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**Positive polynomials for ample vector bundles.**
*(English)*
Zbl 0537.14009

The main goal of the paper under review is to describe the cone of numerically positive polynomials for ample vector bundles. A polynomial \(P\in {\mathbb{Q}}[c_ 1,...,c_ e]\) of degree n in variables \(c_ i\) of weight i is called numerically positive if \(\int_{X}P(c_ 1(E),...,c_ e(E))>0\) for each projective variety X, dim X\(=n\) and each ample vector bundle E on X, rk E\(=e\). The authors show that a polynomial \(P\neq 0\) is numerically positive iff it has non-negative coefficients with respect to the basis of \({\mathbb{Q}}[c_ 1,...,c_ e]\) formed by the so-called Schur polynomials. This generalizes the well known results of Bloch, Gieseker and Kleiman. Furthermore, the authors proceed with proving that their notion of numerical positivity coincides with the notion of positivity introduced by P. A. Griffiths [in Global Analysis, Papers in Honor of K. Kodaira, 185-251 (1969; Zbl 0201.240)] thus establishing the Griffiths conjecture in full generality [a special case of this conjecture was previously proved by S. Usui and H. Tango, J. Math. Kyoto Univ. 17, 151-164 (1977; Zbl 0353.14008)].

The proof of the main theorem is based on a positivity result in the intersection theory for cone classes in ample vector bundles. As an application of their techniques, the authors give a new proof of their theorem to the effect that if \(u:E\to F\) is a map of vector bundles of ranks e and f respectively on a projective variety X and if \(D_ k(u)=\{x\in X| rk u(x)\leq k\},\) then \(D_ k(u)\) meets any subvariety \(Y\subseteq X\) of dimension \(\geq(e-k)(f-k)\) provided that the bundle Hom(E,F) is ample (in particular, \(D_ k(u)\neq \emptyset\) if dim \(X\geq(e-k)(f-k)).\)

Another application is a simple proof of a special case of Hartshorne’s conjecture. Namely, let M be a variety, let \(X\subseteq M\) be a projective local complete intersection of codimension e with ample normal bundle, and let \(Y\subseteq M\) be a subvariety of dimension \(\geq e\) that meets X. Then the intersection class [X]\(\cdot [Y]\) in \(A_*(X)\) is non- zero. In particular, if M is a homogeneous variety, then any subvariety Z of dimension \(\geq e\) meets X [cf. M. Lübke, J. Reine Angew. Math. 316, 215-220 (1980; Zbl 0421.32032)].

The proof of the main theorem is based on a positivity result in the intersection theory for cone classes in ample vector bundles. As an application of their techniques, the authors give a new proof of their theorem to the effect that if \(u:E\to F\) is a map of vector bundles of ranks e and f respectively on a projective variety X and if \(D_ k(u)=\{x\in X| rk u(x)\leq k\},\) then \(D_ k(u)\) meets any subvariety \(Y\subseteq X\) of dimension \(\geq(e-k)(f-k)\) provided that the bundle Hom(E,F) is ample (in particular, \(D_ k(u)\neq \emptyset\) if dim \(X\geq(e-k)(f-k)).\)

Another application is a simple proof of a special case of Hartshorne’s conjecture. Namely, let M be a variety, let \(X\subseteq M\) be a projective local complete intersection of codimension e with ample normal bundle, and let \(Y\subseteq M\) be a subvariety of dimension \(\geq e\) that meets X. Then the intersection class [X]\(\cdot [Y]\) in \(A_*(X)\) is non- zero. In particular, if M is a homogeneous variety, then any subvariety Z of dimension \(\geq e\) meets X [cf. M. Lübke, J. Reine Angew. Math. 316, 215-220 (1980; Zbl 0421.32032)].

Reviewer: F.L.Zak

### MSC:

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |