标题： 关于叶状结构的一些指数及其应用 显示英文标题

标题： On some indices of foliations and applications

摘要： 本文建立了叶状结构关于有效平衡轨线除子的Milnor数、$\chi$-数和Tjurina数之间的关系。此外，利用Gómez-Mont--Seade--Verjovsky指数，我们证明了叶状结构相对于约化曲线的多重性与Tjurina数之差独立于叶状结构本身。我们还推导出了叶状结构关于约化曲线的Tjurina数的伴随公式。这些局部结果具有全局意义；例如，我们给出了Cerveau-Lins Neto关于叶状结构多重性的全局结果的新证明，以及Soares关于叶状结构不变曲线的Milnor数之和的不等式的新证明。此外，我们得到了复射影平面上叶状结构的整体Tjurina数的上下界。最后，我们提供了一个例子，回答了Alcántara--Mozo-Fernández关于复射影平面上具有唯一奇点的叶状结构的一个猜想。 显示英文摘要

摘要： In this paper we establish a relationship between the Milnor number, the $\chi$-number, and the Tjurina number of a foliation with respect to an effective balanced divisor of separatrices. Moreover, using the G\'omez-Mont--Seade--Verjovsky index, we prove that the difference between the multiplicity and the Tjurina number of a foliation with respect to a reduced curve is independent of the foliation. We also derive an adjunction formula for the Tjurina number of a foliation with respect to a reduced curve. These local results have global consequences; for instance, we provide a new proof of a global result regarding the multiplicity of a foliation due to Cerveau-Lins Neto and a new proof of a Soares's inequality for the sum of the Milnor number of an invariant curve of a foliation. Additionally, we obtain upper and lower bounds for the global Tjurina number of a foliation on the complex projective plane. Finally, we provide an example that answers a conjecture posed by Alc\'antara--Mozo-Fern\'andez about foliations on the complex projective plane having a unique singularity.