[注：下文是 3 月 14 日 (1: 07) 发出的群邮件内容，标题为原有的。]
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.
Mathematics can hardly be taught by others; only voluntary self-learning works.
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.
♙ ℂ ℍ ℕ ℙ ℚ ℝ ℤ ℭ ℜ I|φ∪∩∈ ⊆ ⊂ ⊇ ⊃ ⊄ ⊅ ≤ ≥ Γ Θ α Δ δ μ ≠ ⌊ ⌋ ∨∧∞Φ⁺⁻⁰ 1
(continued) A proposition (Pro4.1) is arranged as a special case of Theorem 1.9, which is needed by the proof of the theorem. This proposition introduces two auxiliary items into the statement of the theorem, i.e. a plt-Eve pair and a "moderated" confrontation. One will see, in the final proof of the theorem, near the end there, the two auxiliary items are constructed to have the proposition activated, arriving at the desired conclusion. In this view, the proposition serves as a "sub-routine" in the proof of Theorem 1.9. The two auxiliary items are quoted here ——
* (X, Γ) is plt with S = ⌊Γ⌋, and
* α·M - (Kx + Γ) is ample for some real number α > 0.
One can view the second item here as the interaction between M and (X, Γ), the auxiliary pair. All together, there are eight items for Pro4.1, summarized as a modified* graph ——
.
M
c(·)nb c(α·)Γ
S
Γ pB
XF
.
Illustration for the eight items of Pro4.1.
.
It is interesting to note that, as one sees in the statement/ proof, each object at the vertices interacts with S situated at the centre. I use c(·) and c(α·)Γ to avoid messes in the graph, namely using "·" to replace M in the confrontations. In proper context, the subscript of c(α·)Γ is dropped for shorthand. This ends the description of the statement of Pro4.1. So comes the description of the proof of this proposition ——
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It is needed later that B - Γ has sufficiently small coefficients (positive or negative). But, this is not the case in general. The key idea is to derive another Γ, specialized, satisfying the two auxiliary items as well as the requirement on the coefficients. A simple but useful identity makes this idea possible: B - ((1 - t)B + tΓ) = t(B - Γ) holds for any real number t, viewing (1 - t)B + tΓ as the another Γ. Apparently, one can tuning the real number t to control the coefficients on the left of the identity as desired, for the given B and Γ on the right. The remaining matter is to verify this another Γ satisfies the two auxiliary items ——
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To proceed, it is convenient to denote this another Γ as Γ1 = (1 - t)B + tΓ, avoiding confusions. For one thing, one expects c(β·M)Γ1 ample, where β plays the role of α in c(α·)Γ. Actually, one can verify that the form of Γ1 is originated from the convex combination of c(M) and c(α·M)Γ —— (1 - t)[nb] + t[ample] /ample ==> (1 - t)·c(M)B + t·c(α·M)Γ = c(β·M)Γ1 /ample, where β = (1 - t)·1 + t·α, with any t ∈ (0, 1). Note that β can be viewed as the convex combination of 1 and α, the scaling factors of M. The convex structure on the left is also conveyed to the subscript Γ1 on the right. For a better correspondence, one agrees that c(M) has a default subscript B, or denoted explicitly as c(M)B. The factor β in the consequent c(β·M)Γ1 is subtly related to t and α. As desired, t ∈ (0, 1) is tuned sufficiently small to make B - Γ1 = t(B - Γ) have sufficiently small coefficients (positive or negative). This makes β = (1 - t)·1 + t·α = 1 - t·(1 - α) close to 1. It is "canonical" to require 1 - α fall between -1 and 1, making α ∈ (0, 2), so that β falls into a small neighbourhood of 1. For later usage, α is replaced by some rational number in (0, 2). For the other, as (X, B) is lc and S is a non-klt centre of this pair, (X, Γ1) is plt with S = ⌊Γ1⌋, according to the paper (the detailed deduction on this technical point is left to the future). Since Γ1 fulfills the two auxiliary items exactly as Γ, the subscript of Γ1 is dropped in the later usage. However, one must remember that this another one (Γ1) is not the original one (Γ).
.
Before the next move, it is worthwhile to ask how the thoughts* of such a plt pair originally occurred to the author? One can always ask such a question on the origins of things presented in a mathematical paper. The answer is generally elusive, however, although the concerned matter is an invention of today, instead of handed down from the ancient time. Another strange (modern) phenomenon is that authors feel inconvenient to state things only known to themselves yet not expected in the formal publications. After some while, authors themselves might also forget those information. Here, I'm not asking the origin of the plt pair as a concept. I'm asking the original context for the thoughts. Nevertheless, one can safely expect that, the two auxiliary items on Γ are either deduced from the original six items of Theorem 1.9, or related to some references.
*This question is unexpectedly answered upon the summary for the description of the second move below.
.
As one can see, the objects c(M) and c(α·M) play key roles together in the last move. This is still the case for the second move. Roughly speaking, it is going to have the main pair (X, B) confined on S, so that an n-complement can be found for this confined main pair, with the help of c(M) and c(α·M) confined on S. One can practice informally in this way: c(M)|s = (M - (Kx + B))|s = M|s - (Kx + B)|s = - (Kx + B)|s as M|s ≡ 0 by the sixth item. One can expect c(M)|s is nef and big ("nb" for short) as c(M) is nef and big by the fifth item. That is to say, -(Kx + B)|s is nef and big, expectantly. Similarly, c(α·M)|s = - (Kx + Γ)|s is ample, expectantly. In the formal proof, the equal signs are replaced by "~", as M is semi-ample. The informal practices here might help one think of Th 2.13, aided by the memo phrase "nef-lc-leaf Fano type leads to an n-complement". So, the thought for the second move is to apply Th 2.13 to the confined main pair under consideration. (Hint: here is the right position to review Th 2.13). Another technical point is the role of the plt pair (X, Γ). This "vice" pair (being plt) is to justify the operation of confining the main pair (X, B) on S. The confined main pair is immediately an lc-leaf pair (with references), satisfying two items of Th 2.13. Interestingly, this vice pair can also be confined on S. The confined vice pair is immediately a klt pair. This makes S be of Fano type as c(α·M)|s is ample (thus nef and big). All together, Th 2.13 is activated for the confined main pair (S, Bs), arriving at the desired n-complement. The path of thought (of Step 2) can be illustrated by a path graph ——
.
(B) plc
|
(Γ) plt
↙ ↘
(B)|s lc-leaf (Γ)|s klt
+ +
c(·)|s nb c(α·)|s ample
↓ ↓
(Bs) S
nef-lc-leaf Fano type
↘ ↙
Th 2.13
applied to (Bs)
----
Note: here I use (·) to denote a pair, with "·" refers to the boundary, omitting the default variety. The key nodes are highlighted on the path graph. (It is less relevant, but I guess algorithms of graph theory and combinatorics are used in computer aided proof.)
.
In summary, the main pair and the associated confrontation are confined on the non-klt centre, together with their auxiliary counterparts. The auxiliary plt pair makes this operation legal* under the governing of the main pair which is projective lc, so that Th 2.13 is customized to the confined main pair (S, Bs).
*This answers the question on the arising of the auxiliary plt pair.
.
Before the third move, it is beneficial to illustrate the "ground framework" emerging from the last two moves ——
.
B ~ Γ
| |
M ~ α·M
----
Note: B and Γ may represent the corresponding pairs, while M and α·M are preferably "armored" with c(·). In this view, move one plays the convex combination while move two applies the adjunction ---- both based on the ground framework.
.
The essential outputs of the last move are a natural number n and a greater boundary B⁺s, governed by the concept of n-complement, with the enhanced feature of B⁺s ≥ Bs due to Th 2.13. By "n-complement", here customized to the confined main pair (S, Bs), one has ——
* (S, B⁺s) is lc,
* n(Ks + B⁺s) ~ 0,
* nB⁺s ≥ nTs + ⌊(n + 1)Δs⌋,
where Ts = ⌊Bs⌋ and Δs = Bs - Ts. By the enhanced feature of B⁺s ≥ Bs, one has ⌊(n + 1)Δs⌋ = nΔs due to the logical consistency. This identity gives a hint to the later development. Concerning the two essential outputs of the last move, let us put B⁺s aside for a while, and consider the usage of n which is independent from the adjunction. (Recall that, n depends only on d and ℭ, a finite set of rational numbers in [0, 1]).
.
By observing the items in the definition of n-complement, one agrees that this special natural number n has the tendency to have the concerned divisors scaled, a kind of "mode". It is desirable to have this feature fulfilled in view of the ground framework. In particular, it appears "canonical" to study n·c(M) as the first trial —— n·c(M) = n·(M - (Kx + B)) = nM - nKx - nB. In the context of n-complement, B is divided as B = T + Δ, with T = ⌊B⌋. If one assumes nB is integral, one has ⌊(n + 1)Δ⌋ = nΔ. So, one can write nB = n(T + Δ) = nT + nΔ = nT + ⌊(n + 1)Δ⌋. Now, one has exactly —— n·c(M) = nM - nKx - nT - ⌊(n + 1)Δ⌋ —— One agrees to view ⌊(n + 1)Δ⌋ as the "general" form of nΔ. For some curious reason, author has considered n·c(M) + 2M ——
.
n·c(M) + 2M = (n + 2)M - nKx - nT - ⌊(n + 1)Δ⌋. (%)
.
Now, replace the n-complement division B = T + Δ with the more specialized version B = E + Δ, yet speaking in the space of X': Let E' be the sum of the components of B' which have coefficient 1, and let Δ' = B' - E'. So comes the "magic" of definition ——
.
L': = (n + 2)M' - nKx' - nE' - ⌊(n + 1)Δ'⌋.
.
In view of the ground framework, it is preferable to see c(M) in the space of X'. So comes the primary deduction ——
.
L' = 2M' + nM' - nKx' - nE' - nΔ' + nΔ' - ⌊(n + 1)Δ'⌋
= 2M' + n·c(M') + nΔ' - ⌊(n + 1)Δ'⌋
.
{Added #2: One agrees that ⌊(n + 1)Δ'⌋ is not (⌊(n + 1)Δ⌋)'; the later is equal to nΔ' }. The resulted four items are grouped by underlines for a better view. For some curious reason, I define c'n:= L', n'c: = 2M' + n·c(M'), and d'n: = nΔ' - ⌊(n + 1)Δ'⌋. In the new notations, the above expression is rewritten as c'n = n'c + d'n. Or, in the more “symmetric” form of c'n - n'c = d'n.
.
Discussion: Combined with the equation (%), taking T as E, one sees the correspondence between n·c(M) + 2M and 2M' + n·c(M'). Is not the later the pullback of the former? It appears clearly so. In this view, L' = 2M' + n·c(M'), implying nΔ' - ⌊(n + 1)Δ'⌋ = 0. This later equation might not hold, however. So, one sees the equation (%) does not generally hold in the space of X', very likely due to the floor operation which is not a linear one. Therefore L' = 2M' + n·c(M') does not hold in general. Indeed, the floor operation is introduced to create the discrepancy between L' and 2M' + n·c(M'). These dazzling relations can be clarified by a graph ——
.
n·c(M) + 2M = (n + 2)M - nKx - nE - ⌊(n + 1)Δ⌋ (= L?)
↑ ↗ ↑?
n·c(M') + 2M' ≠ (n + 2)M' - nKx' - nE' - ⌊(n + 1)Δ'⌋ (= L' )
----
Note: the arrow starts from the pullback side. ( I'm a bit suspicious on the pullback on the right-hand side, question marked). {Added #1: actually, one verifies (⌊(n + 1)Δ⌋)' = (nΔ)' = nΔ' ≠ ⌊(n + 1)Δ'⌋. Could L have two distinct pullbacks? Should not.}
.
In summary, move three is to construct L' which evolves from n·c(M) in the context of n-complement customized to a special division of B defined in X'. In view of the ground framework, L' is reorganized to let c(M') appear, resulting in an essential output of nΔ' - ⌊(n + 1)Δ'⌋, or d'n, which carries the messages from n-complement as well as the special division. In mathematics, essential divisions tend to hand themselves down to the end (TOM).
.
So far, the objects n, c(M) and B have participated in the construction of L'. To have the ground framework fulfilled, Γ and c(α·M) are expected to play some (auxiliary) roles. The scene is setup by L', namely the apostrophe space X'. It turns out Γ' appears in the forth move, together with the essential output of the last move, in order to construct an auxiliary divisor. The deeper thought, however, is to construct another plt-Eve pair, so that the Kawamata- Viehweg vanishing theorem is activated (cf. move six). It would be nice if this another plt-Eve pair could be constructed in a simple way, say, by the sum of Γ' and d'n. That is to say, it is a natural wish that (X', Γ' + d'n) formed the desired plt-Eve pair. However, this wish appears to work only for D' = S'. The next trial is to modify Γ' + d'n in a simple way, so that X' and the modified Γ' + d'n form the desired plt-Eve pair. This simple way is just to subtract the floor part from Γ' + d'n ——
.
Γ' + d'n - ⌊Γ' + d'n⌋
.
This falls into the mode of "self-production"* (as I call), a meta form. By "meta", I refer to any thing whose origin is itself. One can write this form as · - ⌊·⌋, where "·" represents any object associated with numbers. For the reason clarified shortly, S' is not desired as a component of the mending part, now denoted as P'. So comes the requirement of μS'P': = 0 (i.e. for D' = S'), taking μD'P': = - ⌊Γ' + d'n⌋ for D' ≠ S'. For the convenience of reference, one can assign a letter, say, Θ' to denote the modified Γ' + d'n which now takes the form of ——
.
Θ': = Γ' + d'n + P'.
.
As stated earlier, it is desired (X', Θ') to be a plt-Eve pair, i.e. a plt pair with ⌊Θ'⌋ = S'. Clearly, S' is not desired as a component of P', as Γ' has already contributed S' as a component of Θ'. Recall, S' is not a component of d'n, either. Therefore, given the coefficient of each component of d'n + P' is greater than zero and smaller than one, one would have ⌊Θ'⌋ = S'. The rigorous proof of this point has not come up to me. Also, the proof for (X', Θ') as a plt pair remains to be shown (as I cannot see). The treatment to the proofs for the properties of P' (i.e. non-negative & exceptional) is left to the future. To conclude move four, I resume the path graph ——
.
Th 2.13
applied to (Bs)
↙ ↘
n B⁺s
↓ ↑ L'+P'
c'(·) c'(α·)
↓ ↑ P', Θ'
B' → Γ'
L', d'n
----
Note: From move two, one obtains the outputs n and B⁺s. Then, n is applied to c'(M) to produce L' and d'n, combined with the n-complement division of B'. Next, d'n is geared on Γ' to produce P' by a self-production skill, aimed to form a plt-Eve pair (X', Θ'). In move five, as one will see, c'(α·M) is used to reorganize L' + P', in order to activate the Kawamata- Viehweg vanishing theorem, a kind of “juridical” procedure driven by the plt-Eve pair (X', Θ'). Inevitably, one meets B⁺s in move six, taking the game back to the realm of S'. (One can see, move one to move five are governed by the ground framework, identified in the description of move two). —— So far I can see.
.
ℭ ℜ I|φ∪∩∈ ⊆ ⊂ ⊇ ⊃ ⊄ ⊅ ≤ ≥ Γ Θ α Δ δ μ ≠ ⌊ ⌋ ⌈ ⌉ ∨∧∞Φ⁺⁻⁰ 1
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]
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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.
.
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.
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