How curved is a random complex curve?

Michele Ancona; Damien Gayet

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arXiv · 2402.11972

How curved is a random complex curve?

Michele Ancona·Damien Gayet

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Abstract

In this paper, we study the curvature properties of random complex plane curves. We bound from below the probability that a uniform proportion of the area of a random complex degree $d$ plane curve has a curvature smaller than $-d/8$. Our lower bound is uniform, in the sense that it does not depend on $d$. We also provide uniform upper bounds for similar probabilities. These results extend to random complex curves of projective surfaces equipped with an ample line bundle. This paper can be viewed as a sequel of [1], where other metric statistics were given. On a larger time scale, it joins the general program initiated in [11] of understanding random complex hypersurfaces of projective manifolds.

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Michele Ancona, Damien Gayet. 2024-02-19. How curved is a random complex curve?. https://arxiv.org/abs/2402.11972

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