Related papers: The nef dimension of log minimal models

A talk presented at International Conference ICMP-2000, London, England

In a recent preprint, H. Tsuji gave a number of interesting assertions on the structure of pseudo-effective line bundles on projective manifolds. In particular, he postulated the existence of an almost-holomorphic "reduction map", whose…

We give a description of the minimal exponent of a hypersurface using higher direct images of suitably twisted sheaves of log forms on a log resolution.

Two lectures given at the UK-Japan Winter School on 'Geometry and Analysis Towards Quantum Theory', Durham, January 2004.

In this expository article we revisit the Bernstein problem for several geometric PDEs including the minimal surface, Monge-Amp\`{e}re, and special Lagrangian equations. We also discuss the minimal surface system where appropriate. The…

We discuss the log minimal model theory for log surfaces. We show that the log minimal model program, the finite generation of log canonical rings, and the log abundance theorem for log surfaces hold true under assumptions weaker than the…

Compact spatial dimension at the TeV scale remain an intriguing possibility that is currently being tested at the LHC. We give an introductory review of extra-dimensional models and ideas, from a phenomenological perspective, but…

We reduce the Abundance Conjecture in dimension 4 to the following numerical statement: if the canonical divisor K is nef and has maximal nef dimension, then K is big. From this point of view, we ``classify'' in dimension 2 nef divisors…

This is the text of the Hermann Weyl Prize lecture given by the author at the XXIV Colloquium on Group Theoretical Methods in Physics, Paris, July 2002 (to appear in the Proceedings of the Colloquium).

In this note we explore the possible marginal deformations of general (0,2) non-linear sigma-models, which arise as descriptions of the weakly-coupled (large radius) limits of four-dimensional $\mathcal{N}= 1$ compactifications of the…

The main purpose of this paper is to study formal deformations of evolution algebras, determining their existence and classifying them up to equivalence. In addition, we examine degenerations in this setting and provide Hasse diagrams that…

These are the notes for the talk "Hodge numbers of a hypothetical complex structure on $S^6$" given by the author at the MAM1 "(Non)-existence of complex structures on $S^6$" held in Marburg in March 2017. They are based on [A. Gray, A…

These are the lecture notes based on earlier papers with some additional new results. New and simple proofs are given for local freeness theorem and the semipositivity theorem. A decomposition theorem for higher direct images of dualizing…

These are the notes of some lectures given by the author for a workshop held at TIFR, Mumbai in December, 2011, giving an exposition of the Deligne-Lusztig theory.

We first introduce a weak type of Zariski decomposition in higher dimensions: an $\R$-Cartier divisor has a weak Zariski decomposition if birationally and in a numerical sense it can be written as the sum of a nef and an effective…

This note is an expository account of the theory of staggered sheaves, based on a series of lectures given by the author at RIMS (Kyoto) in October 2008.

We prove the existence of a smoothing for a toroidal crossing space under mild assumptions. By linking log structures with infinitesimal deformations, the result receives a very compact form for normal crossing spaces. The main approach is…

We discuss the relative log minimal model theory for log surfaces in the analytic setting. More precisely, we show that the minimal model program, the abundance theorem, and the finite generation of log canonical rings hold for log pairs of…

Talk presented at the XXIX Rencontres de Moriond, Electroweak Interactions and Unified Theories, March 12-19, 1994, Meribel

Some elementary considerations are presented concerning Catenoids and their stability, separable minimal hypersurfaces, minimal surfaces obtainable by rotating shapes, determinantal varieties, minimal tori in S3, the minimality in Rnk of…