Let f be a complex polynomial. We relate the behaviour of f “at infinity” to the sheaf of vanishing cycles of the family of projective closures of fibres of f. We show that the absence of such cycles: (i) is equivalent to a condition on the asymptotic behaviour of gradient of f known as Malgrange’s Condition, (ii) implies the -triviality of f. If the support of sheaf of vanishing cycles of is a finite set, then it detects precisely the change of the topology of the fibres of f. Moreover, in this case, the generic fibre of f has the homotopy type of a bouquet of spheres.
@article{bwmeta1.element.bwnjournal-article-bcpv39z1p131bwm,
author = {Parusi\'nski, Adam},
title = {A note on singularities at infinity of complex polynomials},
journal = {Banach Center Publications},
volume = {38},
year = {1997},
pages = {131-141},
zbl = {0882.32018},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv39z1p131bwm}
}
Parusiński, Adam. A note on singularities at infinity of complex polynomials. Banach Center Publications, Tome 38 (1997) pp. 131-141. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv39z1p131bwm/
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