论文标题
关于K3表面模量空间的非理性性
On the irrationality of moduli spaces of K3 surfaces
论文作者
论文摘要
我们研究了极化K3表面的模量空间的非理性程度，相对于$ g $。我们证明，对于任何$ \ varepsilon> 0 $的多项式函数，增长范围$ 14+\ varepsilon $，并且对于三组无限多个属的属，可以将界限用于多项式$ 10 $。我们证明中的主要成分是由于Borcherds而引起的一系列Heegner除数的模块化，以及由于Kudla，Millson，Zhang，Bruinier和Westerholt-Raum而引起的对更高的编码的概括。对于特殊的属，该证明还建立在与某些立方四倍，Gushel-Mukai四倍和Hyperkähler四倍方面的K3表面相关的K3表面的存在基础上。
We study how the degrees of irrationality of moduli spaces of polarized K3 surfaces grow with respect to the genus $g$. We prove that the growth is bounded by a polynomial function of degree $14+\varepsilon$ for any $\varepsilon>0$ and, for three sets of infinitely many genera, the bounds can be refined to polynomials of degree $10$. The main ingredients in our proof are the modularity of the generating series of Heegner divisors due to Borcherds and its generalization to higher codimensions due to Kudla, Millson, Zhang, Bruinier, and Westerholt-Raum. For special genera, the proof is also built upon the existence of K3 surfaces associated Hodge theoretically with certain cubic fourfolds, Gushel-Mukai fourfolds, and hyperkähler fourfolds.
