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János Kollár

Publications and source records attributed to János Kollár.

At least 127 records · Page 7Linked to original sources
arXiv2010

Continuous closure of sheaves

We give a purely algebraic construction of the continuous closure of any finitely generated torsion free module; a concept first studied by H.~Brenner and M.~Hochster. The construction implies that, at least in characteristic 0, taking continuous closure commutes with flat morphisms whose fibers are semi-normal. This implies that the continuous closure of a coherent ideal sheaf is again a coherent ideal sheaf (both in the Zariski and in the étale topologies) and it commutes with field extensions.

math.AC↗
arXiv2010

Lectures on curves on varieties -- Lisbon, 2009

These lectures give a short introduction to the study of curves on algebraic varieties. After an elementary proof of the dimension formula for the space of curves, we summarize the basic properties of uniruled and of rationally connected varieties. At the end, some connections with symplectic geometry are considered.

math.AG↗
arXiv2009

Simultaneous normalization and algebra husks

The aim of this note is to use the concept of algebra husks to prove an analog of the flattening decomposition theorem for simultaneous normalizations. Other applications improve earlier flatness criteria. Parts of this note were contained in the first version of "Hulls and Husks" (arXiv:0805.0576). Dec. 16: new references added.

math.AG↗
arXiv2009

Log canonical singularities are Du Bois

A recurring difficulty in the Minimal Model Program is that while log terminal singularities are quite well behaved (for instance, they are rational), log canonical singularities are much more complicated; they need not even be Cohen-Macaulay. The aim of this paper is to prove that log canonical singularities are Du Bois. The concept of Du Bois singularities, introduced by Steenbrink, is a weakening of rationality. We also prove flatness of the cohomology sheaves of the relative dualizing complex of a projective family with Du Bois fibers. This implies that each connected component of the moduli space of stable log varieties parametrizes either only Cohen-Macaulay or only non-Cohen-Macaulay objects.

math.AG↗
arXiv2009

Hulls and Husks

The aim of this note is to prove an analog of the flattening decomposition theorem for reflexive hulls. The main applications are: the construction of the moduli space of varieties of general type, improved flatness conditions and criteria for simultaneous normalizations.

math.AG↗
arXiv2008

Semi log resolution

The aim of this note is to discuss resolution theorems that are useful in the study of semi log canonical varieties.

math.AG↗
arXiv2008

Cremona transformations and diffeomorphisms of surfaces

We show that the action of Cremona transformations on the real points of quadrics exhibits the full complexity of the diffeomorphisms of the sphere, the torus, and of all non-orientable surfaces. The main result says that if X is rational, then Aut(X), the group of algebraic automorphisms, is dense in Diff(X), the group of self-diffeomorphisms of X.

math.AG↗
arXiv2008

Diophantine subsets of function fields of curves

We consider diophantine subsets of function fields of curves and show, roughly speaking, that they are either very small or very large. In particular, this implies that the ring of polynomials $k[t]$ is a not a diophantine subset of the field of rational functions $k(t)$ for many fields $k$.

math.NT↗
arXiv2008

Is there a topological Bogomolov--Miyaoka--Yau inequality?

The aim of this paper is to consider a possible extension of the Bogomolov--Miyaoka--Yau inequality to differentiable orbifolds. The conjectured extension is related to the Montgomery--Yang problem about circle actions on the 5--sphere and also to the H--cobordism of Seifert fibered 3--manifolds. Related conjectures on algebraic surfaces with quotient singularities which have the same rational homology as the projective plane are also considered. Finally we give such examples which are birational to the projective plane yet have ample canonical class.

math.AG↗
arXiv2007

Looking for rational curves on cubic hypersurfaces

These are the substantially expanded notes of the lectures of JK at the summer school "Higher-Dimensional Geometry over Finite Fields" in Göttingen, June 2007. The first part gives an overview of the methods. The main new result is the construction of rational curves passing through a given collection of points on smooth cubic hypersurfaces over finite fields.

math.AG↗
arXiv2007

Two examples of surfaces with normal crossing singularities

This note gives two examples of surfaces with normal crossing singularities. In the first example the canonical ring is not finitely generated. In the second, the canonical line bundle is not ample but its pull back to the normalization is ample. The latter answers in the negative a problem left unresolved in [EGA,III.2.6.2] and raised again by Viehweg.

math.AG↗