Curvature of vector bundles associated to holomorphic fibrations.

*(English)*Zbl 1195.32012This paper is mainly concerned with the positivity (in the sense of Nakano) of certain Hermitian vector bundles. First, consider a domain \(D=U\times\Omega\) in \({\mathbb C}^m\times{\mathbb C}^n\) and a (strictly) plurisubhamonic function \(\phi\) on \(D\), smooth up to the boundary. For each \(t\in U\), write \(\phi^t\) for the plurisubharmonic function on \(\Omega\) given by \(\phi^t(\cdot):=\phi(t,\cdot)\) and denote by \(A_t^2\) the Bergman space of holomorphic functions on \(\Omega\) with norm given by \(\|h\|^2_t:=\int_\Omega|h|^2e^{-\phi^t}\). The infinite rank vector bundle \(E\) over \(U\) with fibre \(E_t=A_t^2\) is trivial as a bundle but is equipped with a non-trivial metric \(\|\cdot\|\). Then (Theorem 1.1) the Hermitian bundle \((E,\|\cdot\|)\) is (strictly) positive in the sense of Nakano.

There is a natural analogue of this result for holomorphic fibrations with compact fibres. In fact, suppose \(X\) is a Kähler manifold of dimension \(n+m\) which is smoothly fibred with compact fibres over a connected complex manifold \(Y\) of dimension \(m\). This implies that the fibres are all diffeomorphic, but not in general biholomorphic to one another. Now, let \(L\) be a holomorphic Hermitian line bundle over \(X\) and define, for each \(t\in Y\), \(E_t=\Gamma(X_t,L|X_t\otimes K_{X_t})\), where \(X_t\) is the fibre of \(X\) over \(t\) and \(K_{X_t}\) is the canonical bundle of \(X_t\). If \(L\) is (semi)positive, one can then define in a natural way a holomorphic vector bundle \(E\) over \(Y\) whose fibre at \(t\) is \(E_t\) (the author proves the extension result required for this in an appendix). Moreover, \(E\) admits a Hermitian metric \(\|\cdot\|\) by integration over the fibres. Then (Theorem 1.2) \((E,\|\cdot\|)\) is (semi)positive in the sense of Nakano.

An example in which the situation of Theorem 1.2 arises is when \(X={\mathbb P}(V)\) for some holomorphic vector bundle \(V\) of rank \(r\) over \(Y\) and \(L={\mathcal O}_{{\mathbb P}(V)}(r+1)\). In this case, \(E\) is isomorphic (as a holomorphic vector bundle over \(Y\)) to \(V\otimes\det V\). Moreover, \(L\) is positive if and only if \(V\) is ample in the sense of Hartshorne. So we have (Theorem 1.3): Let \(V\) be an ample holomorphic vector bundle of finite rank over a complex manifold \(Y\); then \(V\otimes\det V\) has a smooth Hermitian metric which is strictly positive in the sense of Nakano. Replacing \({\mathcal O}_{{\mathbb P}(V)}(r+1)\) by \({\mathcal O}_{{\mathbb P}(V)}(r+m)\), we obtain also that \(S^m(V)\otimes\det V\) is strictly positive for all \(m\geq0\) (Theorem 7.1). In this context it is known that, if \(V\) is positive in the sense of Griffiths, then \(V\otimes\det V\) is positive in the sense of Nakano. So Theorem 1.3 provides a partial result in the direction of Griffiths’ conjecture that an ample vector bundle is positive in his sense.

There is a natural analogue of this result for holomorphic fibrations with compact fibres. In fact, suppose \(X\) is a Kähler manifold of dimension \(n+m\) which is smoothly fibred with compact fibres over a connected complex manifold \(Y\) of dimension \(m\). This implies that the fibres are all diffeomorphic, but not in general biholomorphic to one another. Now, let \(L\) be a holomorphic Hermitian line bundle over \(X\) and define, for each \(t\in Y\), \(E_t=\Gamma(X_t,L|X_t\otimes K_{X_t})\), where \(X_t\) is the fibre of \(X\) over \(t\) and \(K_{X_t}\) is the canonical bundle of \(X_t\). If \(L\) is (semi)positive, one can then define in a natural way a holomorphic vector bundle \(E\) over \(Y\) whose fibre at \(t\) is \(E_t\) (the author proves the extension result required for this in an appendix). Moreover, \(E\) admits a Hermitian metric \(\|\cdot\|\) by integration over the fibres. Then (Theorem 1.2) \((E,\|\cdot\|)\) is (semi)positive in the sense of Nakano.

An example in which the situation of Theorem 1.2 arises is when \(X={\mathbb P}(V)\) for some holomorphic vector bundle \(V\) of rank \(r\) over \(Y\) and \(L={\mathcal O}_{{\mathbb P}(V)}(r+1)\). In this case, \(E\) is isomorphic (as a holomorphic vector bundle over \(Y\)) to \(V\otimes\det V\). Moreover, \(L\) is positive if and only if \(V\) is ample in the sense of Hartshorne. So we have (Theorem 1.3): Let \(V\) be an ample holomorphic vector bundle of finite rank over a complex manifold \(Y\); then \(V\otimes\det V\) has a smooth Hermitian metric which is strictly positive in the sense of Nakano. Replacing \({\mathcal O}_{{\mathbb P}(V)}(r+1)\) by \({\mathcal O}_{{\mathbb P}(V)}(r+m)\), we obtain also that \(S^m(V)\otimes\det V\) is strictly positive for all \(m\geq0\) (Theorem 7.1). In this context it is known that, if \(V\) is positive in the sense of Griffiths, then \(V\otimes\det V\) is positive in the sense of Nakano. So Theorem 1.3 provides a partial result in the direction of Griffiths’ conjecture that an ample vector bundle is positive in his sense.

Reviewer: P. E. Newstead (Liverpool)