### 4-folds with numerically effective tangent bundles and second Betti numbers greater than one.

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2000 Mathematics Subject Classification: 14C05, 14L30, 14E15, 14J35.When the cyclic group G of order 15 acts with some specific weights on affine four-dimensional space, the G-Hilbert scheme is a crepant resolution of the quotient A^4 / G. We give an explicit description of this resolution using G-graphs.

In this paper we prove that for a nonsingular projective variety of dimension at most 4 and with non-negative Kodaira dimension, the Kodaira dimension of coherent subsheaves of ${\Omega}^{p}$ is bounded from above by the Kodaira dimension of the variety. This implies the finiteness of the fundamental group for such an $X$ provided that $X$ has vanishing Kodaira dimension and non-trivial holomorphic Euler characteristic.

It is proved that there are only finitely many families of codimension two subvarieties not of general type in Q6.

Una contrazione su una varietà proiettiva liscia $X$ è data da una mappa $\phi :X\to Z$ propria, suriettiva e a fibre connesse in una varietà irriducibile normale $Z$. La contrazione si dice di Fano-Mori se inoltre $-{K}_{X}$ è $\phi $-ampio. Nel lavoro, naturale seguito e completamento delle ricerche introdotte in [A-W3], si studiano diversi aspetti delle contrazioni di Fano-Mori attraverso esempi (capitolo 1) e teoremi di struttura (capitoli 3 e 4). Si discutono anche alcune applicazioni allo studio di morfismi birazionali propri...

The main purpose of this paper is twofold. We first analyze in detail the meaningful geometric aspect of the method introduced in [12], concerning families of irreducible, nodal curves on a smooth, projective threefold X. This analysis gives some geometric interpretations not investigated in [12] and highlights several interesting connections with families of other singular geometric objects related to X and to other varieties. Then we use this method to study analogous problems for families of...

O’Grady showed that certain special sextics in ${\mathbb{P}}^{5}$ called EPW sextics admit smooth double covers with a holomorphic symplectic structure. We propose another perspective on these symplectic manifolds, by showing that they can be constructed from the Hilbert schemes of conics on Fano fourfolds of degree ten. As applications, we construct families of Lagrangian surfaces in these symplectic fourfolds, and related integrable systems whose fibers are intermediate Jacobians.

We study the geometry of a general Fano variety of dimension four, genus nine, and Picard number one. We compute its Chow ring and give an explicit description of its variety of lines. We apply these results to study the geometry of non quadratically normal varieties of dimension three in a five dimensional projective space.

We present an example which confirms the assertion of the title.