Memoirs of the American Mathematical Society 2003; 136 pp; softcover Volume: 161 ISBN10: 0821831836 ISBN13: 9780821831830 List Price: US$65 Individual Members: US$39 Institutional Members: US$52 Order Code: MEMO/161/764
 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 \(d:=n(er)(fr)\). In order to study the cohomology of \(Z\), we consider the Grassmannian bundle \[\pi\colon Y:=\mathbb{G}(fr,F)\to X\] of \((fr)\)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 \(n<(er+1)(fr+1)\). (If \(r=0\) then \(W=Z\subseteq X=Y\) is the zerolocus 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 torsionfree cokernel for \(q=d\). This generalizes the Lefschetz hyperplane theorem. We generalize the NoetherLefschetz 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 \(r=f1\) we show that NoetherLefschetz 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 CastelnuovoMumford regularity of \(E\otimes F^\vee\). The examples in the last chapter show that these conditions are quite sharp. Readership Graduate student and research mathematicians. Table of Contents  Introduction
 The Monodromy theorem
 Degeneracy loci of corank one
 Degeneracy loci of arbitrary corank
 Degeneracy loci in projective space
 Examples
 A: On the cohomology of \(\mathbb{G}(s,F)\)
 Frequently used notations
 Bibliography
