Andrea Causin

Every compact Riemann surface X admits a natural projective structure $$p_u$$ p u as a consequence of the uniformization theorem. In this work we describe the construction of another natural projective structure on X, namely the Hodge projective structure $$p_h$$ p h , related to the second fundamental form of the period map. We then describe how projective structures correspond to (1, 1)-differential forms on the moduli space of projective curves and, from this correspondence, we deduce that $$p_u$$ p u and $$p_h$$ p h are not the same structure.

We exhibit new examples of double Kodaira fibrations by using finite Galois covers of a product $\Sigma_b \times \Sigma_b$, where $\Sigma_b$ is a smooth projective curve of genus $b \geq 2$. Each cover is obtained by providing an explicit group epimorphism from the pure braid group $\mathsf{P}_2(\Sigma_b)$ to some finite Heisenberg group. In this way, we are able to show that every curve of genus $b$ is the base of a double Kodaira fibration; moreover, the number of pairwise non-isomorphic Kodaira fibred surfaces fibering over a fixed curve $\Sigma_b$ is at least $\boldsymbol{\omega}(b+1)$, where $\boldsymbol{\omega} \colon \mathbb{N} \to \mathbb{N}$ stands for the arithmetic function counting the number of distinct prime factors of a positive integer. As a particular case of our general construction, we obtain a real $4$-manifold of signature $144$ that can be realized as a real surface bundle over a surface of genus $2$, with fibre genus $325$, in two different ways. This provides (to our knowledge) the first "double" solution to a problem from Kirby's list in low-dimensional topology.

Given $n$ integer, let $X$ be either the set of hermitian or real $n\times n$ matrices of rank at least $n-1$. If $n$ is even, we give a sharp estimate on the maximal dimension of a real vector subspace of $X\cup\{0\}$. The rusults are obtained, via K-theory, by studying a bundle map induced by the adjugation of matrices