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Let P Ω d M 0 , n (κ) , where κ = (k 1 , ⋯ , k n) , be a stratum of (projectivized) d-differentials in genus 0. We prove a recursive formula which relates the volume of P Ω d M 0 , n (κ) to the volumes of other strata of lower dimensions in the case where none of the k i is divisible by d. As an application, we give a new proof of the Kontsevich’s formula for the volumes of strata of quadratic differentials with simple poles and zeros of odd order, which was originally proved by Athreya–Eskin–Zorich. In another application, we show that up to some power of π , the volume of the moduli spaces of flat metrics on the sphere with prescribed cone angles is a continuous piecewise polynomial with rational coefficients function of the angles, provided none of the angles is an integral multiple of 2 π . This generalizes the results of Koziarz and Nguyen (Ann Sci l’Éc Normale Supér 51(6):1549–1597, 2018) and McMullen (Am J Math 139(1):261–291, 2017). [ABSTRACT FROM AUTHOR]

Carlson and Toledo conjectured that if an infinite group Γ is the fundamental group of a compact Kähler manifold, then virtually $${H^{2}(\Gamma, {\mathbb R}) \ne 0}$$ . We assume that Γ admits an unbounded reductive rigid linear representation. This representation necessarily comes from a complex variation of Hodge structure ( $${\mathbb{C}}$$ -VHS) on the Kähler manifold. We prove the conjecture under some assumption on the $${\mathbb{C}}$$ -VHS. We also study some related geometric/topological properties of period domains associated to such a $${\mathbb{C}}$$ -VHS. [ABSTRACT FROM AUTHOR]

We prove an extension theorem of 'Ohsawa-Takegoshi type' for Dolbeault q-classes of cohomology ( q ≥ 1) on smooth compact hypersurfaces in a weakly pseudoconvex Kähler manifold. [ABSTRACT FROM AUTHOR]

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