曲线上平坦丛的李雅普诺夫指数的下界
Lower bounds for Lyapunov exponents of flat bundles on curves
摘要：
考虑复曲线上的平坦丛。本文证明了费玉的一个猜想：平坦丛的前k个李雅普诺夫指数之和总是大于或等于任意秩k全纯子丛的次数。我们推广原来的背景下，从Teichmueller曲线的任何局部系统的曲线与非扩张尖点monodromies。作为应用，我们得到了阿贝尔微分超椭圆层中单个李雅普诺夫指数的大亏格极限。理解与来自霍奇过滤的子丛的度相等的情况似乎具有挑战性，例如对于Calabi-Yau型族。我们猜想，李雅普诺夫指数之和与度相等与单值群是其Zapriki闭包的薄子群有关。
英文摘要：
Consider a flat bundle over a complex curve. We prove a conjecture of Fei Yu that the sum of the top k Lyapunov exponents of the flat bundle is always greater or equal to the degree of any rank k holomorphic subbundle. We generalize the original context from Teichmueller curves to any local system over a curve with non-expanding cusp monodromies. As an application we obtain the large genus limits of individual Lyapunov exponents in hyperelliptic strata of Abelian differentials. Understanding the case of equality with the degrees of subbundle coming from the Hodge filtration seems challenging, e.g. for Calabi-Yau type families. We conjecture that equality of the sum of Lyapunov exponents and the degree is related to the monodromy group being a thin subgroup of its Zariski closure.
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