Seen through the lens of the Minimal Model Program, the classification of algebraic varieties can be summarised into two main steps: firstly, the algorithm of the MMP allows to decompose a variety with mild singularities, birationally, into a tower of fibrations whose general fibres have ample, anti-ample, or numerically trivial canonical divisor. In view of this decomposition, the natural second step to take is to study these 3 classes of algebraic varieties in detail, for example, studying their moduli theory and/or any other property that could shed light on ways to understand all possible elements that belong to such classes. It turns out a very important property to understand in this process is boundedness: a collection of varieties is bounded when the elements of the given collection can be parametrised using a finite type geometric space. The property of boundedness plays a crucial role in the construction of proper moduli spaces of finite type. Moreover, if a given collection of algebraic varieties is bounded (in char 0), then the topological types of their underlying analytic spaces belong to only finitely many homeomorphism classes: hence, all of their topological invariants come in just finitely many possible different versions. While over the past 15 years, several breakthroughs have completely settled the question of boundedness (and the subsequent construction of moduli spaces) in the case of log canonical models (varieties/pairs with ample canonical divisor) and Fano varieties (those with anti-ample canonical divisor), the situation is still quite unclear in the case of trivial numerical divisor. In this seminar, I will try to explain what is known, or not, and what those challenges are that make the situation quite more complicated than the other 2 cases. I will moreover explain how we can overcome most of the issues if we assume that a K-trivial variety is endowed with a fibration structure of relative dimenesion one. The seminar includes results from various works I developed over the past 10 years with G. Di Cerbo, C. Birkar, S. Filipazzi, and C. Hacon. Moreover, I will talk about current work in progress with P. Engel, S. Filipazzi, F. Greer, M. Mauri were we show various new boundedness results for K-trvial varieties fibered in K3 surfaces or abelian varieties.
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Meeting ID: 9086116889
Passcode: 362880
Website: http://math.ipm.ac.ir/agnt/