On the number and boundedness of log minimal models of general type
On the number and boundedness of log minimal models of general type
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DOI：
10.24033/asens.2443
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发表时间：
2016-10
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影响因子：
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通讯作者：
Diletta Martinelli;Stefan Schreieder;L. Tasin
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文献类型：
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作者：
Diletta Martinelli;Stefan Schreieder;L. Tasin
我们证明了一般类型的n维光滑复射影簇的标记极小模型的数目可以用其体积来定界，并且，如果n=3，也可以用其Betti数来定界。对于一个n维射影klt对（X，D），当$KX +D$大时，我们更一般地证明了它的弱对数正则模型的个数可以用D的系数和$KX +D$的体积有界.我们进一步证明了所有n维射影klt对（X，D），使得$KX +D$是大的且nef是固定体积的，并且使得D的系数包含在给定的DCC集中，形成一个有界族.由此可见，在任何维数上，一般类型和有界体积的极小模型构成一个有界族。
We show that the number of marked minimal models of an n-dimensional smooth complex projective variety of general type can be bounded in terms of its volume, and, if n=3, also in terms of its Betti numbers. For an n-dimensional projective klt pair (X,D) with $K_X+D$ big, we show more generally that the number of its weak log canonical models can be bounded in terms of the coefficients of D and the volume of $K_X+D$. We further show that all n-dimensional projective klt pairs (X,D), such that $K_X+D$ is big and nef of fixed volume and such that the coefficients of D are contained in a given DCC set, form a bounded family. It follows that in any dimension, minimal models of general type and bounded volume form a bounded family.