Noether-Lefschetz Problems for Degeneracy Loci (Memoirs of the American Mathematical Society)

Noether-Lefschetz Problems for Degeneracy Loci (Memoirs of the American Mathematical Society)


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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.


Introduction The Monodromy theorem Degeneracy loci of corank one Degeneracy loci of arbitrary corank Degeneracy loci in projective space Examples A: On the cohomology of $\mathbb{G}(s,F)$ Frequently used notations Bibliography.

Product Details

  • ISBN13: 9780821831830
  • Format: Paperback
  • ID: 9780821831830
  • ISBN10: 0821831836

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