Classical (real) hedgehogs can be regarded as the geometrical realizations of formal differences of convex bodies in $\mathbb{R}^{n+1}$. Like convex bodies, hedgehogs can be identified with their support functions. Adopting a projective viewpoint, we prove that any holomorphic function $h : \mathbb{C}^n \to \mathbb{C}$ can be regarded as the ‘support function’ of a complex hedgehog $\mathcal{H}_h$ in $\mathbb{C}^{n+1}$. In the same vein, we introduce the notion of evolute of such a hedgehog $\mathcal{H}_h$ in $\mathbb{C}^2$, and a natural (but apparently hitherto unknown) notion of complex curvature, which allows us to interpret this evolute as the locus of the centers of complex curvature. It is of course permissible to think that the development of a ‘Brunn–Minkowski theory for complex hedgehogs’ (replacing Euclidean volumes by symplectic ones) might be a promising way of research. We give first two results in this direction. We next return to real hedgehogs in $\mathbb{R}^{2n}$ endowed with a linear complex structure. We introduce and study the notion of evolute of a hedgehog. We particularly focus our attention on $\mathbb{R}^4$ endowed with a linear Kähler structure determined by the datum of a pure unit quaternion. In parallel, we study the symplectic area of the images of the oriented Hopf circles under hedgehog parametrizations and introduce a quaternionic curvature function for such an image. Finally, we consider briefly the convolution of hedgehogs, and the particular case of hedgehogs in $\mathbb{R}^{4n}$ regarded as a hyperkähler vector space.

中文翻译：

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真实和复杂的刺猬，它们的辛面积、曲率和演化

经典（真实）刺猬可以看作是 $\mathbb{R}^{n+1}$ 中凸体形式差异的几何实现。像凸体一样，刺猬可以通过它们的支持功能来识别。采用射影的观点，我们证明了任何全纯函数 $h : \mathbb{C}^n \to \mathbb{C}$ 都可以看作是复杂刺猬 $\mathcal{H}_h$ 的“支持函数”在 $\mathbb{C}^{n+1}$ 中。同样，我们在 $\mathbb{C}^2$ 中引入了这种刺猬 $\mathcal{H}_h$ 的进化概念，以及复杂曲率的自然（但显然迄今为止未知）概念，它允许我们将此演化解释为复曲率中心的轨迹。认为发展“复杂刺猬的布伦-闵可夫斯基理论”（用辛体积代替欧几里德体积）可能是一种有前途的研究方式，这当然是允许的。我们给出了这个方向的前两个结果。我们接下来回到 $\mathbb{R}^{2n}$ 中真正的刺猬，它具有线性复杂结构。我们介绍并研究了刺猬进化的概念。我们特别关注 $\mathbb{R}^4$ ，它具有由纯单位四元数的数据决定的线性 Kähler 结构。同时，我们研究了刺猬参数化下定向 Hopf 圆图像的辛面积，并为此类图像引入了四元曲率函数。最后，我们简要地考虑刺猬的卷积，