[按：导言下方是群邮件的内容，标题未改动。略有修订。]

导言：去年11月1日重启菲奖论文学习，半道发现出来个 v2 (12月1日贴出)，改动较大，内容扩充了 50% (30页--> 45页)。当时正在写 v1的第三轮笔记，逐行注释某个定理，坚持做完了 (弄出个"礼乐传")。到了12月31日，转到 v2，重新写那个定理的笔记，今天完成该定理v2版的最后一篇笔记。(后续打算重写该定理的证明 —— 按我的理解和风格)。

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This is coming to you from Yiwei LI (PhD, Applied math), Taiyuan University of Science and Technology  (TYUST) Taiyuan, China

It's going on here for the third round of learning of Birkar's BAB-paper (v2), with scenarios of chess stories. No profession implications. 

Textbook thinking is able to transform one into a textbook writer who tends to deliver others such a message that "Party is over". 

Th 2.15    Th 1.8            ♖    ♘

     ↓   ↖     ↓

Th 1.1      Th 1.6            ♔    ♗

    Mathematics vs Palace stories.(v2)

Note: technical theorem is not on the board.

M          Λ           ♖          ♘

       S                          ☂

XF        pB          ♔          ♗

Note: the upper right/left corner is of output.

♙ ℂ ℍ ℕ ℙ ℚ ℝ ℤ ℭ ℜ I|φ∪∩∈ ⊆ ⊂ ⊇ ⊃ ⊄ ⊅ ≤ ≥ Γ Θ α Δ δ μ ≠ ⌊ ⌋ ∨∧∞Φ⁺⁻⁰ 1

Step 7, Para three ——

Assume now that (X, B⁺) is not lc over z = f(S).

---- This is to setup the assumption for a proof by contradiction.

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By the previous paragraph, (X, B⁺) is lc near S and S is a non-klt centre of this pair.

---- This is a nice summary, so that one can judge what is "lc near S".

---- Basically, for certain pair, Say (X, B), one can define Ks + Bs by (Kx + B)|s.

---- Or, given Ks + Bs, one can recover the original pair by "inversion of adjunction", to show Ks + Bs = (Kx + B)|s.

---- The divisor S is required to be a "non-klt centre" of the original pair.

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On the other hand, (X, Γ) is plt with ⌊Γ⌋ = S, so if u > 0 is sufficiently small, then (X, (1 - u)B⁺ + uΓ) is plt near S and S is a non-klt centre of this pair and no other non-klt centre intersects S.

---- In combination of last sentence, if (X, B⁺) is lc near S and (X, Γ) is plt near S, then the pair of (small) convex combination is still plt near S.

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Comment (wrong intuition): At the first glance, one might feel ⌊(1 - u)B⁺⌋ = 0 and ⌊uΓ⌋ = 0, for their coefficients are smaller than 1 ——

---- This is really the case. (No kidding).

---- But, there is a re-organization matter, however, for the sum of the two terms, before taking the floor operation.

---- That is, (1 - u)S + uS = S, while S has the coefficient 1.

---- For this aspect, u can be any number in [0, 1].

---- The small positive u is required by some other aspects.(to be hunted).

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Then since (X, B⁺) is not lc over z, the non-klt locus of (X, (1 - u)B⁺ + uΓ) has at least two connected components (one of which is S) near the fibre f ^⁻1 {z}.

---- Translation: if the original pair is not lc over z, then the (plt convex) combined pair has its non-klt locus possessed at least two connected components near f⁻1 {z}. 

---- To illustrate this in a simple way, take the boundary as the pair ——

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      B⁺   ~   u(B⁺)Γ 

  not | lc       not | simple    

      z     ~   f ^⁻1 {z}

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Note: u(B⁺)Γ, understood in context, is the homemade notation for (X, (1 - u)B⁺ + uΓ).

Note: I call a (plt) pair is not simple, if its non-klt locus has least two connected components near f⁻1 {z}.

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This contradicts the connectedness principle [25, Theorem 17.4] as - (Kx + (1 - u)B⁺ + uΓ) = - (1 - u) (Kx + B⁺) - u(Kx + Γ) ~R -u(Kx + Γ) ~R uαM - u(Kx + Γ)/Z is ample over Z.

---- That is, the defence form of u(B⁺)Γ is ample over Z, which expects u(B⁺)Γ  simple near f ^⁻1 {z} by the connectedness principle.

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Therefore, (X, B⁺) is lc over z.

---- This is the close statement.

---- It's needed.

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Summary comment: This is the end of proof of Pro4.1, the special case of Th1.9 (i.e. v1 Th1.7).

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I give an instant praise, inspired by a recent blog article* out of a theoretical physicist ——

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Math and Matics have laid hid in the darkness: 

Fears of contempt propagate among folks.

So has formed the culture to puzzle, blind and block others...

Consciences yell: let birational algebraic geometry be.

And all becomes primary.

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Calling graph for the technical theorem (Th1.9) ——

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Th1.9

    |

[5, 2.13(7)]   Lem 2.26   Pro4.1   Lem2.7

                                                          |

.......................................................Lem2.3   

Note: Th1.9 is only called by Pro.5.11, one of the two devices for Th1.8, the executing theorem.

Pro4.1                                                    

    |

[5, ?]   [37, Pro3.8]   [5, Lem3.3]   Th2.13[5, Th1.7]   [16, Pro2.1.2]  [20]  [25, Th17.4]

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Special note: Original synthesized scenarios in Chinese for the whole proof of v1 Th1.7, the technical theorem.

*It's now largely revised* due to new understandings.

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See also: Earlier comments in Chinese* (v1).

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It is my hope that this action would not be viewed from the usual perspective that many adults tend to hold.