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CHARACTERIZATIONS OF PROJECTIVE SPACES AND HYPERQUADRICS

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Niveau: Supérieur, Doctorat, Bac+8
CHARACTERIZATIONS OF PROJECTIVE SPACES AND HYPERQUADRICS STEPHANE DRUEL AND MATTHIEU PARIS Abstract. In this paper we prove that if the r-th tensor power of the tangent bundle of a smooth projective variety X contains the determinant of an ample vector bundle of rank at least r, then X is isomorphic either to a projective space or to a smooth quadric hypersurface. Our result generalizes Mori's, Wahl's, Andreatta-Wisniewski's and Araujo-Druel-Kovacs's characterizations of projective spaces and hyperquadrics. 1. Introduction Projective spaces and hyperquadrics are the simplest projective algebraic varieties, and they can be characterized in many ways. The aim of this paper is to provide a new characterization of them in terms of positivity properties of the tangent bundle. We refer the reader to the article [ADK08] which reviews these matters. Notice that our results generalize Mori's (see [Mor79]), Wahl's (see [Wah83] and [Dru04]), Andreatta-Wisniewski's (see [AW01] and [Ara06]), Araujo-Druel-Kovacs's (see [ADK08]) and Paris's (see [Par10]) characterizations of projective spaces and hyperquadrics. K. Ross recently posted a somewhat related result (see [ROS10]). In this paper, we prove the following theorems. Here Qn denotes a smooth quadric hypersurface in Pn+1, and OQn(1) denotes the restriction of OPn+1(1) to Qn.

  • stephane druel

  • araujo-druel-kovacs's

  • nef

  • projective variety

  • vector bundle

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  • over curves

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CHARACTERIZATIONS OF PROJECTIVE SPACES AND
´ STEPHANE DRUEL AND MATTHIEU PARIS
HYPERQUADRICS
Abstract.In this paper we prove that if the r-th tensor power of the tangent bundle of a smooth projective variety X contains the determinant of an ample vector bundle of rank at least r, then X is isomorphic either to a projective space or to a smooth quadric hypersurface. Our result generalizes Moris,Wahls,Andreatta-Wi´sniewskisandAraujo-Druel-Kova´csscharacterizationsofprojective spaces and hyperquadrics.
1.Introduction
Projective spaces and hyperquadrics are the simplest projective algebraic varieties, and they can be characterized in many ways. The aim of this paper is to provide a new characterization of them in terms of positivity properties of the tangent bundle. We refer the reader to the article [ADK08] which reviews these matters. Notice that our results generalize Mori’s (see [Mor79]), Wahl’s (see [Wah83]and[Dru04]),Andreatta-Wis´niewskis(see[AW01]and[Ara06]),Araujo-Druel-Kov´acss (see [ADK08]) and Paris’s (see [Par10]) characterizations of projective spaces and hyperquadrics. K. Ross recently posted a somewhat related result (see [ROS10]). In this paper, we prove the following theorems. HereQndenotes a smooth quadric hypersurface n+1 restriction ofO(1) toQ. Whenn= 1, (Q ,O(1)) is just inP, andOQn(1) denotes thePn1Q1 n+1 1 (P,O1(2)). P Theorem A.LetXbe a smooth complex projectiven-dimensional variety andEan ample vector 0r⊗−1 bundle onXof rankr+kwithr>1andk>1. Ifh(X, Tdet(E) )6= 0, then(X,det(E))' X r(n+1) n n (P,OP(l))withr+k6l6. n
Theorem B.LetXbe a smooth complex projectiven-dimensional variety andEan ample vector 0r⊗−1n bundle onXof rankr>1. Ifh(X, Tdet(E) )6= 0, then either(X,det(E))'(P,OP(l)) n X r(n+1) r withr6l6, or(X,E)'(Qn,OQn(1) )andr= 2i+njwithi>0andj>0. n The line of argumentation follows [AW01] (see also [ADK08] and [Par10]). We first prove The-orem A and Theorem B for Fano manifolds with Picard numberρ(X) = 1 (see Proposition 14). Then the argument for the proof of the main Theorem goes as follows. We argue by induction on dim(X). We may assumeρ(X)>the2. Hence H-rationally connected quotient ofXwith respect to an unsplit covering familyHof rational curves onXis non-trivial. It can be ex-tended in codimension one so that we can produce a normal varietyXBequipped with a surjective 1 morphismπBwith integral fibers onto a smooth curveBsuch that eitherB'P,XBB d0i∗ ⊗ri⊗−1 is aP-bundle for somed>1 andh(XB, T1πGdet(E) )6= 0 for some in-|XB XB/P 1teger 16i6rwhereGbe a vector bundle onPsuch thatG(2) is nef, orXBBis a
2000Mathematics Subject Classification.14M20. The first named author was partially supported by the A.N.R. 1