On The Decomposition Theorem For Intersection De Rham Complexes

Hodge had proved a very fundamental and important result on compact Kahler manifolds,namely the Hodge decomposition theorem.The proof of the Hodge decomposition theorem implies the Hodge to de Rham spectral sequence of a compact Kahler manifold degenerates at E1.In particular,by GAGA we know that the E1-degeneration property on complex projective manifolds can be described by the language of algebraic geometry.Hence,a natural question arise:whether there exists a purely algebraic proof for the E1-degeneration property on complex projective manifolds.The work of Deligne-Illusie gave the above question a positive answer.This dissertation is my Ph.D work,which is a joint work with Prof.Mao Sheng who plays the roles of supervisor and collaborator.My Ph.D work focus on the generalization of the theory of DeligneIllusie in their spirit.As applications:i)We give a mod p proof for the intersection version of E1-degeneration property on complex projective manifolds;ii)We obtain an intersection version of vanishing theorem over a field of characteristic zero.More concretely,my work contains three parts.First,we generalize DeligneIllusie's decomposition theorem by their spirit,i.e.,generalize their explicit construction.Let X be a smooth scheme over a perfect field k of characteristic p of finite type,D a SNCD on X such that the pair(X,D)has a W2(k)-lifting.In the derived category of X',Deligne-Illusie have constructed an isomorphism between following complexes:i)The classical logarithmic Higgs complex on X';?)The relative Frobenius pushforward of the classical logarithmic de Rham complex on X.We show that Deligne-Illusie's construction can be realized in a more general setting,namely we constructed an isomorphism between following complexes:i)The logarithmic Higgs complex of a logarithmic Higgs sheaf subject to some nilpotent condition;ii)The relative Frobenius pushforward of the logarithmic de Rham complex of the inverse Cartier transformation of the logarithmic Higgs sheaf.Our construction enjoys the merit that it preserves the intersection condition.Second,we show that for a given logarithmic Fontaine-Faltings module which satisfies some numerical condition,its intersection complex degenerates at E1 under the natural filtration(additionally assume X proper over k).In particular,we obtain two interesting results:i)Intersection-adaptedness theorem;ii)All the filtered residue maps are bundle maps.Third,we give two applications for the Hodge theory in positive characteristics which is established by the above two Parts.i)Let X be a smooth projective variety over a field k of characteristic zero with a SNCD,then we can show an intersection version of Kodaira-Akizuki-Nakano vanishing theorem on X.ii)Let f:(Y,E)?(X,D)be a semi-stable family of relative dimension n over C,where D,E are SNCDs on X,Y respectively with E=f-1D.Let V=Rif*an QYan-Ean,0?i?n,then the Hodge complex(IC(Xan,V),Fil)defines a pure Hodge structure on Hm(Xan,V?)of weight i+m.By our E1-degeneration theorem in positive characteristics,we can provide a mod p proof for the E1-degeneration property of(IC(Xan,V),Fil)by spread-out technical in algebraic geometry.