eBook ISBN:  9781470403621 
Product Code:  MEMO/161/764.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $39.00 
eBook ISBN:  9781470403621 
Product Code:  MEMO/161/764.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $39.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 161; 2003; 136 ppMSC: Primary 14
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.
ReadershipGraduate student and research mathematicians.

Table of Contents

Chapters

1. Introduction

2. The monodromy theorem

3. Degeneracy loci of corank one

4. Degeneracy loci of arbitrary corank

5. Degeneracy loci in projective space

6. Examples


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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.
Graduate student and research mathematicians.

Chapters

1. Introduction

2. The monodromy theorem

3. Degeneracy loci of corank one

4. Degeneracy loci of arbitrary corank

5. Degeneracy loci in projective space

6. Examples