论文标题
预期的随机复杂子延伸物的局部拓扑
Expected local topology of random complex submanifolds
论文作者
论文摘要
令$ n \ geq 2 $和$ r \ in \ {1，\ cdots，n-1 \} $为整数，$ m $是一种紧凑的光滑k \''ah'ah ah ah ah ah ah ahler ahler ahler ahler complieve $ n $，$ e $ $ $ e $ as a holomorphic vector vector bundle a plange conppplecept $ r $ r $ r $ r $ r $ u $ r $，是$ bunmitian $ $ $ $ h_ $ h_e e n y l h_e e n y l。 $ m $配备了带正弯曲形式的度量$ h $。对于任何足够大的$ d \ in \ mathbb {n} $，我们认可与$ h_e $，$ h $相关的天然高斯措施$ h^0（m，m，e \ otimes l^d）$。令$ u \子集M $为具有光滑边界的开放子集。我们证明，$（n-r）$ - th betti的平均数量在随机的$ u $ $ u $ $ h^0（m，e \ otimes l^d）$的$ u $中是渐近的，均为$ {n-1 \ select r-1} d^n \ int_u c_1（l）$ d $。另一方面，其他贝蒂号的平均值为$ O（d^n）$。第一个渐近恢复经典的确定性全局代数计算。此外，在这些平均值的增长顺序上，这种差异是新的，并且与所有其他已知的平滑高斯模型，尤其是真正的代数差异。我们证明了Aggine Complex Bargmann-Fock模型的类似结果。
Let $n\geq 2$ and $r\in \{1, \cdots, n-1\}$ be integers, $M$ be a compact smooth K\''ahler manifold of complex dimension $n$, $E$ be a holomorphic vector bundle with complex rank $r$ and equipped with an hermitian metric $h_E$, and $L$ be an ample holomorphic line bundle over $M$ equipped with a metric $h$ with positive curvature form. For any $d\in \mathbb{N}$ large enough, we endorse the space of holomorphic sections $H^0(M,E\otimes L^d)$ with the natural Gaussian measure associated to $h_E$ , $h$ and its curvature form. Let $U\subset M$ be an open subset with smooth boundary. We prove that the average of the $(n-r)$-th Betti number of the vanishing locus in $U$ of a random section $s$ of $H^0(M,E\otimes L^d)$ is asymptotic to ${n-1 \choose r-1} d^n\int_U c_1(L)^n$ for large $d$. On the other hand, the average of the other Betti numbers are $o(d^n)$. The first asymptotic recovers the classical deterministic global algebraic computation. Moreover, such a discrepancy in the order of growth of these averages is new and constrasts with all known other smooth Gaussian models, in particular the real algebraic one. We prove a similar result for the affine complex Bargmann-Fock model.
