Coherence of local and global hulls
We give necessary and sufficient conditions for the hull of a coherent sheaf to be coherent.
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arXiv subjects
János Kollár
Publications and source records attributed to János Kollár.
Variants of normality for Noetherian schemes
This note presents a uniform treatment of normality and three of its variants---topological, weak and seminormality---for Noetherian schemes. The key is to define these notions for pairs $(Z, X)$ consisting of a (not necessarily reduced) scheme $X$ and a closed, nowhere dense subscheme $Z$. An advantage of the new definitions is that, unlike the usual absolute ones, they are preserved by completions. This shortens some of the proofs and leads to more general results. Version 2: small changes.
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How much of the Hilbert function do we really need to know?
We describe several examples where the leading coefficient of a Hilbert function tells us everything we need. Based on my lectures at Oberwolfach and Stony Brook.
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Volume and Hilbert function of R-divisors
If D is a big divisor and E is an effective divisor, then vol(D-E)<=vol(D)<=vol(D+E). We discuss when each inequality is an equality. Surprisingly, the answer is that the asymptotic equality vol(D-E)=vol(D) is equivalent to the equality of the Hilbert functions of D-E and D. The analogous result holds when vol(D)=vol(D+E).
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Szemerédi--Trotter-type theorems in dimension 3
We estimate the number of incidences in a configuration of $m$ lines and $n$ points in dimension 3. The main term is $mn^{1/3}$ if we work over the real or complex numbers but $mn^{2/5}$ over finite fields. Both of these are optimal, aside from a multiplicative constant that is at most 5.
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Durfee's conjecture on the signature of smoothings of surface singularities
In 1978 Durfee conjectured various inequalities between the signature and the geometric genus of a normal surface singularity. Since then a few counter examples have been found and positive results established in some special cases. We prove a `strong' Durfee--type inequality for any smoothing of a Gorenstein singularity, provided that the intersection form the resolution is unimodular, and the conjectured `weak' inequality for all hypersurface singularities and for sufficiently large multiplicity strict complete intersections. The proofs establish general inequalities valid for any normal surface singularity.
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Maps between local Picard groups
We study the local Picard group of singularities. We study those elements in the local Picard group that become trivial either when pulled-back to the normalization or when restricted to a hyperplane section. In the second case we improve the similar results of SGA2. Version2: Many small changes, new Theorem 4 added.
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Incidences of lines in $P^3$ and the arithmetic genus of curves
Guth and Katz proved that, as conjectured by Elekes and Sharir, $m$ lines in 3-space have at most constant times $ m^{3/2}$ intersection points, aside from some obvious counter examples. We give an explicit bound for the constant, using the arithmetic genus of the union of the lines.
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The dual complex of singularities
The dual complex of a singularity is defined, up-to homotopy, using resolutions of singularities. In many cases, for instance for isolated singularities, we identify and study a "minimal" representative of the homotopy class that is well defined up-to piecewise linear homeomorphism. This is derived from a more global result concerning dual complexes of dlt pairs. As an application, we also show that the dual complex of a log terminal singularity as well as the one of a simple normal crossing degeneration of a family of rationally connected manifolds are contractible.
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The Lefschetz property for families of curves
By the Lefschetz hyperplane theorem, if X is a smooth quasi-projective variety and C a general curve section of X then the fundamental group of C surjects onto the fundamental group of X. Here we consider when this conclusion holds for a general member of an arbitrary family of curves that covers X. The most interesting case is families of rational curves.
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Esnault-Levine-Wittenberg indices
These are extended notes of a lecture about the papers 1207.1883 by Esnault-Levine-Wittenberg and 1308.3024 by Wittenberg. The aim is to define the Esnault-Levine-Wittenberg indices, establish their basic properties amd to pose several questions. v.2: references updated.
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Examples of vanishing Gromov--Witten--Welschinger invariants
We give examples of real enumerative problems without real solutions. Most of the examples concern rational curves in ${\mathbb C}{\mathbb P}^3$ passing through a real set of points and lines.
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Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds
We introduce complex singularity exponents of plurisubharmonic functions and prove a general semi-continuity result for them. This concept contains as a special case several similar concepts which have been considered e.g. by Arnold and Varchenko, mostly for the study of hypersurface singularities. The plurisubharmonic version is somehow based on a reduction to the algebraic case, but it also takes into account more quantitative informations of great interest for complex analysis and complex differential geometry. We give as an application a new derivation of criteria for the existence of Kähler-Einstein metrics on certain Fano orbifolds, following Nadel's original ideas (but with a drastic simplication in the technique, once the semi-continuity result is taken for granted). In this way, 3 new examples of rigid Kähler-Einstein Del Pezzo surfaces with quotient singularities are obtained.
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Approximating curves on real rational surfaces
We give necessary and sufficient topological conditions for a simple closed curve on a real rational surface to be approximable by smooth rational curves. We also study approximation by smooth rational curves with given complex self-intersection number.
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Continuous rational functions on real and $p$-adic varieties II
This note replaces two earlier preprints (1101.3737 by Kollár) and (1211.6681 by Nowak). It studies, and partially solves, 3 elementary questions about continuous rational functions on real (and p-adic) algebraic varieties: Can one restrict such a function to a subvariety? Can one extend such a function from a subvariety? Which linear equations can be solved using such functions? v.2: Section 3 simplified and typos corrected.
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Neighborhoods of subvarieties in homogeneous spaces
We study the holomorphic/meromorphic function theory and the fundamental group of Euclidean open neighborhoods of compact subvarieties in homogeneous spaces; building on results of Hironaka, Hartshorne, Napier and Ramachandran in the ample normal bundle case.
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Holomorphic arcs on analytic spaces
Let X be a complex analytic space. A short analytic arc is a holomorphic map of the closed unit disc to X such that only the origin is mapped to a singular point. In contrast with the space of formal arcs studied by Nash, the moduli space of short analytic arcs usually has infinitely many connected components. We describe these for surface singularities, in terms of certain conjugacy classes of the fundamental group of the link. For quotient singularities (in any dimension), this gives a concrete realization of the McKay correspondence. Our results also give new connections between a surface cusp singularity, its dual and hyperbolic Inoue surfaces. version 2: References added.
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