We prove the local hard Lefschetz theorem and local Hodge–Riemann bilinear relations for Soergel bimodules. Using results of Soergel and Kübel, one may deduce an algebraic proof of the Jantzen conjectures. We observe that the Jantzen filtration may depend on the choice of non-dominant regular deformation direction.

We obtain a system of relations between linear Hodge integrals. As an application, we show that its first non-trivial relation implies the Witten's Conjecture/Kontsevich Theorem.

This note answers some questions on holomorphic curves and their distribution in an algebraic surface of positive index. More specifically, we exploit the existence of natural negatively curved "pseudo-Finsler" metrics on a surface S of general type whose Chern numbers satisfy c(2)1>2c2 to show that a holomorphic map of a Riemann surface to S whose image is not in any rational or elliptic curve must satisfy a distance decreasing property with respect to these metrics. We show as a consequence that such a map extends over isolated punctures. So assuming that the Riemann surface is obtained from a compact one of genus q by removing a finite number of points, then the map is actually algebraic and defines a compact holomorphic curve in S. Furthermore, the degree of the curve with respect to a fixed polarization is shown to be bounded above by a multiple of q - 1 irrespective of the map.

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We verify a conjecture of Lutwak, Yang, and Zhang about the equality case in the Orlicz-Petty projection inequality, and provide an essentially optimal stability version.