81 research outputs found
On the connectedness principle and dual complexes for generalized pairs
No full textLet be a pair, and let be a contraction
with nef over . A conjecture, known as the Shokurov-Koll\'{a}r
connectedness principle, predicts that has
at most two connected components, where is an arbitrary schematic
point and denotes the non-klt locus of . In this
work, we prove this conjecture, characterizing those cases in which
fails to be connected, and we extend these same results
also to the category of generalized pairs. Finally, we apply these results and
the techniques to the study of the dual complex for generalized log Calabi-Yau
pairs, generalizing results of Koll\'{a}r-Xu and Nakamura.Comment: Final version, to appear in "Forum of Mathematics, Sigma
Hyperbolicity for log canonical pairs and the cone theorem
No full textGiven a log canonical pair (X, Delta), we show that K-X + Delta is nef assuming there is no non-constant map from the affine line with values in the open strata of the stratification induced by the non-klt locus of (X, Delta). This implies a generalization of the Cone Theorem where each K-X + Delta-negative extremal ray is spanned by a rational curve that is the closure of a copy of the affine line contained in one of the open strata of Nklt(X, Delta). Moreover, we give a criterion of Nakai type to determine when under the above condition K-X + Delta is ample and we prove some partial results in the case of arbitrary singularities
A geometric characterization of toric singularities
No full textGiven a projective contraction and a log canonical pair such that is nef over a neighborhood of a closed point , one can define an invariant, the complexity of over , comparing the dimension of and the relative Picard number of with the sum of the coefficients of those components of intersecting the fibre over . We prove that the complexity of over is non-negative and that when it is zero then is formally isomorphic to a morphism of toric varieties around . In particular, considering the case when is the identity morphism, we get a geometric characterization of singularities that are formally isomorphic to toric singularities. This gives a positive answer to a conjecture due to Shokurov.Etant donné une contraction projective et une paire log canonique telle que soit numeriquement effectif sur un voisinage d'un point fermé , on peut définir un invariant, la complexité de sur , en comparant la dimension de et le nombre de Picard relatif de avec la somme des coefficients des composantes de qui intersectent la fibre sur .
Nous demonstrons que, dans les hypothèses ci-dessus, la complexité de la paire logarithmique sur est non-négative, et que, lorsqu'elle est nulle, alors est formellement isomorphe à un morphisme de variétés toriques autour de .
En particulier, en considérant le cas où est le morphisme d'identité, on obtient une caractérisation géométrique des singularités qui sont formellement isomorphes aux singularités toriques, résolvant ainsi une conjecture de Shokurov
Effective algebraic integration in bounded genus
No full textWe introduce and study birational invariants for foliations on
projective surfaces built from the adjoint linear series of
positive powers of the canonical bundle of the foliation.
We apply the results in order to investigate the effective
algebraic integration of foliations on the projective plane.
In particular, we describe the Zariski closure of the set of foliations on
of degree admitting rational first integrals with
fibers having geometric genus bounded by
Birational boundedness of low-dimensional elliptic Calabi-Yau varieties with a section
No full textWe prove that there are finitely many families, up to isomorphism in codimension one, of elliptic Calabi-Yau manifolds Y -> X with a rational section, provided that dim(Y) <= 5 and Y is not of product type. As a consequence, we obtain that there are finitely many possibilities for the Hodge diamond of such manifolds. The result follows from log birational boundedness of Kawamata log terminal pairs (X, Delta) with K-X + Delta numerically trivial and not of product type, in dimension at most four
Local and global applications of the Minimal Model Program for co-rank one foliations on threefolds
No full textWe provide several applications of the minimal model program to the local and
global study of co-rank one foliations on threefolds. Locally, we prove a
singular variant of Malgrange's theorem, a classification of terminal foliation
singularities and the existence of separatrices for log canonical
singularities. Globally, we prove termination of flips, a connectedness theorem
on lc centres, a non-vanshing theorem and some hyperbolicity properties of
foliations.Comment: 55 pages; new version taking into account referee comments, to appear
in JEM
Rational curves and strictly nef divisors on Calabi--Yau threefolds
No full textWe give a criterion for a nef divisor D to be semi-ample on a Calabi-Yau threefold X when D3=0=c2(X)⋅D and c3(X)≠0. As a direct consequence, we show that on such a variety X, if D is strictly nef and ν(D)≠1, then D is ample; we also show that if there exists a Cariter divisor D≢0 in the boundary of the nef cone of X, then X contains a rational curve when its topological Euler characteristic is not 0
Boundedness of elliptic Calabi-Yau varieties with a rational section
No full textWe show that for each fixed dimension , the set of -dimensional klt elliptic varieties with numerically trivial canonical bundle is bounded up to isomorphism in codimension one, provided that the torsion index of the canonical class is bounded and the elliptic fibration admits a rational section. This case builds on an analogous boundedness result for the set of rationally connected log Calabi-Yau pairs with bounded torsion index. In dimension , we prove the more general statement that the set of -lc pairs with nef and rationally connected is bounded up to isomorphism in codimension one
Effective generation for foliated surfaces: Results and applications
No full textWe prove some results on the birational structure and invariants of a foliated surface in terms of the adjoint divisor , 0 < \epsilon \ll 1. Several applications of these ideas are considered as well
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