a singular metric h such that Θ h ( K X ) ≥ 0 ( by the result of Graber-Harris-Starr ) . 11 / 21

  62. Introduction Main results Strategy of Thm B Details of proof Related topics Step 1: MRC fibration • ∃ φ : X ��� Y : MRC fibration (Campana, Koll´ ar-Miyaoka-Mori): (1) φ is a rational map whose general fiber is compact. (2) A general fiber F is RC. (3) ∄ horizontal rational curve at “general points”. • For example, X := P ( E ) → Y is always an RC fibration, but it is not necessarily an MRC fibration (consider Y = P m ). • X := P ( E ) → Y is MRC when Y has no rational curve. Remark (1) MRC fibrations are not uniquely determined, but they are unique up to birational models of Y . (2) X is RC i ff dim Y = 0 . (3) X is not uniruled i ff dim Y = dim X. ( 4 ) If MRC is non-trivial ( that is, 0 < dim Y < dim X ) , then K Y is always pseudo-e ff ective ( psef ) , namely, K Y has a singular metric h such that Θ h ( K X ) ≥ 0 ( by the result of Graber-Harris-Starr ) . 11 / 21

  63. Introduction Main results Strategy of Thm B Details of proof Related topics Step 1: MRC fibration • ∃ φ : X ��� Y : MRC fibration (Campana, Koll´ ar-Miyaoka-Mori): (1) φ is a rational map whose general fiber is compact. (2) A general fiber F is RC. (3) ∄ horizontal rational curve at “general points”. • For example, X := P ( E ) → Y is always an RC fibration, but it is not necessarily an MRC fibration (consider Y = P m ). • X := P ( E ) → Y is MRC when Y has no rational curve. Remark (1) MRC fibrations are not uniquely determined, but they are unique up to birational models of Y . X (2) X is RC i ff dim Y = 0 . (3) X is not uniruled i ff dim Y = dim X. idx MRI ¥ ( 4 ) If MRC is non-trivial ( that is, 0 < dim Y < dim X ) , then K Y is always pseudo-e ff ective ( psef ) , namely, K Y has a singular metric h such that Θ h ( K X ) ≥ 0 ( by the result of Graber-Harris-Starr ) . 11 / 21

  64. Introduction Main results Strategy of Thm B Details of proof Related topics Step 1: MRC fibration • ∃ φ : X ��� Y : MRC fibration (Campana, Koll´ ar-Miyaoka-Mori): (1) φ is a rational map whose general fiber is compact. (2) A general fiber F is RC. (3) ∄ horizontal rational curve at “general points”. • For example, X := P ( E ) → Y is always an RC fibration, but it is not necessarily an MRC fibration (consider Y = P m ). • X := P ( E ) → Y is MRC when Y has no rational curve. Remark (1) MRC fibrations are not uniquely determined, but they are unique up to birational models of Y . (2) X is RC i ff dim Y = 0 . (3) X is not uniruled i ff dim Y = dim X. ( 4 ) If MRC is non-trivial ( that is, 0 < dim Y < dim X ) , then K Y is always pseudo-e ff ective ( psef ) , namely, K Y has a singular metric h such that Θ h ( K X ) ≥ 0 ( by the result of Graber-Harris-Starr ) . 11 / 21

  65. Introduction Main results Strategy of Thm B Details of proof Related topics Step 1: MRC fibration • ∃ φ : X ��� Y : MRC fibration (Campana, Koll´ ar-Miyaoka-Mori): (1) φ is a rational map whose general fiber is compact. (2) A general fiber F is RC. (3) ∄ horizontal rational curve at “general points”. • For example, X := P ( E ) → Y is always an RC fibration, but it is not necessarily an MRC fibration (consider Y = P m ). • X := P ( E ) → Y is MRC when Y has no rational curve. Remark (1) MRC fibrations are not uniquely determined, but they are unique up to birational models of Y . (2) X is RC i ff dim Y = 0 . (3) X is not uniruled i ff dim Y = dim X. ( 4 ) If MRC is non-trivial ( that is, 0 < dim Y < dim X ) , then K Y is always pseudo-e ff ective ( psef ) , namely, K Y has a singular metric h such that Θ h ( K X ) ≥ 0 ( by the result of Graber-Harris-Starr ) . 11 / 21

  66. Introduction Main results Strategy of Thm B Details of proof Related topics • If HSC g ≡ 0, then X is torus up to ´ etale covers. We may assume: HSC g ≥ 0 and HSC g ̸≡ 0 Claim Then X is uniruled. In particular, K Y is psef. Theorem (BDPP) X is not uniruled if and only if K X is psef. Proof. Assume K X is psef. Then c 1 ( K X ) ∧ ω n − 1 = − 1 � � Scal g ω n . 0 ≤ π n X X On the the hand, we have � Scal g ( p ) = HSC ( v ) dV FS . v ∈ P ( T X , p ) Hence Scal g is a quasi-positive function on X . 12 / 21

  67. Introduction Main results Strategy of Thm B Details of proof Related topics • If HSC g ≡ 0, then X is torus up to ´ etale covers. We may assume: HSC g ≥ 0 and HSC g ̸≡ 0 Claim Then X is uniruled. In particular, K Y is psef. Theorem (BDPP) X is not uniruled if and only if K X is psef. Proof. Assume K X is psef. Then c 1 ( K X ) ∧ ω n − 1 = − 1 � � Scal g ω n . 0 ≤ π n X X On the the hand, we have � Scal g ( p ) = HSC ( v ) dV FS . v ∈ P ( T X , p ) Hence Scal g is a quasi-positive function on X . 12 / 21

  68. Introduction Main results Strategy of Thm B Details of proof Related topics • If HSC g ≡ 0, then X is torus up to ´ etale covers. We may assume: HSC g ≥ 0 and HSC g ̸≡ 0 Claim Then X is uniruled. In particular, K Y is psef. Theorem (BDPP) X is not uniruled if and only if K X is psef. Proof. Assume K X is psef. Then c 1 ( K X ) ∧ ω n − 1 = − 1 � � Scal g ω n . 0 ≤ π n X X On the the hand, we have � Scal g ( p ) = HSC ( v ) dV FS . v ∈ P ( T X , p ) Hence Scal g is a quasi-positive function on X . 12 / 21

  69. Introduction Main results Strategy of Thm B Details of proof Related topics • If HSC g ≡ 0, then X is torus up to ´ etale covers. We may assume: HSC g ≥ 0 and HSC g ̸≡ 0 Claim Then X is uniruled. In particular, K Y is psef. Theorem (BDPP) X is not uniruled if and only if K X is psef. Proof. Assume K X is psef. Then c 1 ( K X ) ∧ ω n − 1 = − 1 � � Scal g ω n . 0 ≤ π n X X On the the hand, we have � Scal g ( p ) = HSC ( v ) dV FS . v ∈ P ( T X , p ) Hence Scal g is a quasi-positive function on X . 12 / 21

  70. Introduction Main results Strategy of Thm B Details of proof Related topics • If HSC g ≡ 0, then X is torus up to ´ etale covers. We may assume: HSC g ≥ 0 and HSC g ̸≡ 0 Claim Then X is uniruled. In particular, K Y is psef. Theorem (BDPP) X is not uniruled if and only if K X is psef. Proof. Assume K X is psef. Then c 1 ( K X ) ∧ ω n − 1 = − 1 � � Scal g ω n . 0 ≤ π n X X On the the hand, we have � Scal g ( p ) = HSC ( v ) dV FS . v ∈ P ( T X , p ) Hence Scal g is a quasi-positive function on X . 12 / 21

  71. Introduction Main results Strategy of Thm B Details of proof Related topics • If HSC g ≡ 0, then X is torus up to ´ etale covers. We may assume: HSC g ≥ 0 and HSC g ̸≡ 0 Claim Then X is uniruled. In particular, K Y is psef. Theorem (BDPP) X is not uniruled if and only if K X is psef. Proof. Assume K X is psef. Then c 1 ( K X ) ∧ ω n − 1 = − 1 � � Scal g ω n . 0 ≤ π n X X On the the hand, we have � Scal g ( p ) = HSC ( v ) dV FS . v ∈ P ( T X , p ) Hence Scal g is a quasi-positive function on X . 12 / 21

  72. Introduction Main results Strategy of Thm B Details of proof Related topics • If HSC g ≡ 0, then X is torus up to ´ etale covers. We may assume: HSC g ≥ 0 and HSC g ̸≡ 0 Claim Then X is uniruled. In particular, K Y is psef. Theorem (BDPP) X is not uniruled if and only if K X is psef. Proof. Assume K X is psef. Then c 1 ( K X ) ∧ ω n − 1 = − 1 � � Scal g ω n . 0 ≤ π n X X On the the hand, we have � Scal g ( p ) = HSC ( v ) dV FS . v ∈ P ( T X , p ) Hence Scal g is a quasi-positive function on X . 12 / 21

  73. Introduction Main results Strategy of Thm B Details of proof Related topics • If HSC g ≡ 0, then X is torus up to ´ etale covers. We may assume: HSC g ≥ 0 and HSC g ̸≡ 0 Claim Then X is uniruled. In particular, K Y is psef. Theorem (BDPP) X is not uniruled if and only if K X is psef. Proof. Assume K X is psef. Then c 1 ( K X ) ∧ ω n − 1 = − 1 � � Scal g ω n . 0 ≤ π n X X On the the hand, we have � Scal g ( p ) = HSC ( v ) dV FS . v ∈ P ( T X , p ) Hence Scal g is a quasi-positive function on X . 12 / 21

  74. Introduction Main results Strategy of Thm B Details of proof Related topics • If HSC g ≡ 0, then X is torus up to ´ etale covers. We may assume: HSC g ≥ 0 and HSC g ̸≡ 0 Claim Then X is uniruled. In particular, K Y is psef. Theorem (BDPP) X is not uniruled if and only if K X is psef. Proof. Assume K X is psef. Then c 1 ( K X ) ∧ ω n − 1 = − 1 � � Scal g ω n . 0 ≤ π n X X On the the hand, we have � Scal g ( p ) = HSC ( v ) dV FS . v ∈ P ( T X , p ) Hence Scal g is a quasi-positive function on X . 12 / 21

  75. Introduction Main results Strategy of Thm B Details of proof Related topics Step 2: Splitting of T X • Assume that MRC φ : X → Y is holo in Step 2 and Step 3. In Step 2, we will prove the followings: • φ : X → Y is smooth (that is, any fiber F is non-singular). In particular, we have the exact sequence of the tangent bundles: 0 → T X / Y → T X → φ ∗ T Y → 0 . g �� g Q • The sequence above admits the holo and orthogonal decomposition: Namely, a bundle morphism ∃ j : φ ∗ T Y → T X satisfying that T X = T X / Y ⊕ j ( φ ∗ T Y ) , T X / Y and j ( φ ∗ T Y ) is orthogonal w.r.t. g . • a metric ∃ g Y on Y s.t. g Q = φ ∗ g Y and R g Y ≡ 0 . 13 / 21

  76. Introduction Main results Strategy of Thm B Details of proof Related topics Step 2: Splitting of T X • Assume that MRC φ : X → Y is holo in Step 2 and Step 3. In Step 2, we will prove the followings: • φ : X → Y is smooth (that is, any fiber F is non-singular). In particular, we have the exact sequence of the tangent bundles: 0 → T X / Y → T X → φ ∗ T Y → 0 . g �� g Q • The sequence above admits the holo and orthogonal decomposition: Namely, a bundle morphism ∃ j : φ ∗ T Y → T X satisfying that T X = T X / Y ⊕ j ( φ ∗ T Y ) , T X / Y and j ( φ ∗ T Y ) is orthogonal w.r.t. g . • a metric ∃ g Y on Y s.t. g Q = φ ∗ g Y and R g Y ≡ 0 . 13 / 21

  77. Introduction Main results Strategy of Thm B Details of proof Related topics Step 2: Splitting of T X • Assume that MRC φ : X → Y is holo in Step 2 and Step 3. In Step 2, we will prove the followings: • φ : X → Y is smooth (that is, any fiber F is non-singular). In particular, we have the exact sequence of the tangent bundles: 0 → T X / Y → T X → φ ∗ T Y → 0 . g �� g Q • The sequence above admits the holo and orthogonal decomposition: Namely, a bundle morphism ∃ j : φ ∗ T Y → T X satisfying that T X = T X / Y ⊕ j ( φ ∗ T Y ) , T X / Y and j ( φ ∗ T Y ) is orthogonal w.r.t. g . • a metric ∃ g Y on Y s.t. g Q = φ ∗ g Y and R g Y ≡ 0 . 13 / 21

  78. Introduction Main results Strategy of Thm B Details of proof Related topics Step 2: Splitting of T X • Assume that MRC φ : X → Y is holo in Step 2 and Step 3. In Step 2, we will prove the followings: • φ : X → Y is smooth (that is, any fiber F is non-singular). In particular, we have the exact sequence of the tangent bundles: 0 → T X / Y → T X → φ ∗ T Y → 0 . g �� g Q • The sequence above admits the holo and orthogonal decomposition: Namely, a bundle morphism ∃ j : φ ∗ T Y → T X satisfying that T X = T X / Y ⊕ j ( φ ∗ T Y ) , T X / Y and j ( φ ∗ T Y ) is orthogonal w.r.t. g . • a metric ∃ g Y on Y s.t. g Q = φ ∗ g Y and R g Y ≡ 0 . 13 / 21

  79. Introduction Main results Strategy of Thm B Details of proof Related topics Step 2: Splitting of T X • Assume that MRC φ : X → Y is holo in Step 2 and Step 3. In Step 2, we will prove the followings: • φ : X → Y is smooth (that is, any fiber F is non-singular). In particular, we have the exact sequence of the tangent bundles: 0 → T X / Y → T X → φ ∗ T Y → 0 . g �� g Q • The sequence above admits the holo and orthogonal decomposition: Namely, a bundle morphism ∃ j : φ ∗ T Y → T X satisfying that T X = T X / Y ⊕ j ( φ ∗ T Y ) , T X / Y and j ( φ ∗ T Y ) is orthogonal w.r.t. g . • a metric ∃ g Y on Y s.t. g Q = φ ∗ g Y and R g Y ≡ 0 . 13 / 21

  80. Introduction Main results Strategy of Thm B Details of proof Related topics Step 2: Splitting of T X • Assume that MRC φ : X → Y is holo in Step 2 and Step 3. In Step 2, we will prove the followings: • φ : X → Y is smooth (that is, any fiber F is non-singular). In particular, we have the exact sequence of the tangent bundles: 0 → T X / Y → T X → φ ∗ T Y → 0 . g �� g Q • The sequence above admits the holo and orthogonal decomposition: Namely, a bundle morphism ∃ j : φ ∗ T Y → T X satisfying that T X = T X / Y ⊕ j ( φ ∗ T Y ) , T X / Y and j ( φ ∗ T Y ) is orthogonal w.r.t. g . • a metric ∃ g Y on Y s.t. g Q = φ ∗ g Y and R g Y ≡ 0 . 13 / 21

  81. Introduction Main results Strategy of Thm B Details of proof Related topics Step 2: Splitting of T X • Assume that MRC φ : X → Y is holo in Step 2 and Step 3. In Step 2, we will prove the followings: • φ : X → Y is smooth (that is, any fiber F is non-singular). In particular, we have the exact sequence of the tangent bundles: 0 → T X / Y → T X → φ ∗ T Y → 0 . g �� g Q • The sequence above admits the holo and orthogonal decomposition: Namely, a bundle morphism ∃ j : φ ∗ T Y → T X satisfying that T X = T X / Y ⊕ j ( φ ∗ T Y ) , T X / Y and j ( φ ∗ T Y ) is orthogonal w.r.t. g . • a metric ∃ g Y on Y s.t. g Q = φ ∗ g Y and R g Y ≡ 0 . 13 / 21

  82. Introduction Main results Strategy of Thm B Details of proof Related topics Step 2: Splitting of T X • Assume that MRC φ : X → Y is holo in Step 2 and Step 3. In Step 2, we will prove the followings: • φ : X → Y is smooth (that is, any fiber F is non-singular). In particular, we have the exact sequence of the tangent bundles: 0 → T X / Y → T X → φ ∗ T Y → 0 . g �� g Q • The sequence above admits the holo and orthogonal decomposition: Namely, a bundle morphism ∃ j : φ ∗ T Y → T X satisfying that T X = T X / Y ⊕ j ( φ ∗ T Y ) , T X / Y and j ( φ ∗ T Y ) is orthogonal w.r.t. g . • a metric ∃ g Y on Y s.t. g Q = φ ∗ g Y and R g Y ≡ 0 . 13 / 21

  83. Introduction Main results Strategy of Thm B Details of proof Related topics Step 2: Splitting of T X • Assume that MRC φ : X → Y is holo in Step 2 and Step 3. In Step 2, we will prove the followings: • φ : X → Y is smooth (that is, any fiber F is non-singular). In particular, we have the exact sequence of the tangent bundles: 0 → T X / Y → T X → φ ∗ T Y → 0 . g �� g Q • The sequence above admits the holo and orthogonal decomposition: Namely, a bundle morphism ∃ j : φ ∗ T Y → T X satisfying that T X = T X / Y ⊕ j ( φ ∗ T Y ) , T X / Y and j ( φ ∗ T Y ) is orthogonal w.r.t. g . • a metric ∃ g Y on Y s.t. g Q = φ ∗ g Y and R g Y ≡ 0 . 13 / 21

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