# 接近结束啦...

This is an in-mail from TYUST.

(接前: 22 21 20) 引理3.2.
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Since (Y, BY + sLY) is klt weak log Fano, Y is Fano type.
---- 条件和结论并不显然.(?)
---- klt 须满足 a(D, Y, BY + sLY) > 0, 但看不出.
(可能引用了某个现成的结论)
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Run an MMP on -T and let Y' --> Z' be the resulting Mori fibre space.
---- 在 -T 上做MMP似乎是罕见的.
(此句作为方法记住)
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Then -(KY' + BY' + sLY') ~R (1 - s)LY'   0.
---- 左端是带符号的“国”, 右端是带尺度因子的.
---- 此结论并不显然 (?).
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Moreover, (Y', BY' + sLY') is eps'-lc because (Y, BY + sLY) is eps'-lc and -(KY + BY + sLY) is semi-ample.
---- 这里可以作为结论记住:
---- 配对(eps-lc) 及 带符“国”(丰) ===> 像配对(eps-lc).
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If dim Z' > 0, then restricting to a general fibre of Y' --> Z' and applying Theorem 1.4 in lower dimension by induction (or applying Theorem 1.1) shows that the coeffcients of the horizontal/Z' components of (1 - s)LY' are bounded from above.
---- 此结论并不显然.(?)
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In particular, μT'(1 - s)LY' is bounded from above.
---- 推出 T' 在 (1-s)LY' 有上界.
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Thus from the inequality μT'(1 - s)LY'  (1 - s)(eps - eps')/s, we deduce that s is bounded from below away from zero.
---- 上述不等式该是由第一段的落点得出.
---- 两边约去(1 - s), 然后两边同乘以 s.
---- 这里只是给 T 和 Y 带上了撇.
---- 特别地, 若 s -> 0, 则右端无穷大, 这与左端有上界矛盾(见上一句).
---- 于是 s 必须有正的下界.
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Therefore, we can assume Z' is a point and that Y' is a Fano variety with Picard number one.
---- 后半句不好理解(好似凭空假设).
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Now -KY' ~R (1 - s)LY' + BY' + sLY'≥ (1 - s)LY', ...
---- 此等价关系是怎么得到的?
---- 明白了, 是从上面第三句移项得到的.
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...so by Proposition 3.1, μT'(1 - s)LY' is bounded from above which again gives a lower bound for s as before.
---- 按命题3.1, 此处对应的条件是 -KY' ~R (1 - s)LY' + BY' + sLY'.
---- 即 那里的 L 相当于此处的 L':=(1 - s)LY' + BY' + sLY'.
---- 但此处的T' 跟命题中的 T 来源方式不同... (?)
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符号大全上下标.|| 常用：↑↓ πΓΔΛΘΩμφΣ∈  ∪ ∩ ⊆ ⊇ ⊂ ⊃ ≤ ≥ ⌊ ⌋ ⌈ ⌉ ≠ ≡ ⁰ ¹ ² ³ ᵈ ₀ ₁ ₂ ₃ ᵢ .

#### Glossary(AG)

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Introduction
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http://blog.sciencenet.cn/blog-315774-1190790.html

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