Seminar

The `standard' pairs of LMMP, seen as virtual ramified covers of varieties $X$, carry naturally the usual geometric invariants of
varieties, including morphisms and birational transformations. In this extended category, the birational decomposition into parts of `pure' geometry (according to whether the (refined) Kodaira dimension is $-\infty$, 0, or dim) can be performed in a ransparent and meaningful way, using the new class of `special' pairs. Like varieties, pairs with Kodaira dimension $-\infty$ are conjectured to be uniruled (ie: covered by $\Delta$-rational curves, defined in an appropriate sense, natural in this category). Which leads to develop a geometry of such $\Delta$-rational curves, expected to be analogous, but richer, to the one
for usual rational curves on varieties. Conjectures can be also formulated for fundamental groups, universal covers, arithmetics and complex hyperbolicity in this category.