论文标题
等级2吸引者和费马特类型CY $ n $折。
Rank-2 attractors and Fermat type CY $n$-folds
论文作者
论文摘要
Fermat类型的Calabi-yau $ n $ - fold，用$ \ Mathscr {f} _n $表示，是$ \ Mathbb {p}^{n+1} $由$ \ sum_ = 0}^i = 0}^{ $ψ= 0 $的fermat铅笔$$ \ sum_ {i = 0}^{n+1} x^{n+2} _i-（n+2）\，ψ\，\ prod_ {i = 0}^{i = 0}^{n+1} x_i = 0。 $$在$ \ Mathscr {f} _n $上无处消失的holomorphic $ n $ - 定义$（h^n（h^n（\ Mathscr {f} _n，\ Mathbb {q}））的$ n+1 $ dimensional sub-hodge结构，f_p）$。在本文中，我们将制定一个猜想，该猜想说，这种$ n+1 $尺寸的子蜂蜜结构完全分成了具有尺寸$ \ leq 2 $的纯hodge结构的直接总和，其中是直接的summand $ \ mathbf {h}^n_ {a，1} $ $ \ Mathbf {h}^n_ {a，1} = h^{n，0}（\ Mathscr {f} _n）\ oplus h^{0，n}（\ Mathscr {f} _n）。 $$使用数值方法，我们能够在$ n = 3,4,6 $的情况下明确构建这种分裂，而我们还为$ n = 8,10 $的情况下构建了部分分配。对于$ n = 3,4,6,8,10 $，我们从数字上发现，镜像映射$ t $的fermat铅笔的值在费尔马特点$ψ= 0 $是$ t $ t | _ {ψ= 0} = \ frac {1} {1} {2} {2} {2} {2}+ξ\，i，i，ut的utter aut ant的utter at a utter的uttra $ n+2 $。此外，我们还从数值上发现，商$ c^+（\ mathbf {h}^n_ {a，1}）/c^ - （\ nybf { $ n = 3,4,6,8,10 $，实际上，我们将提出更强有力的猜想，从而推广这一观察结果。我们还将证明$ \ Mathbf {h}^4_ {a，1} $满足了Deligne猜想的预测。
The Fermat type Calabi-Yau $n$-fold, denoted by $\mathscr{F}_n$, is the hypersurface of $\mathbb{P}^{n+1}$ defined by $\sum_{i=0}^{n+1}x_i^{n+2}=0$, which is the smooth fiber over the Fermat point $ψ=0$ of the Fermat pencil $$ \sum_{i=0}^{n+1} x^{n+2}_i -(n+2)\, ψ\, \prod_{i=0}^{n+1} x_i =0. $$ The nowhere vanishing holomorphic $n$-form on $\mathscr{F}_n$ defines an $n+1$ dimensional sub-Hodge structure of $(H^n(\mathscr{F}_n,\mathbb{Q}),F_p)$. In this paper, we will formulate a conjecture which says that this $n+1$ dimensional sub-Hodge structure splits completely into the direct sum of pure Hodge structures with dimensions $\leq 2$, among which is a direct summand $\mathbf{H}^n_{a,1}$ whose Hodge decomposition is $$ \mathbf{H}^n_{a,1}=H^{n,0}(\mathscr{F}_n) \oplus H^{0,n}(\mathscr{F}_n). $$ Using numerical methods, we are able to explicitly construct such a split for the cases where $n=3,4,6$, while we also construct a partial split for the cases where $n=8,10$. For $n=3,4,6,8,10$, we have numerically found that the value of the mirror map $t$ for the Fermat pencil at the Fermat point $ψ=0$ is of the form $$ t|_{ψ=0}=\frac{1}{2}+ξ\,i, $$ where $ξ$ is a real algebraic number that intuitively depends on the integer $n+2$. Furthermore, we have also numerically found that the quotient $c^+(\mathbf{H}^n_{a,1})/c^-(\mathbf{H}^n_{a,1})$ of the Deligne's periods of $\mathbf{H}^n_{a,1}$ is an algebraic number for the cases where $n=3,4,6,8,10$, and in fact we will formulate a stronger conjecture generalizing this observation. We will also show that $\mathbf{H}^4_{a,1}$ satisfies the prediction of Deligne's conjecture.
