69 research outputs found
Relatively Minimal Quasihomogeneous Projective 3-Folds
In the present work we classify the relatively minimal 3-dimensional
quasihomogeneous complex projective varieties under the assumption that the
automorphism group is not solvable. By relatively minimal we understand
varieties X having at most Q-factorial terminal singularities and allowing an
extremal contraction X to Y, where dim Y <3
Projective bundles of singular plane cubics
Classification theory and the study of projective varieties which are covered
by rational curves of minimal degrees naturally leads to the study of families
of singular rational curves. Since families of arbitrarily singular curves are
hard to handle, it has been shown in a previous paper that there exists a
partial resolution of singularities which transforms a bundle of possibly badly
singular curves into a bundle of nodal and cuspidal plane cubics.
In cases which are of interest for classification theory, the total spaces of
these bundles will clearly be projective. It is, however, generally false that
an arbitrary bundle of plane cubics is globally projective. For that reason the
question of projectivity seems to be of interest, and the present work gives a
characterization of the projective bundles
Pull-back Morphisms for Reflexive Differential Forms
Let f : X -> Y be a morphism between normal complex varieties, and assume
that Y is Kawamata log terminal. Given any differential form, defined on the
smooth locus of Y, we construct a "pull-back form" on X. The pull-back map
obtained by this construction is O_Y-linear, uniquely determined by natural
universal properties and exists even in cases where the image of f is entirely
contained in the singular locus of the target variety Y.
One relevant setting covered by the construction is that where f is the
inclusion (or normalisation) of the singular locus of Y. As an immediate
corollary, we show that differential forms defined on the smooth locus of Y
induce forms on every stratum of the singularity stratification. The same
result also holds for many Whitney stratifications.Comment: Final version, to appear in Advances in Mathematic
Boundedness results for singular Fano varieties and applications to Cremona groups
This survey paper reports on work of Birkar, who confirmed a long-standing
conjecture of Alexeev and Borisov-Borisov: Fano varieties with mild
singularities form a bounded family once their dimension is fixed. Following
Prokhorov-Shramov, we explain how this boundedness result implies that
birational automorphism groups of projective spaces satisfy the Jordan
property, answering a question of Serre in the positive.Comment: Final version. To appear in the proceedings of the January 2019
edition of the S\'eminaire Nicolas Bourbak
Families of singular rational curves
Let X be a projective variety which is covered by a family of rational curves
of minimal degree. The classic bend-and-break argument of Mori asserts that if
x and y are two general points, then there are at most finitely many curves in
that family which contain both x and y. In this work we shed some light on the
question as to whether two sufficiently general points actually define a unique
curve.
As an immediate corollary to the results of this paper, we give a
characterization of projective spaces which improves on the known
generalizations of Kobayashi-Ochiai's theorem.Comment: Reason for resubmission: improved expositio
On the Classification of 3-dimensional SL_2(C)-varieties
In the present work we describe 3-dimensional complex SL_2-varieties where
the generic SL_2-orbit is a surface. We apply this result to classify the
minimal 3-dimensional projective varieties with Picard-number 1 where a
semisimple group acts such that the generic orbits are 2-dimensional.
This is an ingredient of the classification [Keb98, math.AG/9805042] of the
3-dimensional relatively minimal quasihomogeneous varieties where the
automorphism group is not solvable
Differential forms on singular spaces, the minimal model program, and hyperbolicity of moduli stacks
This survey discusses hyperbolicity properties of moduli stacks and
generalisations of the Shafarevich Hyperbolicity Conjecture to higher
dimensions. It concentrates on methods and results that relate moduli theory
with recent progress in higher dimensional birational geometry.Comment: 45 pages, 3 figures. Final version, to appear in the Handbook of
Moduli, in honour of David Mumford, to be published by International press,
editors Gavril Farkas and Ian Morriso
A refinement of Stein factorization and deformations of surjective morphisms
This paper is concerned with a refinement of the Stein factorization, and
with applications to the study of deformations of surjective morphisms. We show
that every surjective morphism f:X->Y between normal projective varieties
factors canonically via a finite cover of Y that is etale in codimension one.
This "maximally etale factorization" is characterized in terms of positivity of
the push-forward of the structure sheaf and satisfies a functorial property.
It turns out that the maximally etale factorization is stable under
deformations, and naturally decomposes an etale cover of the Hom-scheme into a
torus and into deformations that are relative with respect to the rationally
connected quotient of the target Y. In particular, we show that all
deformations of f respect the rationally connected quotient of Y
Extending holomorphic forms from the regular locus of a complex space to a resolution of singularities
We investigate under what conditions holomorphic forms defined on the regular
locus of a reduced complex space extend to holomorphic (or logarithmic) forms
on a resolution of singularities. We give a simple necessary and sufficient
condition for this, whose proof relies on the Decomposition Theorem and Saito's
theory of mixed Hodge modules. We use it to generalize the theorem of
Greb-Kebekus-Kov\'acs-Peternell to complex spaces with rational singularities,
and to prove the existence of a functorial pull-back for reflexive
differentials on such spaces. We also use our methods to settle the "local
vanishing conjecture" proposed by Musta\c{t}\u{a}, Olano, and Popa.Comment: Final version. To appear in slightly shortened version in J. Amer.
Math. So
Deformation along subsheaves, II
Let Y be a compact reduced subspace of a complex manifold X, and let F be a
subsheaf of the tangent bundle T_X which is closed under the Lie bracket. This
paper discusses criteria to guarantee that infinitesimal deformations of the
inclusion morphism Y -> X give rise to positive-dimensional deformation
families, deforming the inclusion map "along the sheaf F". In case where X is
complex-symplectic and F is the sheaf of Hamiltonian vector fields, this
partially reproduces known results on unobstructedness of deformations of
Lagrangian submanifolds.
Written for the IMPANGA Lecture Notes series, this paper aims at simplicity
and clarity of argument. It does not strive to present the shortest proofs or
most general results available. The proof is rather elementary and geometric,
constructing higher-order liftings of a given infinitesimal deformation using
flow maps of carefully crafted time-dependent vector fields
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