A Schwarz lemma for K\"ahler affine metrics and the canonical potential of a proper convex cone

This is an account of some aspects of the geometry of K\"ahler affine metrics based on considering them as smooth metric measure spaces and applying the comparison geometry of Bakry-Emery Ricci tensors. Such techniques yield a version for K\"ahler affine metrics of Yau's Schwarz lemma for volume forms. By a theorem of Cheng and Yau there is a canonical K\"ahler affine Einstein metric on a proper convex domain, and the Schwarz lemma gives a direct proof of its uniqueness up to homothety. The potential for this metric is a function canonically associated to the cone, characterized by the property that its level sets are hyperbolic affine spheres foliating the cone. It is shown that for an $n$-dimensional cone a rescaling of the canonical potential is an $n$-normal barrier function in the sense of interior point methods for conic programming. It is explained also how to construct from the canonical potential Monge-Amp\`ere metrics of both Riemannian and Lorentzian signatures, and a mean curvature zero conical Lagrangian submanifold of the flat para-K\"ahler space.Comment: Minor corrections. References adde

Gene expression predictions and networks in natural populations supports the omnigenic theory

Background: Recent literature on the differential role of genes within networks distinguishes core from peripheral genes. If previous works have shown contrasting features between them, whether such categorization matters for phenotype prediction remains to be studied.Results: We measured 17 phenotypic traits for 241 cloned genotypes from a Populus nigra collection, covering growth, phenology, chemical and physical properties. We also sequenced RNA for each genotype and built co-expression networks to define core and peripheral genes. We found that cores were more differentiated between populations than peripherals while being less variable, suggesting that they have been constrained through potentially divergent selection. We also showed that while cores were overrepresented in a subset of genes statistically selected for their capacity to predict the phenotypes (by Boruta algorithm), they did not systematically predict better than peripherals or even random genes.Conclusion: Our work is the first attempt to assess the importance of co-expression network connectivity in phenotype prediction. While highly connected core genes appear to be important, they do not bear enough information to systematically predict better quantitative traits than other gene sets.Université Fédérale de ToulouseUne approche de biologie intégrative pour améliorer le peuplier en vue de sa valorisation en bio-raffinerie grâce à une meilleure compréhension de l'architecture génétique de la production et de la qualité de la biomasse lignocellulosiqu