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We compute the algebraic equation of the universal family over the Kenyon-Smillie (2, 3, 4)-Teichmüller curve and we prove that the equation is correct in two different ways.Firstly, we prove it in a constructive way via linear conditions imposed by three special points of the Teichmüller curve. Secondly, we verify that the equation is correct by computing its associated Picard-Fuchs equation.We also notice that each point of the Teichmüller curve has a hyperflex and we see that the torsion map is a central projection from this point.

For the moduli spaces of Abelian differentials, the Euler characteristic is one of the most intrinsic topological invariants. We give a formula for the Euler characteristic that relies on intersection theory on the smooth compactification by multi-scale differentials. It is a consequence of a formula for the full Chern polynomial of the cotangent bundle of the compactification. The main new technical tools are an Euler sequence for the cotangent bundle of the moduli space of multi-scale differentials and computational tools in the Chow ring, such as a description of normal bundles to boundary divisors.

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In the first part we extend the construction of the smooth normalcrossing divisors compactification of projectivized strata of abelian differentials given by Bainbridge, Chen, Gendron, Grushevsky and Möller to the case of kdifferentials. Since the generalized construction is closely related to the original one, we mainly survey their results and justify the details that need to be adapted in the more general context.In the second part we show that the flat area provides a canonical hermitian metric on the tautological bundle over the projectivized strata of finite area kdifferentials whose curvature form represents the first Chern class. This result is useful in order to apply Chern-Weil theory tools. It has already been used as an assumption in the work of Sauvaget for abelian differentials and is also used in a paper of Chen, Möller and Sauvaget for quadratic differentials.Résumé. -Dans la première partie de cet article nous étendons aux k-différentielles la construction d'une compactification lisse avec un bord a croisements normaux des strates projectivisées de différentielles abéliennes introduite par Bainbridge, Chen, Gendron, Grushevsky et Moeller. Comme cette construction est très liée à la construction originale, nous ne présentons qu'un survol de celle-ci en soulignant les points qui nécessitent des modifications dans ce contexte plus général.Dans la deuxième partie nous démontrons que l'aire fournit une métrique canonique hermitienne sur le fibré tautologique au-dessus des strates projectivisées dont la courbure représente la première classe de Chern. Ce résultat est utile pour pouvoir appliquer les outils de la théorie de Chern-Weyl. Ce résultat a déjà été utilisé comme une hypothèse dans les travaux de Sauvaget sur les différentielles abéliennes et dans les travaux de Chen, Moeller et Sauvaget sur les différentielles quadratiques.