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November 2021 Generalized polarized manifolds with low second class
Antonio Lanteri, Andrea Luigi Tironi
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Hiroshima Math. J. 51(3): 301-334 (November 2021). DOI: 10.32917/h2020070


On a smooth complex projective variety $X$ of dimension $n$, consider an ample vector bundle $\mathscr E$ of rank $r \leq n - 2$ and an ample line bundle $H$. A numerical character $m_2 = m_2(X,\mathscr E, H)$ of the triplet $(X,\mathscr E, H)$ is defined, extending the well-known second class of a polarized manifold $(X, H)$, when either $n = 2$ or $H$ is very ample. Under some additional assumptions on $\mathscr F:= \mathscr E \oplus H^{\oplus(n-r-2)}$, triplets $(X,\mathscr E, H)$ as above whose $m_2$ is small with respect to the invariants $d := c_{n-2}(\mathscr F)H^2$ and $g := 1 + \frac{1}{2}(K_X + c_1(\mathscr F)+H)\cdot c_{n-2}(\mathscr F)\cdot H$ are studied and classified.

Funding Statement

The first author is a member of G.N.S.A.G.A. of the Italian INdAM. He would like to thank the Projects PRIN 2010-11 and 2015 Geometry of Algebraic Varieties and the University of Milano for partial support.
The second author was partially supported by the National Project Anillo ACT 1415 PIA CONICYT and the Proyecto VRID N.219.015.023-INV of the University of Concepción.


The authors are grateful to Professor Y. Fukuma for calling [10] to their attention. They are also grateful to the referee for pointing out some gaps in the earlier version of the paper.


Download Citation

Antonio Lanteri. Andrea Luigi Tironi. "Generalized polarized manifolds with low second class." Hiroshima Math. J. 51 (3) 301 - 334, November 2021.


Received: 27 July 2020; Revised: 18 June 2021; Published: November 2021
First available in Project Euclid: 1 December 2021

Digital Object Identifier: 10.32917/h2020070

Primary: 14F05
Secondary: 14J25 , 14J60 , 14N30

Keywords: ample vector bundle , special variety , surface

Rights: Copyright © 2021 Hiroshima University, Mathematics Program


Vol.51 • No. 3 • November 2021
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