The chapter "Théorie de Hodge, II" by Pierre Deligne is structured into several sections, covering filtrations, Hodge structures, and applications. The introduction explains that the cohomology of a compact Kähler manifold is equipped with a "Hodge structure" of weight \( n \), which is a natural bigrading. The main result shows that the complex cohomology of a non-singular algebraic variety, possibly non-compact, admits a more general Hodge structure, where \( H^n(X, \mathbf{C}) \) is an "iterated extension" of decreasing weight Hodge structures, with Hodge numbers \( h^{p q} \) being zero for \( p > n \) or \( q > n \). The first section introduces filtrations, defining them as families of nested subobjects in an abelian category. Key results include the theorem on mixed Hodge structures and the "two filtrations lemma." The second section recalls the classical Hodge theory and introduces mixed Hodge structures, with a focus on defining the mixed Hodge structure of \( H^n(X, \mathbf{C}) \) and establishing spectral sequence degeneracies. The final section discusses various applications, including the fixed part theorem, semi-simplicity theorem, and homomorphisms of abelian schemes. The chapter relies on both Hodge theory and Hironaka's resolution of singularities to express the cohomology of a quasi-projective non-singular variety in terms of projective non-singular varieties.The chapter "Théorie de Hodge, II" by Pierre Deligne is structured into several sections, covering filtrations, Hodge structures, and applications. The introduction explains that the cohomology of a compact Kähler manifold is equipped with a "Hodge structure" of weight \( n \), which is a natural bigrading. The main result shows that the complex cohomology of a non-singular algebraic variety, possibly non-compact, admits a more general Hodge structure, where \( H^n(X, \mathbf{C}) \) is an "iterated extension" of decreasing weight Hodge structures, with Hodge numbers \( h^{p q} \) being zero for \( p > n \) or \( q > n \). The first section introduces filtrations, defining them as families of nested subobjects in an abelian category. Key results include the theorem on mixed Hodge structures and the "two filtrations lemma." The second section recalls the classical Hodge theory and introduces mixed Hodge structures, with a focus on defining the mixed Hodge structure of \( H^n(X, \mathbf{C}) \) and establishing spectral sequence degeneracies. The final section discusses various applications, including the fixed part theorem, semi-simplicity theorem, and homomorphisms of abelian schemes. The chapter relies on both Hodge theory and Hironaka's resolution of singularities to express the cohomology of a quasi-projective non-singular variety in terms of projective non-singular varieties.