L2 Methods and Effective Results in Algebraic Geometry

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L2-Methods and Effective Results in Algebraic Geometry Jean-Pierre Demailly Universite de Grenoble I, Institut Fourier URA 188 du CNRS, BP74, F-38402 Saint-Martin d'Heres, France Abstract. One important problem arising in algebraic geometry is the computation of effective bounds for the degree of embeddings in a projective space of given algebraic varieties. This problem is intimately related to the following question: Given a positive (or ample) line bundle L on a projective manifold X, can one compute explicitly an integer m0 such that mL is very ample for m > m0 ? It turns out that the answer is much easier to obtain in the case of adjoint line bundles 2(KX + mL), for which universal values of m0 exist. We indicate here how such bounds can be derived by a combination of powerful analytic methods: theory of positive currents and plurisubharmonic functions (Lelong), L2 estimates for ∂ (Andreotti-Vesentini, Hormander, Bombieri, Skoda), Nadel vanishing theorem, Aubin-Calabi-Yau theorem, and holomorphic Morse inequalities. 1. Basic concepts of hermitian differential geometry Let X be a complex manifold of dimension n and let F be a C∞ complex vector bundle of rank r over X . A connection D on F is a linear differential operator D acting on spaces C∞(X,?p,qT ?X ? F ) of F -valued differential forms, increasing the degree by 1 and satisfying Leibnitz' rule D(f ? u) = df ? u+ (?1)deg ff ?Du for all

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2 LMethods and Eﬀective Results in Algebraic Geometry

JeanPierre Demailly

Universit´edeGrenobleI,InstitutFourier URA188duCNRS,BP74,F38402SaintMartind’H`eres,France

Abstract.One important problem arising in algebraic geometry is the computation of eﬀective bounds for the degree of embeddings in a projective space of given algebraic varieties. This problem is intimately related to the following question: Given a positive (or ample) line bundleLon a projective manifoldX, can one compute explicitly an integerm0such thatmLis very ample form>m0turns out that? It the answer is much easier to obtain in the case of adjoint line bundles 2(KX+mL), for which universal values ofm0exist. We indicate here how such bounds can be derived by a combination of powerful 2 analytic methods: theory of positive currents and plurisubharmonic functions (Lelong),Lestimates for∂meA,ehroClabunielva,NadingtnishreibmoB,)adokS,iH¨i,innterndmaorittseVeAn(eodriabuYa theorem, and holomorphic Morse inequalities.

1. Basic concepts of hermitian diﬀerential geometry ∞ LetXbe a complex manifold of dimensionnand letFbe aCcomplex vector bundle of rankroverX. AconnectionDonFis a linear diﬀerential operatorDacting ∞p,q ⋆ on spacesC(X,ΛT⊗F) ofFvalued diﬀerential forms, increasing the degree by 1 X and satisfying Leibnitz’ rule degf D(f∧u) =df∧u+ (−1)f∧Du

∞a,b ⋆∞p,q ⋆ for all formsf∈C(X,ΛT),u∈C(X,ΛT⊗F). As usual, we split X X ′ ′′ D=D+Dinto its (1,0) and (0,1) parts, where ′ ′′ ∞p,q ⋆∞p+1,q ⋆∞p,q+1⋆ D+D:C(X,ΛT⊗F)−→C(X,ΛT⊗F)⊕C(X,ΛT⊗F). X X X ≃ r With respect to a trivializationτ:F↾Ω−→Ω×C, a connectionDcan be written ′ ′′ Du≃τdu+ Γ∧uis an arbitrary (+ Γ , where Γ = Γ r×r)matrix of 1forms and 2 dacts componentwise. A standard computation shows thatD u≃τΘ(D)∧u, where Θ(D) =dΓ + Γ∧Γ is a global 2form onXwith values in Hom(F, Fform is called). This thecurvature tensorofFthe important case of rank 1 bundles, Θ(. In F) =dΓ is a dclosed form with complex values; it is well known that the De Rham cohomology class i i 2 ofθ(FΘ() := F) =Dis the image in De Rham cohomology of the ﬁrst Chern class 2π2π 2 c1(F)∈H(M,Z). For any line bundlesF1, . . . , FponXand any compactpdimensional analytic setYinX, we set Z F1∙. . .∙Fp∙Y=c1(F1)∧. . .∧c1(Fp). Y

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