# Noether-Lefschetz Problems for Degeneracy Loci

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*J. Spandaw*

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.

#### Table of Contents

# Table of Contents

## Noether-Lefschetz Problems for Degeneracy Loci

- Contents ix10 free
- 1 Introduction 114 free
- 2 The Monodromy Theorem 1326
- 3 Degeneracy Loci of Corank One 2639
- 4 Degeneracy Loci of Arbitrary Corank 3649
- 4.1 The basic construction 3649
- 4.2 The cohomology of the pair (Y, W) 3952
- 4.3 A Noether–Lefschetz theorem 4154
- 4.4 The infinitesimal method 4255
- 4.5 The reduction step 4356
- 4.6 Cohomology on Grassmannians (I) 4760
- 4.7 Pieri's rule 5164
- 4.8 Miscellaneous preparations 5265
- 4.9 Surjectivity of δ 5366
- 4.10 Surjectivity of ε 5770
- 4.11 Injectivity of β[sub(1)]for s = 1 6275
- 4.12 Cohomology on Grassmannians (II) 6578
- 4.13 Vanishing Lemma 6780
- 4.14 Injectivity of β[sub(1)] for s ≥ 2 7083
- 4.15 Injectivity of β[sub(3)] 7083
- 4.16 Surjectivity of γ 7386

- 5 Degeneracy Loci in Projective Space 7790
- 6 Examples 100113
- Appendix A: On the Cohomology of G(s, F) 128141
- Frequently Used Notations 132145
- Bibliography 133146