ScalingStacks

Case where ∩ i { t i = 0 } = ∅ [02E5]

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Case where ∩i{ti=0}=∅\cap_{i}\{t_{i}=0\}=\emptyset

88 8 It suffices to consider this case for constructing singular KE metrics on algebraic varieties with canonical singularities

We study here the equation

(ωo+t​Ω+d​dc​φt)n=Ct​(|s1|2​k+…+|sp|2​k)​eF​Ωn(\omega_{o}+t\Omega+dd^{c}\varphi_{t})^{n}=C_{t}(|s_{1}|^{2k}+\ldots+|s_{p}|^{2k})e^{F}\Omega^{n}

The first few steps of the preceding argument can be repeated without changes. Next we apply formula (2.22) in [Y] as earlier, except that we set F=Fε,t+log⁡‖s‖(2​k)F=F_{\varepsilon,t}+\log||s||^{(2k)}, where ‖s‖(2​k):=|s1|2​k+…+|sp|2​k||s||^{(2k)}:=|s_{1}|^{2k}+\ldots+|s_{p}|^{2k}. This yields

eC​φt​Δ′​(e−C​φt​(n+Δ​φt))\displaystyle e^{C\varphi_{t}}\Delta^{\prime}(e^{-C\varphi_{t}}(n+\Delta\varphi_{t})) ≥\displaystyle\geq Δ​Fε,t+Δ​log⁡‖s‖(2​k)−n2​C′−C​n​(n+Δ​φt)\displaystyle\Delta F_{\varepsilon,t}+\Delta\log||s||^{(2k)}-n^{2}C^{\prime}-Cn(n+\Delta\varphi_{t})
+(e−Fε,t‖s‖(2​k))1/(n−1)​(n+Δ​φt)nn−1\displaystyle+\left(\frac{e^{-F_{\varepsilon,t}}}{||s||^{(2k)}}\right)^{1/(n-1)}(n+\Delta\varphi_{t})^{\frac{n}{n-1}}

We recall the two preceding inequalities and observe two new ones that are available:

Δ​Fε,t≥C1\displaystyle\Delta F_{\varepsilon,t}\geq C_{1}\; and e−Fε,t/n−1≥C3>0;\displaystyle\;e^{-F_{\varepsilon,t}/n-1}\geq C_{3}>0;
Δ​log⁡‖s‖(2​k)≥C2\displaystyle\Delta\log||s||^{(2k)}\geq C_{2} and C4≥‖s‖(2​k).\displaystyle C_{4}\geq||s||^{(2k)}.

Setting as earlier y=n+Δ​φty=n+\Delta\varphi_{t}, we get

eC​φt​Δ′​(e−C​φt​(n+Δ​φt))≥C5+C6​y+eC7​ymm−1.e^{C\varphi_{t}}\Delta^{\prime}(e^{-C\varphi_{t}}(n+\Delta\varphi_{t}))\geq C_{5}+C_{6}y+e^{C_{7}}y^{\frac{m}{m-1}}.

After this point, the proof is entirely the same as before.

Remark 3.7.

In order to carry out the second order a priori estimate, one needs information that only depend on supXF\sup_{X}F and infXΔ​F\inf_{X}\Delta F. This is pointed out in [Y], p. 351, and it is the basis for the proof of [Y], Thm 3.

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