Holomorphic 1-forms on the moduli space of curves

Since the sixties it is well known that there are no non-trivial closed holomorphic $1$-forms on the moduli space $\mathcal{M}_g$ of smooth projective curves of genus $g>2$. In this paper, we strengthen such result proving that for $g\geq 5$ there are no non-trivial holomorphic $1$-forms. With this aim, we prove an extension result for sections of locally free sheaves $\mathcal{F}$ on a projective variety $X$. More precisely, we give a characterization for the surjectivity of the restriction map $\rho_D:H^0(\mathcal{F})\to H^0(\mathcal{F}|_{D})$ for divisors $D$ in the linear system of a sufficiently large multiple of a big and semiample line bundle $\mathcal{L}$. Then, we apply this to the line bundle $\mathcal{L}$ given by the Hodge class on the Deligne Mumford compactification of $\mathcal{M}_g$.Comment: New version with several improvement

Relabelling in Bayesian mixture models by pivotal units

A simple procedure based on relabelling to deal with label switching when exploring complex posterior distributions by MCMC algorithms is proposed. Although it cannot be generalized to any situation, it may be handy in many applications because of its simplicity and low computational burden. A possible area where it proves to be useful is when deriving a sample for the posterior distribution arising from finite mixture models when no simple or rational ordering between the components is available

Quillen connection and the uniformization of Riemann surfaces

The Quillen connection on ${\mathcal L} \rightarrow {\mathcal M}_g$, where ${\mathcal L}^*$ is the Hodge line bundle over the moduli stack of smooth complex projective curves curves ${\mathcal M}_g$, $g \geq 5$, is uniquely determined by the condition that its curvature is the Weil--Petersson form on ${\mathcal M}_g$. The bundle of holomorphic connections on ${\mathcal L}$ has a unique holomorphic isomorphism with the bundle on ${\mathcal M}_g$ given by the moduli stack of projective structures. This isomorphism takes the $C^\infty$ section of the first bundle given by the Quillen connection on ${\mathcal L}$ to the $C^\infty$ section of the second bundle given by the uniformization theorem. Therefore, any one of these two sections determines the other uniquely

Fostering Social Impact Through Corporate Implementation of the SDGs: Transformative Mechanisms Towards Interconnectedness and Inclusiveness

The United Nations (UN) 2030 Agenda for Sustainable Development has considerable potential for achieving a more sustainable future. However, the concrete realisation of Sustainable Development Goals (SDGs) is impeded by how they are implemented by a diverse set of competent agents. This conceptual paper draws on social impact theory to investigate how businesses can utilise the SDG framework to achieve positive social outcomes. We identify two pathways that can guide businesses to improve their SDGs interventions, which entail considering the interconnections between the goals that are directly or indirectly affected by the initiative at stake and the inclusiveness of the actors affected by the SDGs. Building on the literature on hybrid organising (to frame interconnectedness) and the literature on multi-stakeholder partnerships and deliberative governance (to frame inclusiveness), we discuss a set of organisational mechanisms and transformations that can help businesses ensure that their SDGs interventions are more socially impactful. By doing so, this paper extends the literature on the role of companies for sustainable development and provides some practical implications

Caught while Dissolving: Revealing the Interfacial Solvation of the Mg2+ Ions on the MgO Surface

Interfaces between water and materials are ubiquitous and are crucial in materials sciences and in biology, where investigating the interaction of water with the surface under ambient conditions is key to shedding light on the main processes occurring at the interface. Magnesium oxide is a popular model system to study the metal oxide-water interface, where, for sufficient water loadings, theoretical models have suggested that reconstructed surfaces involving hydrated Mg2+ metal ions may be energetically favored. In this work, by combining experimental and theoretical surface-selective ambient pressure X-ray absorption spectroscopy with multivariate curve resolution and molecular dynamics, we evidence in real time the occurrence of Mg2+ solvation at the interphase between MgO and solvating media such as water and methanol (MeOH). Further, we show that the Mg2+ surface ions undergo a reversible solvation process, we prove the dissolution/redeposition of the Mg2+ ions belonging to the MgO surface, and we demonstrate the formation of octahedral [Mg(H2O)6]2+ and [Mg(MeOH)6]2+ intermediate solvated species. The unique surface, electronic, and structural sensitivity of the developed technique may be beneficial to access often elusive properties of low-Z metal ion intermediates involved in interfacial processes of chemical and biological interest