论文标题
平滑平面网的曲率和应用到$ \ mathbb {p}^{2} _ {\ mathbb {c}} $上的均匀叶子的双网的曲率的标准
A criterion for the holomorphy of the curvature of smooth planar webs and applications to dual webs of homogeneous foliations on $\mathbb{P}^{2}_{\mathbb{C}}$
论文作者
论文摘要
令$ d \ geq3 $为整数。对于一个复杂的表面$ m $上的全态$ d $ -web $ \ nathcal {w} $，沿其判别$δ的不可约组$ d $平稳，我们建立了一个有效的标准，我们建立了一个有效的标准，用于$ \ nater $ \ nater $ \ nath $ \ natercal $ \ mathcal $ \ w} w} w} $ dep dep dep dep， Mar \'ın，Pereira和Pirio。作为一个应用程序，我们针对Legendre Transform（双Web）的曲率的全体形状进行了完整的特征概括我们以前的一些结果。然后，这使我们能够研究$ d $ -web $ \ mathrm {leg} \ mathcal {h} $的平坦度，在叶片$ \ mathcal {h} $的特殊情况下。当$ \ mathcal {h} $的Galois Group是循环的时，我们表明$ \ mathrm {leg} \ mathcal {h} $在且仅当$ \ mathcal {h} $时才是平坦的$ω_1^{\ hspace {0.2mm} d} = y^d \ mathrm {d} x-x^d \ mathrm {d} y $，$ω_2^{\ hspace {\ hspace {0.2mm} d} d} d} = x^d \ \ m \ m mathrm { $ \ MATHCAL {H} $是非循环的，我们得到$ \ Mathrm {leg} \ Mathcal {h} $始终是平坦的。
Let $d\geq3$ be an integer. For a holomorphic $d$-web $\mathcal{W}$ on a complex surface $M$, smooth along an irreducible component $D$ of its discriminant $Δ(\mathcal{W}),$ we establish an effective criterion for the holomorphy of the curvature of $\mathcal{W}$ along $D,$ generalizing results on decomposable webs due to Mar\'ın, Pereira and Pirio. As an application, we deduce a complete characterization for the holomorphy of the curvature of the Legendre transform (dual web) $\mathrm{Leg}\mathcal{H}$ of a homogeneous foliation $\mathcal{H}$ of degree $d$ on $\mathbb{P}^{2}_{\mathbb{C}},$ generalizing some of our previous results. This then allows us to study the flatness of the $d$-web $\mathrm{Leg}\mathcal{H}$ in the particular case where the foliation $\mathcal{H}$ is Galois. When the Galois group of $\mathcal{H}$ is cyclic, we show that $\mathrm{Leg}\mathcal{H}$ is flat if and only if $\mathcal{H}$ is given, up to linear conjugation, by one of the two 1-forms $ω_1^{\hspace{0.2mm}d}=y^d\mathrm{d}x-x^d\mathrm{d}y$, $ω_2^{\hspace{0.2mm}d}=x^d\mathrm{d}x-y^d\mathrm{d}y.$ When the Galois group of $\mathcal{H}$ is non-cyclic, we obtain that $\mathrm{Leg}\mathcal{H}$ is always flat.
