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<p>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 $e\ge f &gt; r\ge 0$. 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 $n&lt;(e-r+1)(f-r+1)$. (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 $q &lt; d$ and is injective wi

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Author

Jeroen Spandaw

Publisher

American Mathematical Society

Published

2003

Language

EN

ISBN

9780821831830

Category

nonfiction

Subjects

Computer Science, Mathematics, Science