Mathematics; rationality problems in algebraic geometry
Final Report Abstract
One of the main surprises and results in the course of this project was the discovery of new unexpected structures in and properties of semi-orthogonal decompositions of derived categories of smooth projective varieties. Informally, an algebraic variety is a geometric object de_ned by solutions to polynomial equations, and its derived category is a sort of non-commutative or quantum way to think of the variety; there are several reasons why this is useful: in the non-commutative setting there are non-classical categories as well, hence more exibility in the deformation theory; moreover, derived categories can be said to provide a type of linearization of the varieties; derived categories can frequently be chopped up into simpler pieces via semiorthogonal decompositions. The _rst discovery was that there can be pieces of semi-orthogonal decompositions which are so-called quasi-phantom or phantom categories on which all natural additive invariants, Hochschild homology, Grothendieck group etc. are zero, although they resp. the way they are coupled to the remaining pieces in the decomposition encode all the moduli (parameters) on which the algebraic varieties depend. This answered conjectures of Bondal and Kuznetsov in the negative and came as a surprise to many of the experts. One may think of this phenomenon as a failure of a Torelli type theorem, or simply as saying that additive invariants are too crude to reect whether a non-commutative space is trivial or not. The second discovery was that for semi-orthogonal decompositions an analogue of unique prime factorization fails, more precisely, maximal semiorthogonal decompositions (those for which the individual pieces are indecomposable) are not necessarily unique up to reordering of the pieces and equivalences of categories. This answered a question of Kuznetsov in the negative. It also showed that an approach to prove irrationality of very general cubic fourfolds via the construction of a categorical analogue of the intermediate Jacobian will need substantial modi_cation if it is to work at all. The main motivation and point of departure for much of this work were rationality questions for algebraic varieties, in particular for very general cubic fourfolds (the zero-sets of a cubic homogeneous polynomial in six variables). They are concerned with deciding when a solution set of a bunch of polynomial equations admits a parametrization by rational functions of a set of independent variables which is generically one-to-one. For cubic fourfolds it turned out in the course of the project that other approaches based on mimicking the construction of the intermediate Jacobian for cubic threefolds fail as well for reasons connected to problems with unique prime factorization, in particular a Hodge-theoretic approach due to Kulikov. Recently, Borisov has shown that a motivic approach by Galkin-Shinder fails for similar reasons. A new approach to this problem based on entropy-type invariants of birational automorphism groups was developed towards the end of the project and will be pursued further.
