论文标题
立方体表面模量空间的非同态平滑压缩
Non-isomorphic smooth compactifications of the moduli space of cubic surfaces
论文作者
论文摘要
复杂立方体表面的模量空间具有三种不同但同构的，紧凑的实现：作为git商，作为球bail的 - 孔的压实，是球商的压实，作为压实的$ k $ -moduli空间。从所有三个角度来看，都有一个与非稳定表面相对应的唯一边界点。从GIT的角度来看，要处理这一点，自然要考虑Kirwan爆炸，而从球商的角度来看，考虑环形紧凑型是很自然的。这两个空间都具有相同的同胞学，因此自然要问它们是否是同构。在这里，我们表明事实并非如此。确实，我们表明了更精致的陈述，即两个空间在Grothendieck戒指中都是等效的，但不是$ k $等效的。在此过程中，我们建立了许多结果和技术，用于处理Kirwan爆炸的奇异和规范类别的球员和环形的压缩。
The moduli space of complex cubic surfaces has three different, but isomorphic, compact realizations: as a GIT quotient, as a Baily--Borel compactification of a ball quotient, and as a compactified $K$-moduli space. From all three perspectives, there is a unique boundary point corresponding to non-stable surfaces. From the GIT point of view, to deal with this point, it is natural to consider the Kirwan blowup, while from the ball quotient point of view it is natural to consider the toroidal compactification. Both these spaces have the same cohomology and and it is therefore natural to ask whether they are isomorphic. Here we show that this is in fact not the case. Indeed, we show the more refined statement that both spaces are equivalent in the Grothendieck ring, but not $K$-equivalent. Along the way, we establish a number of results and techniques for dealing with singularities and canonical classes of Kirwan blowups and toroidal compactifications of ball quotients.
