Летняя математическая школа «Алгебра и геометрия»
23 - 31 июля, 2018
Научный комитет: Фёдор Богомолов (Courant Institute, НИУ ВШЭ), Михаил Вербицкий (НИУ ВШЭ, ULB), Александр Кузнецов (МИАН, НИУ ВШЭ), Константин Шрамов (МИАН, НИУ ВШЭ)
We decompose any complex quasi-projective manifold $X$, by means of a single functorial fibration $c:$$X\to C$ (its `Core map') into parts of opposite geometry: `special' (the fibres) vs `general type' (the `orbifold base'). `Special' manifolds are the ones such that $\kappa(X,L)\neq p$, for any $p>0, L\subset \Omega^p_X$ of rank $1$; they generalise rational and elliptic curves, as opposed to hyperbolic curves.
The Core map appears to be relevant in topics others than birational classification: it permits to formulate for arbitrary $X's$ the conjectures of Lang-Vojta, about hyperbolicity and arithmetics for $X's$ of general type. It also allows to formulate (and prove) the higher dimensional version of Shafarevich's `hyperbolicity conjecture' for families of canonically polarised manifolds ( strengthening Viehweg's conjecture, which deals with bases of Log-general type).
The core map is (conditionally in a weak form of abundance) canonically decomposed as $c=(j\circ r)^n$, with $j$ (resp. $r$) orbifold versions of the Iitaka (resp. weak MRC) fibration. In particular: Special manifolds are just towers of fibrations with general fibres either with $\kappa=0$, or weakly rationally connected.
Материалы по курсу: Записки по курсу.