Tag - Fano varieties

Fano manifolds are complex projective manifolds having positive first Chern class. The positivity condition on the first Chern class has far reaching geometric and arithmetic implications. For instance, Fano manifolds are covered by rational curves, and families of Fano manifolds over one dimensional bases always admit holomorphic sections. In recent years, there has been great effort towards defining suitable higher analogues of the Fano condition. Higher Fano manifolds are expected to enjoy stronger versions of several of the nice properties of Fano manifolds. For instance, they should be covered by higher dimensional rational varieties, and families of higher Fano manifolds over higher dimensional bases should admit meromorphic sections (modulo Brauer obstruction). In this talk, I will discuss a possible notion of higher Fano manifolds in terms of positivity of higher Chern characters, and describe special geometric features of these manifolds.

Several years ago, Chi Li introduced the local volume of a klt singularity in his work on K-stability. The local-global analogy between klt singularities and Fano varieties, together with recent study in K-stability lead to the conjecture that klt singularities whose local volumes are bounded away from zero are bounded up to special degeneration. In this talk, I will discuss some recent work on this conjecture through the minimal log discrepancies of Kollár components.

In this talk, we will introduce the coregularity of Fano varieties.

This invariant measures how large of a dual complex can we find among log Calabi-Yau structures on a Fano variety. The coregularity relates to log canonical thresholds, existence of complements, and the index of log Calabi-Yau pairs. In this talk, we will discuss some recent results about this invariant and other future directions. The results of this talk are joint work with Fernando Figueroa, Stefano Filipazzi, Mirko Mauri, and Junyao Peng.

I will report on a joint work (in progress) with Hamid Abban, Erroxe Etxabarri-Alberdi, Dongchen Jiao, Anne-Sophie Kaloghiros, Jesus Martinez-Garcia, Elena Denisova, and Theo Papazachariou about the K-moduli of smooth Fano threefolds in deformation families 2-22, 2-24, 2-25, 3-12, 3-13, 4-13 (see this website for the description of these families). These are all 1-dimensional families of smooth Fano threefolds for which K-moduli exist. We know all smooth K-polystable Fano threefolds in these families. I will explain how to find their K-polystable singular limits.

This seminar is about the K-moduli of smooth Fano threefolds in deformation families 2-22, 2-24, 2-25, 3-12, 3-13, 4-13. These are all one-dimensional families of smooth Fano threefolds for which K-moduli exist. We know all smooth K-polystable Fano threefolds in these families. I will explain how to find their K-polystable singular limits.

Consider a positively monotone (Fano) closed symplectic manifold M and a symplectic simple crossings divisor D in it. Assume that the Poincare dual of the anti-canonical class is a positive
rational linear combination of the classes [D_i], where D_i are the components of D with their symplectic orientation. A choice of such coefficients, called the weights, (roughly speaking) equips M - D with a Liouville structure. I will start by discussing results relating the components of D with their symplectic orientation. A choice of such coefficients, called the weights, (roughly speaking) equips M - D with a Liouville structure. I will start by discussing results relating the symplectic cohomology of M - D with quantum cohomology of M. These results are particularly sharp when the weights are all at most 1 (hypothesis A). Then, I will discuss certain rigidity results (inside M) for
skeleton type subsets of M - D, which will also demonstrate the geometric meaning of hypothesis A in examples.