Title of Seminar: Algebraic Geometry Preprint Seminar 2023
Title of Talk: Geometry of Canonical Extensions
Speaker: Priyankur Chaudhuri, TIFR, Mumbai
Date: September 5, 2023
Time: 16:00:00 Hours
Venue: AG-77
Abstract: This talk is based on the recent works Fano threefolds with affine canonical extensions (https://arxiv.org/abs/2211.11261), Stein complements in compact Kähler manifolds (https://arxiv.org/abs/2111.03303) by Horing and Peternell as well as earlier work Canonical complex extensions of Kähler manifolds (https://arxiv.org/abs/1807.01223)by Greb and Wong. It will be centered around the study of complements of smooth codimension 1 subvarieties in smooth projective varieties. It is well known that the complement of an ample subvariety of a projective variety is affine. Conversely, specifying that a smooth divisor has affine (or more generally, Stein) complement places strong restrictions on its normal bundle. A particularly interesting example of such complements is the so-called canonical extensions, first introduced by Greb and Wong. They are defined as follows: any hyperplane class on a smooth projective variety corresponds to a nontrivial extension E of its tangent bundle TX by the structure sheaf. Letting PE and P(TX) denote the respective projectisations, the complement of P(TX) in PE is known as the canonical extension corresponding to the given class. Varieties with affine canonical extensions (such as flag varieties) have big tangent bundles. Conversely, Horing and Peternell conjecture that any projective manifold having an affine canonical extension is rational homogeneous. We will discuss this along with a string of related conjectures and present evidence in lower dimensions.
Title of Talk: Geometry of Canonical Extensions
Speaker: Priyankur Chaudhuri, TIFR, Mumbai
Date: September 5, 2023
Time: 16:00:00 Hours
Venue: AG-77
Abstract: This talk is based on the recent works Fano threefolds with affine canonical extensions (https://arxiv.org/abs/2211.11261), Stein complements in compact Kähler manifolds (https://arxiv.org/abs/2111.03303) by Horing and Peternell as well as earlier work Canonical complex extensions of Kähler manifolds (https://arxiv.org/abs/1807.01223)by Greb and Wong. It will be centered around the study of complements of smooth codimension 1 subvarieties in smooth projective varieties. It is well known that the complement of an ample subvariety of a projective variety is affine. Conversely, specifying that a smooth divisor has affine (or more generally, Stein) complement places strong restrictions on its normal bundle. A particularly interesting example of such complements is the so-called canonical extensions, first introduced by Greb and Wong. They are defined as follows: any hyperplane class on a smooth projective variety corresponds to a nontrivial extension E of its tangent bundle TX by the structure sheaf. Letting PE and P(TX) denote the respective projectisations, the complement of P(TX) in PE is known as the canonical extension corresponding to the given class. Varieties with affine canonical extensions (such as flag varieties) have big tangent bundles. Conversely, Horing and Peternell conjecture that any projective manifold having an affine canonical extension is rational homogeneous. We will discuss this along with a string of related conjectures and present evidence in lower dimensions.