Abstract
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.
Acknowledgement
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.
Citation
Antonio Lanteri. Andrea Luigi Tironi. "Generalized polarized manifolds with low second class." Hiroshima Math. J. 51 (3) 301 - 334, November 2021. https://doi.org/10.32917/h2020070
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