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Abstract des Vortrages von Andreas Kappes

Kontsevich's formula for the sum of Lyapunov exponents

There is a remarkable connection between dynamical systems and algebraic geometry first discovered by M. Kontsevich. Given a flat normed vector bundle over a hyperbolic curve, one can lift the geodesic flow by parallel transport to a linear flow on the bundle. The Lyapunov exponents of this dynamical system measure the logarithmic growth rate of sections, when they are dragged along a generic geodesic. Interesting flat bundles that come up in algebraic geometry are the relative cohomology bundles of families of algebraic varieties. Over the complex numbers, these are naturally endowed with a variation of Hodge structures, a holomorphically varying flag of subbundles. Kontsevich's formula relates the sum of the Lyapunov exponents of a weight 1-variation of Hodge structures to the degrees of certain line bundles. In my talk, I will explain the ingredients of Kontsevich's formula and I will shed some light on the proof as well as on applications of this formula.