论文标题
可能的交叉点
Likely intersections
论文作者
论文摘要
我们证明了一个可能的相交定理，这是与Zilber-pink猜想的对应物，这是在假设轴链曲叶特性和一些温和的附加条件下，已知在给定的一类复杂商类别中，可以在某些固定的O-Minimal膨胀中确定的实数阶数中的固定O-Minimal膨胀。 对于我们一般结果的一个例子，请考虑Shimura品种的亚地区的情况。让$ s $成为Shimura品种。令$π：d \toγ\ backslash d = s $实现$ s $作为$ d $的商，这是一个真实代数$ g $的同质空间，该空间是$γ<g $，一个算术子组。令$ s'\ subseteq s $是$ s $的特殊子各种，以$ d'\ subseteq d $ $ s $实现为$ d'\ subseteq d $一个代数子组的同质空间。令$ x \ subseteq s $是$ s $不包含的$ s $的不可约的亚变量。假设$ x $与$ s'$的交集持续意味着每当$ζ：s_1 \ to s $和$ξ：s_1 \至s_2 $是Shimura品种的地图（这意味着定期由$ qu fimitie tim $ fimite fim fim fim files t y lim fim fim files tim j $ fim fim fim fim fim fim fim fim fim fim fim fim fim fim fim fim fim fim fim fim fim fim fim fim fim fim fim fim fim fim fim fim fim fim fim- ξζ^{ - 1} s'\ geq \ dim dimξs_1$。然后，$ x \ cap \ bigcup_ {g \ in g，π（g d'）\ text {special}}}π（g d'）$在欧几里得拓扑的$ x $中密集。
We prove a general likely intersections theorem, a counterpart to the Zilber-Pink conjectures, under the assumption that the Ax-Schanuel property and some mild additional conditions are known to hold for a given category of complex quotient spaces definable in some fixed o-minimal expansion of the ordered field of real numbers. For an instance of our general result, consider the case of subvarieties of Shimura varieties. Let $S$ be a Shimura variety. Let $π:D \to Γ\backslash D = S$ realize $S$ as a quotient of $D$, a homogeneous space for the action of a real algebraic group $G$, by the action of $Γ< G$, an arithmetic subgroup. Let $S' \subseteq S$ be a special subvariety of $S$ realized as $π(D')$ for $D' \subseteq D$ a homogeneous space for an algebraic subgroup of $G$. Let $X \subseteq S$ be an irreducible subvariety of $S$ not contained in any proper weakly special subvariety of $S$. Assume that the intersection of $X$ with $S'$ is persistently likely meaning that whenever $ζ:S_1 \to S$ and $ξ:S_1 \to S_2$ are maps of Shimura varieties (meaning regular maps of varieties induced by maps of the corresponding Shimura data) with $ζ$ finite, $\dim ξζ^{-1} X + \dim ξζ^{-1} S' \geq \dim ξS_1$. Then $X \cap \bigcup_{g \in G, π(g D') \text{ is special }} π(g D')$ is dense in $X$ for the Euclidean topology.
