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Vladimir Lazić

Publications and source records attributed to Vladimir Lazić.

At least 19 recordsLinked to original sources
arXiv2026

Generalised Cone Conjecture, II: Generic Sakai surfaces

In this paper we prove that the Generalised Cone Conjecture holds for generic Sakai surfaces, which completes the proof of the Generalised Cone Conjecture in dimension two. Furthermore, we show that there exists a non-generic Sakai surface for which no meaningful variant of the conjecture holds.

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arXiv2025

Nakayama-Zariski decomposition and the termination of flips

We show that for pseudoeffective projective pairs the termination of one sequence of flips implies the termination of all flips, assuming a natural conjecture on the behaviour of the Nakayama-Zariski decomposition under the operations of a Minimal Model Program.

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arXiv2025

Nonvanishing results for Kähler varieties

Nonvanishing theorems play a central role in birational geometry, since they derive geometric consequences from numerical information and constitute a crucial step towards abundance and semiampleness problems. General nonvanishing statements remain rare, especially in the Kähler setting. We present two types of nonvanishing results for compact Kähler varieties. First, on non-uniruled varieties with nonzero Euler-Poincaré characteristic, we prove nonvanishing for adjoint bundles of numerical dimension one on Kähler klt pairs, as well as nonvanishing for nef line bundles of numerical dimension one on $K$-trivial varieties. Second, on hyperkähler manifolds we study line bundles $\mathcal L$ which are nef but not big, and establish a dichotomy: either nonvanishing holds for $\mathcal L$, or any closed positive current in the cohomology class of $\mathcal L$ has maximal Lelong components with a rather restricted geometry. We obtain much stronger abundance-type results in dimension $4$.

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arXiv2024

Metrics with minimal singularities and the Abundance conjecture

The Abundance conjecture predicts that on a minimal projective klt pair $(X,Δ)$, the adjoint divisor $K_X+Δ$ is semiample. When $χ(X,\mathcal O_X)\neq0$, we give a necessary and sufficient condition for the conjecture to hold in terms of the asymptotic behaviour of multiplier ideals of currents with minimal singularities of small twists of $K_X+Δ$. Furthermore, we prove fundamental structural properties as well as regularity and weak convergence behaviour of an important class of currents with minimal singularities: the supercanonical currents. The results of the paper indicate strongly that supercanonical currents are central to the completion of the proof of the Abundance conjecture for minimal klt pairs $(X,Δ)$ with $χ(X,\mathcal O_X)\neq0$.

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arXiv2024

Rigid currents in birational geometry

A rigid current on a compact complex manifold is a closed positive current whose cohomology class contains only one closed positive current. Rigid currents occur in complex dynamics, algebraic and differential geometry. The goals of the present paper are: (a) to give a systematic treatment of rigid currents, (b) to demonstrate how they appear within the Minimal Model Program, and (c) to give many new examples of rigid currents.

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arXiv2023

Abundance for uniruled pairs which are not rationally connected

One of the central aims of the Minimal Model Program is to show that a projective log canonical pair $(X,Δ)$ with $K_X+Δ$ pseudoeffective has a good model, i.e.\ a minimal model $(Y,Δ_Y)$ such that $K_Y+Δ_Y$ is semiample. The goal of this paper is to show that this holds if $X$ is uniruled but not rationally connected, assuming the Minimal Model Program in dimension $\dim X-1$. Moreover, if $X$ is rationally connected, then we show that the existence of a good minimal model for $(X,Δ)$ follows from a nonexistence conjecture for a very specific class of rationally connected pairs of Calabi--Yau type.

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arXiv2022

Special MMP for log canonical generalised pairs

We show that minimal models of $\mathbb{Q}$-factorial NQC log canonical generalised pairs exist, assuming the existence of minimal models of smooth varieties. More generally, we prove that on a $\mathbb{Q}$-factorial NQC log canonical generalised pair $ (X,B+M) $ we can run an MMP with scaling of an ample divisor which terminates, assuming that it admits an NQC weak Zariski decomposition or that $K_X+B+M$ is not pseudoeffective. As a consequence, we establish several existence results for minimal models and Mori fibre spaces.

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arXiv2021

On Nonvanishing for uniruled log canonical pairs

We prove the Nonvanishing conjecture for uniruled projective log canonical pairs of dimension $n$, assuming the Nonvanishing conjecture for smooth projective varieties in dimension $n-1$. We also show that the existence of good minimal models for non-uniruled projective klt pairs in dimension $n$ implies the existence of good minimal models for projective log canonical pairs in dimension $n$.

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arXiv2019

On the B-Semiampleness Conjecture

The B-Semiampleness Conjecture of Prokhorov and Shokurov predicts that the moduli part in a canonical bundle formula is semiample on a birational modification. We prove that the restriction of the moduli part to any sufficiently high divisorial valuation is semiample, assuming the conjecture in lower dimensions.

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