In this monograph we study the cohomology of degeneracy loci of the following type. Let \(X\) be a complex projective manifold of dimension \(n\), let \(E\) and \(F\) be holomorphic vector bundles on \(X\) of rank \(e\) and \(f\), respectively, and let \(\psi\colon F\to E\) be a holomorphic homomorphism of vector bundles. Consider the degeneracy locus \[Z:=D_r(\psi):=\{x\in X\colon \mathrm{rk} (\psi(x))\le r\}.\] We assume without loss of generality that \(e\ge f > r\ge 0\). We assume furthermore that \(E\otimes F^\vee\) is ample and globally generated, and that \(\psi\) is a general homomorphism. Then \(Z\) has dimension \(d:=n-(e-r)(f-r)\).

In order to study the cohomology of \(Z\), we consider the Grassmannian bundle \[\pi\colon Y:=\mathbb{G}(f-r,F)\to X\] of \((f-r)\)-dimensional linear subspaces of the fibres of \(F\). In \(Y\) one has an analogue \(W\) of \(Z\): \(W\) is smooth and of dimension \(d\), the projection \(\pi\) maps \(W\) onto \(Z\) and \(W\stackrel{\sim}{\to} Z\) if \(n<(e-r+1)(f-r+1)\). (If \(r=0\) then \(W=Z\subseteq X=Y\) is the zero-locus of \(\psi\in H^0(X,E\otimes F^\vee)\).) Fulton and Lazarsfeld proved that \[ H^q(Y;\mathbb{Z}) \to H^q(W;\mathbb{Z}) \] is an isomorphism for \(q < d\) and is injective with torsion-free cokernel for \(q=d\). This generalizes the Lefschetz hyperplane theorem. We generalize the Noether-Lefschetz theorem, i.e. we show that the Hodge classes in \(H^d(W)\) are contained in the subspace \(H^d(Y)\subseteq H^d(W)\) provided that \(E\otimes F^\vee\) is sufficiently ample and \(\psi\) is very general.

The positivity condition on \(E\otimes F^\vee\) can be made explicit in various special cases. For example, if \(r=0\) or \(r=f-1\) we show that Noether-Lefschetz holds as soon as the Hodge numbers of \(W\) allow, just as in the classical case of surfaces in \(\mathbb{P}^3\). If \(X=\mathbb{P}^n\) we give sufficient positivity conditions in terms of Castelnuovo-Mumford regularity of \(E\otimes F^\vee\). The examples in the last chapter show that these conditions are quite sharp.