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.
Readership
Graduate student and research mathematicians.
