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菲文笔记 | Technical theorem (v2) --- mathematics and capitalist

已有 1086 次阅读 2021-3-21 21:06 |个人分类:师生园地|系统分类:科研笔记

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. 

One can learn a new theorem from its original place; or one can re-assemble the theorem in the context where it arises...

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

(continuedKawamata- Viehweg vanishing theorem assembled (nominally) ——

Assume

*  (X, ฿) is plt with S = ⌊฿⌋,

*  (X, ฿ - S) is klt,

*  A is nef and big,

*  O is an integral divisor satisfying O - A = Kx + ฿,

Then

* h1(O - S) = 0, and

* H⁰(O) --> H⁰(O|S) is surjective.

.

O    -     A

        \

(฿)   -    S

Note: It is possible that one of the first two items in the condition is redundant.

---- Another possibility is that, an projective lc pair (X, B) with non-klt centre S is mandatory, with (X, ฿ - S) being klt automatically.

---- One can use Γ instead of ฿, dismissing the smoke of bit coin*.

---- In the context of the paper, L' + P' takes the role of O (in the X' space).

*One will see, plt is of "capitalist".

.

Memo: the first item is also referred as "plt-Eve pair" (as I call).

---- Let plt be a default attribution, acquiring a shorter name of "Eve pair".

---- In combination with the second item, one has a (plt-)Eve-klt pair.

---- I call an integral divisor O is "plt type", if there is a "nef and big" divisor A, such that the forth item holds.

---- That is, Eve-klt plt type lead to the KV- vanishing theorem.

---- cf. nef-lc-leaf Fano type lead to the bounded n-complement. (Memo for Th2.13).  

.

Comments: Looking at the conclusions, one knows this theorem is about a game between O and S.

---- For one thing, O - S forms the core/ root of the special map/ function h1.

---- For another, O has the potential to be recovered from its adjunction on S.

---- KV vanishing theory has strong implication on the adjunction approach.

---- Note that, both O and S are integral divisors.

---- In particular, KV suggests O and A be constructed.

.

Special comments: In retrospect, the plt pair arises largely due to the demand of the KV vanishing theory.

---- The raw thought of the technical theorem is to have Th2.13 customized to the adjuction governed by the main pair, acquiring the "key" (n) to activate the KV vanishing theorem, a set of “juridical” system  to help form the "product" (L') suggested by the adjunction (Bs) and convey it to the whole world ---- along the way the plt system plays the role of "capitalist".

---- The remaining myth is on the invention of c(M), i.e. the special structure of M - (Kx + B). (?)

---- 此过程很像政!&府主导的产业:进口现成的方案(Th2.13),交由体制内研发机构 “二次开发”,最后提交给国有企业生产及发布 ---- 这一切都离不开资!&本(家)。

n⁺          plt

          

KV         adj

.

In summary, the technical theorem is of the KV vanishing theorem in the context of n-complement (KV vs n⁺).

.

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]

.

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.




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