Поступления иностранных книг в библиотеки ННЦ в 2003 г.
Институт математики |  |
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Spandaw J. Noether-Lefschetz problems for degeneracy loci. - Providence: Amer. math. soc., 2003. - XI, 136 p. - (Memoirs of the Amer. math. soc., ISBN 0-8218-3183-6; N 764). - Задачи Нетера-Лефшеца для вырожденных геометрических мест точек
Оглавление
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- 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 $q\lt d$ 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.
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