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Noether-Lefschetz Problems for Degeneracy Loci
J. Spandaw, Institüut fur Mathematik, Universität Hannover, Germany
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Memoirs of the American Mathematical Society
2003; 136 pp; softcover
Volume: 161
ISBN-10: 0-8218-3183-6
ISBN-13: 978-0-8218-3183-0
List Price: US$61 Individual Members: US$36.60
Institutional Members: US\$48.80
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 $$d:=n-(e-r)(f-r)$$.

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.

• A: On the cohomology of $$\mathbb{G}(s,F)$$