Ricci-flat manifolds of generalized ALG asymptotics
| Autor: | Wang, Yuanqi |
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| Rok vydání: | 2022 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | In complex dimensions $\geq 3$, we provide a geometric existence for generalized ALG complete non-compact Ricci flat K\"ahler 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 $\sigma$ of finite order, a K\"ahler crepant resolution of the orbifold $\frac{\mathbb{C} \times K3}{\langle \sigma \rangle}$ admits ALG Ricci-flat K\"ahler metrics with Schwartz decay. It is known that K\"ahler crepant resolution exists in our case. Hence there are $39$ integers, such that $2\pi$ divided by each of them is the asymptotic angle of an ALG Ricci-flat K\"ahler $3-$fold with Schwartz decay. We also exhibit a 1638 parameters family of ALG Ricci-flat K\"ahler $3-$folds with asymptotic angle $\pi$ 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 $\geq 2$. Comment: 47 pages. 5 figures, 3 tables. Revisions are carried out and presentation improved. Inaccuracy in Proposition 1.6 is fixed: k^2_{0} is the eigen-value, not k_{0}. Definition of weakly distinct is revised. Typos are corrected. Comments are welcome |
| Databáze: | arXiv |
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