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Vladimir Lazić

Publications and source records attributed to Vladimir Lazić.

34 records · Page 2Linked to original sources
arXiv2019

On Generalised Abundance, II

In our previous work, we introduced the Generalised Nonvanishing Conjecture, which generalises several central conjectures in algebraic geometry. In this paper, we derive some surprising nonvanishing results for pluricanonical bundles which were not predicted by the Minimal Model Program, by making progress towards the Generalised Nonvanishing Conjecture in every dimension. The main step is to establish that a somewhat stronger version of the Generalised Nonvanishing Conjecture holds almost always in the presence of metrics with generalised algebraic singularities, assuming the Minimal Model Program in lower dimensions.

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arXiv2018

On Generalised Abundance, I

The goal of this paper is to make a surprising connection between several central conjectures in algebraic geometry: the Nonvanishing Conjecture, the Abundance Conjecture, and the Semiampleness Conjecture for nef line bundles on K-trivial varieties.

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arXiv2018

Nef line bundles on Calabi-Yau threefolds, I

We prove that a nef line bundle $\mathcal L$ with $c_1(\mathcal L)^2 \ne 0$ on a Calabi-Yau threefold $X$ with Picard number $2$ and with $c_3(X) \ne 0$ is semiample, that is, some multiple of $\mathcal L$ is generated by global sections.

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arXiv2018

Maps from K-trivial varieties and connectedness problems

In this paper we study varieties covered by rational or elliptic curves. First, we show that images of Calabi-Yau or irreducible symplectic varieties under rational maps are almost always rationally connected. Second, we investigate elliptically connected and elliptically chain connected varieties, and varieties swept out by a family of elliptic curves. Among other things, we show that Calabi-Yau or hyperkähler manifolds which are covered by a family of elliptic curves contain uniruled divisors and that elliptically chain connected varieties of dimension at least two contain a rational curve, and so do K-trivial varieties with finite fundamental group which are covered by elliptic curves.

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arXiv2018

Abundance for varieties with many differential forms

We prove that the abundance conjecture holds on a variety $X$ with mild singularities if $X$ has many reflexive differential forms with coefficients in pluricanonical bundles, assuming the Minimal Model Program in lower dimensions. This implies, for instance, that under this condition, hermitian semipositive canonical divisors are almost always semiample, and that klt pairs whose underlying variety is uniruled have good models in many circumstances. When the numerical dimension of $K_X$ is $1$, our results hold unconditionally in every dimension. We also treat a related problem on the semiampleness of nef line bundles on Calabi-Yau varieties.

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arXiv2014

A note on the abundance conjecture

We prove that the abundance conjecture for non-uniruled klt pairs in dimension $n$ implies the abundance conjecture for uniruled klt pairs in dimension $n$, assuming the Minimal Model Program in lower dimensions.

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arXiv2013

On the Cone conjecture for Calabi-Yau manifolds with Picard number two

Following a recent work of Oguiso, we calculate explicitly the groups of automorphisms and birational automorphisms on a Calabi-Yau manifold with Picard number two. When the group of birational automorphisms is infinite, we prove that the Cone conjecture of Morrison and Kawamata holds.

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arXiv2012

Finite generation and geography of models

There are two main examples where a version of the Minimal Model Program can, at least conjecturally, be performed successfully: the first is the classical MMP associated to the canonical divisor, and the other is Mori Dream Spaces. In this paper we formulate a framework which generalises both of these examples. Starting from divisorial rings which are finitely generated, we determine precisely when we can run the MMP, and we show why finite generation alone is not sufficient to make the MMP work.

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arXiv2012

New outlook on the Minimal Model Program, II

We prove that the finite generation of adjoint rings proved in [Cascini and Lazić] implies all the foundational results of the Minimal Model Program: the Rationality, Cone and Contraction theorems, the existence of flips, and termination of flips with scaling in the presence of a big boundary.

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arXiv2012

New outlook on the Minimal Model Program, I

We give a new and self-contained proof of the finite generation of adjoint rings with big boundaries. As a consequence, we show that the canonical ring of a smooth projective variety is finitely generated.

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arXiv2012

The Minimal Model Program Revisited

We give a light introduction to some recent developments in Mori theory, and to our recent direct proof of the finite generation of the canonical ring.

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