Recollection and reflection on minimal models.
Tag - Minimal model programme
I will survive some recent results on the study of the birational geometry of foliations over complex projective varieties, focusing, in particular, on the case of algebraically integrable foliations.
Kawamata log terminal (klt) singularities form an important class of singularities due to its fundamental roles in MMP, Kähler-Einstein geometry, and K-stability. Recently, Chi Li invented a new invariant called the local volume of a klt singularity which encodes lots of interesting geometric and topological information. In this talk, we will explore the relation between local volumes and certain boundedness condition of singularities related to the existence of ε-plt blow-ups. As a main result, we show that the set of local volumes of klt singularities is discrete away from zero (resp. satisfies ACC) if the coefficient set is finite (resp. satisfies DCC) and the ambient spaces are analytically bounded.
We explain some new vanishing theorems in a complex analytic setting. We will use them for the study of the minimal model programme for projective morphisms between complex analytic spaces.
Skew Calabi-Yau algebras are generalizations of Calabi-Yau algebras due to Reyes, Rogalski, and Zhang. Within the graded (associative and unital) algebras over a field k, they form the non-commutative analogues of the regular algebras. As a special feature, such an algebra A is equipped with its so-called Nakayama automorphism φ. The talk will present ongoing investigations on the presentations of these algebras by generators and relations taking into account their homological specificities. Such presentations are well-known for Calabi-Yau algebras (after Ginzburg, Bocklandt and van den Bergh) and also for Koszul skew Calabi-Yau algebras (after Bocklandt, Wemyss and Schedler). The general situation involves the interaction of the A∞-Yoneda algebra E(A) := ExtA(k,k) with the Nakayama automorphism φ, and also the A∞-Yoneda algebra E(A[x,φ]) of the Ore extension A[x,φ] of A by φ. More precisely, one is particularly intereseted in minimal models of these A∞-algebras. After having presented all these concepts, I will discuss the relationship between these minimal models as well as consequences in terms of presentations of A.
I hope to talk more about how to find generators for Fukaya categories using symplectic version of the minimal model programme in examples such as symplectic quotients of products of spheres and moduli spaces of parabolic bundles.
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