Noether-Lefschetz Problems for Degeneracy Loci Book

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 $psicolon Fto E$ be a holomorphic homomorphism of vector bundles. Consider the degeneracy locus $$Z:=D_r(psi):={xin Xcolon mathrm{rk} (psi(x))le r}.$$ We assume without loss of generality that $ege f > rge 0$. We assume furthermore that $Eotimes F^vee$ 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 $$picolon Y:=mathbb{G}(f-r,F)to X$$ 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 $Wstackrel{sim}{to} Z$ if $n<(e-r+1)(f-r+1)$. (If $r=0$ then $W=Zsubseteq X=Y$ is the zero-locus of $psiin H^0(X,Eotimes F^vee)$.) 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 $Eotimes F^vee$ is sufficiently ample and $psi$ is very general. The positivity condition on $Eotimes F^vee$ 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 $Eotimes F^vee$. The examples in the last chapter show that these conditions are quite sharp.