Through tilting objects, we construct complete cotorsion pairs for specific hereditary abelian categories, such as the category of modules that are finitely generated over a finite-dimensional hereditary algebra as well as the category of coherent sheaves over weighted projective lines. We prove that a complete cotorsion pair exists in the category of coherent sheaves over a weighted projective curve $\mathbb{X}$ if and only if $\mathbb{X}$ is a weighted projective line. We also characterize the canonical tilting cotorsion pair for any weighted projective line and obtain Hovey triples in the category of vector bundles over a weighted projective line.

In this paper, we mainly focus on extension-closed subcategories of extriangulated categories. Let $\mathcal{X}$ be an extension-closed subcategory. We show that if $C$ is $\mathcal{X}$-projective and there is a minimal right almost split deflation in $\mathcal{X}$ ending by $C$, then there is an $\mathfrak{s}$-triangle ending by $C$ which is very similar to an Auslander-Reiten triangle in $\mathcal{X}$. We also show that if the extriangulated category admits a negative first extension ${\mathbb{E}}^{-1}$, and $\mathcal{X}$ is self-orthogonal with respect to ${\mathbb{E}}^{-1}$, then $\mathcal{X}$ has an exact structure.