论文标题
广义ALG渐近学的RICCI-FLAT歧管
Ricci-flat manifolds of generalized ALG asymptotics
论文作者
论文摘要
在复杂的尺寸中，$ \ geq 3 $，我们为通用的alg完全非紧密的ricci flatkähler歧管提供了几何存在，并带有schwartz衰变，即任何多项式速率的度量衰减，降低了对alg型$ \ mathbb {c} \ mathbb {c} \ times y $ $ modulo y $ modulo fimulo fimulo fimulo fimulo for y y y $ y $ y $ y $ y y y y y y y y y y yes calabi。 因此，对于任何$ k3 $的表面表面，有限订单的纯粹非成直型自晶$σ$，kählercrepant的解决方案是Orbifold $ \ frac {\ Mathbb {c} \ times k3} \ langleσ\ rangleσ\ rangle} $ schwers and gung ricci-decarwers a gung ricci-rec.众所周知，在我们的情况下，KählerCrepant决议存在。因此，有$ 39 $的整数，因此每个划分的$2π$是ALG RICCI-FLATKähler的渐近角度与Schwartz Decay $ 3- $ fold。我们还展示了Alg ricci-flatKähler的1638个参数家族$ 3- $折叠，并带有渐近角$π$，该$π$ $π$实现了$ 64 $ $ nifting的Betti数字。它们在$ k3 $表面具有非透明质尼古拉蛋白的相关性。 $ h^{1,1} $/local $ i \ partial \ partial \ poartline {\ partial} - $ lemma的简单版本在schwartz decay中起作用，构建ANSATZ，以及平等的Ansatz，与Ricci Flat Alg模型相等的ANSATZ（iSotriv set a iSotrivial anssatz anssatz anssatz）。 Schwartz衰减的证明取决于牛顿电位的不集中，并且由于存在$ l^{2} $ l^{2} $归一化特征函数的集中序列在（真实的）维度$ \ egeq 2 $的单位圆球上的浓度序列，因此无法立即将其概括为具有较高尺寸的纤维化。
In complex dimensions $\geq 3$, we provide a geometric existence for generalized ALG complete non-compact Ricci flat Kähler manifolds with Schwartz decay i.e. metric decay in any polynomial rate to an ALG model $\mathbb{C}\times Y$ modulo finite cyclic group action, where $Y$ is Calabi-Yau. Consequently, for any $K3$ surface with a purely non-symplectic automorphism $σ$ of finite order, a Kähler crepant resolution of the orbifold $\frac{\mathbb{C} \times K3}{\langle σ\rangle}$ admits ALG Ricci-flat Kähler metrics with Schwartz decay. It is known that Kähler crepant resolution exists in our case. Hence there are $39$ integers, such that $2π$ divided by each of them is the asymptotic angle of an ALG Ricci-flat Kähler $3-$fold with Schwartz decay. We also exhibit a 1638 parameters family of ALG Ricci-flat Kähler $3-$folds with asymptotic angle $π$ that realize $64$ distinct triples of Betti numbers. They are iso-trivially fibred by $K3$ surface with a non-symplectic Nikulin involution. A simple version of local Kunneth formula for $H^{1,1}$/local $i\partial\overline{\partial}-$lemma plays a role in both the Schwartz decay, and the construction of ansatz that equals a Ricci flat ALG model outside a compact set (isotrivial ansatz). The proof of Schwartz decay relies on a non-concentration of the Newtonian potential, and can not be immediately generalized to fibration with higher dimensional base, due to existence of concentrating sequence of $L^{2}$ normalized eigen-functions on unit round spheres of (real) dimension $\geq 2$.
