Finsler集的零点平均数与混合辛体积
Average number of zeros and mixed symplectic volume of Finsler sets
摘要：
设$X$是$n$维流形，$V_1, \ldots, V_n \subset C^\infty(X, \mathbb R)$是具有欧氏度量的有限维向量空间.我们给每个$V_i$分配一个芬斯勒椭球，即，$X$余切束纤维中的一类椭球体。证明了$f_1 \in V_1, \ldots, f_n \in V_n$的孤立公共零点的平均个数等于这些Finsler椭球的混合辛体积。若$X$是紧李群的齐性空间，且所有向量空间$V_i$及其欧氏度量不变，则其零点的平均个数满足类似于射影簇上因子的相交数的Hodge不等式.这适用于不变黎曼度量的拉普拉斯算子的特征空间。的证明是基于建设环的正常密度$X$，一个模拟环的微分形式。特别地，这种构造用于将Crofton公式推广到球面的乘积。
英文摘要：
Let $X$ be an $n$-dimensional manifold and $V_1, \ldots, V_n \subset C^\infty(X, \mathbb R)$ finite-dimensional vector spaces with Euclidean metric. We assign to each $V_i$ a Finsler ellipsoid, i.e., a family of ellipsoids in the fibers of the cotangent bundle of $X$. We prove that the average number of isolated common zeros of $f_1 \in V_1, \ldots, f_n \in V_n$ is equal to the mixed symplectic volume of these Finsler ellipsoids. If $X$ is a homogeneous space of a compact Lie group and all vector spaces $V_i$ and their Euclidean metrics are invariant, then the average numbers of zeros satisfy the inequalities, similar to Hodge inequalities for intersection numbers of divisors on a projective variety. This is applied to the eigenspaces of Laplace operator of an invariant Riemannian metric. The proofs are based on a construction of the ring of normal densities on $X$, an analogue of the ring of differential forms. In particular, this construction is used for a generalization of Crofton formula to the product of spheres.
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