[注：下文是刚发出的群邮件内容，原标题“the art of replacement”。最近一直在学习。]

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. 

Is not motion just an illusion, like time ?

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 α ⌊ ⌋ ∨∧∞Φ⁺⁻⁰ 1

4. Complements in a neighbourhood of a non-klt centre

---- "neighbourhood" is a topological concept.

---- "non-klt centre" (of a pair) is a "divisor"; it might be viewed as a "point".

---- "divisor" has components with coefficients.

---- Say, S = s·U + t·V.

---- Take (s, t) as a centre, one may draw a circle.

---- That circle defines the boundary of an area.

---- Divisors "over" the area might form a "neighbourhood" of S.

---- An understanding of the moment.

.

In this section, we prove our main result on the existence of complements (Theorem 1.9).

---- This is the target.

.

It does not follow directly from [5] but the proofs in [5] work with appropriate modifications.

---- Indicate the origin of the proof, a work by the same author.

---- [5] is [3] of v1.

.

First we treat a special case of the theorem.

---- This is to treat the issues of "we can assume" of v1 Step 1c.

.

Comment: "we can assume" might represent the true spirit of algebra.

---- If the big structure is captured, local issues become "preys".

.

Proposition 4.1. Theorem 1.9 holds under the additional assumption that there is a boundary Γ such that

• (X, Γ) is plt with S = ⌊Γ⌋, and

• α·M - (Kx + Γ) is ample for some real number α > 0.

---- That is, two items are added to the six items of Th1.9.

---- In the proof of Th1.9, near the end of Step 4 there, a "Γ" is defined (in an image space) and proved to satisfy the additional two items here.

---- After that, this pro4.1 is activated to end the proof of Th1.9.

.

Proof. Step 1. In this step we reduce to the case when α ∈ (0, 2) and when B - Γ has small (positive or negative) coefficients.

---- The range of α is specified, while the coefficient of B - Γ are qualitatively constrained.

---- This "Γ" here is nominal.

---- It is attribution that actually works.

.

Since M - (Kx + B) is nef and big and α·M - (Kx + Γ) is ample,...

---- Two "buttons" are pressed...

.

(1 - t + t·α)·M - (Kx + (1 - t)·B + t·Γ) = (1 - t + t·α)·M - (1 - t)·(Kx + B) - t·(Kx + Γ) = (1 - t)(M - (Kx + B)) + t(α·M - (Kx + Γ))

is ample for any t ∈ (0, 1).

---- The actual starting position is to setup a convex combination of M - (Kx + B) and α·M - (Kx + Γ).

----| If I may, to use the notations of mine...

----| (1 - t)·[nb] + t·[ample] ==> (1 - t)·c(M) + t·c(αM)Γ is ample ==> the left-hand side is ample.

----| [·] contains the attribution of the object.

----| "nb" is the shorthand of "nef and big".

---- Now, I check 1 - t + t·α = 1 - t·(1 - α).

----| As the range of (0, 2) is desired for α, the range for 1 - α is (-1, 1).

----| For small t, the value of 1 - t + t·α is around 1, closely.

.

Comment: the expression (1 - t)·B + t·Γ on the left-hand side is of author's concern, as one sees soon.

---- The left-hand side has the form of M1 - (Kx + Γ1)...

---- with (1 - t + t·α)·M as M1, and (1 - t)·B + t·Γ as Γ1.

---- In this way, M1 - (Kx + Γ1) falls into the mode "α·M - (Kx + Γ)" which is ample.

.

Thus replacing Γ with (1 - t)·B + t·Γ for some sufficiently small real number t > 0, we can replace α by some rational number in (0, 2).

---- Now, author takes (1 - t)·B + t·Γ as Γ for small t (> 0).

---- As analyzed above, the value of 1 - t + t·α is around 1 closely for small t, when α falling in (0, 2).

---- For later convenience (?), I guess, author considers a rational setting for α in (0, 2).

---- So, "some real number α > 0" is replaced by "some rational number in (0, 2)".

---- That rational number is stilled denoted as α.

.

Comment: when stating "replace", author refers to the second item in Pro4.1.

---- After replacement, the notates are still kept as the original ones.

---- A common practice known to programmers.

.

Note that since (X, B) is lc and since S is a non-klt centre of this pair, we have S ≤ ⌊B⌋...

---- As no reference is given, it should be light. 

.

and the above change of Γ preserves the plt property of (X, Γ) and the condition S = ⌊Γ⌋.

---- It should be light.

.

It is also clear that if we choose t small enough then we can ensure that B - Γ has sufficiently small positive or negative coefficients (we will use this in steps below).

---- Note that, Γ here is the replaced one.

---- That is, B - Γ refers to B - ((1 - t)·B + t·Γ) = t(B - Γ).

---- By original items, the coefficients of B - Γ are not treated.

---- That is, the signs are generally indefinite.

---- Clearly, t(B - Γ) has sufficiently small coefficients (generally sign-indefinite) for small t.

.

Special comment: "B - Γ has sufficiently small coefficients" might serve as the "matics" behind the art of replacement.

.

Skill: (B - Γ1 =) B - ((1 - t)·B + t·Γ) = t(B - Γ)

.

.

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

.

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]

.

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.

.

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

.

.

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