Title: Realizable classes in Grassmannians
Abstract: Given a class in the cohomology of a projective manifold, one can ask whether the class can be represented by an irreducible subvariety. If the class is represented by an irreducible subvariety, we say that the class is realizable. One can further ask whether the subvariety can be taken to satisfy additional properties such as smooth, nondegenerate, rational, etc. These questions are closely related to central problems in algebraic geometry such as the Hodge Conjecture or the Hartshorne Conjecture. Recently, June Huh and collaborators have made significant progress in understanding realizable classes in products of projective spaces. In this talk, I will discuss joint work with Julius Ross on realizable classes in Grassmannians.
Title: Baily--Borel compactifications of period images and the b-semiampleness conjecture
Abstract: A fundamental tool to study algebraic varieties is given by morphisms to projective space. A line bundle is called semiample if some positive tensor multiple provides a morphism to projective space. Unlike the case of ample line bundles, there are no general numerical criteria to determine whether a line bundle is semiample. In particular, proving the semiampleness of line bundles can be a challenging task. In this talk, we will report on recent developments regarding the semiampleness of line bundles coming from Hodge theory. As a consequence, we will obtain functorial compactifications of general period images, generalizing classic work of Baily--Borel and thus confirming a conjecture of Griffiths. This talk is based on joint work with Bakker, Mauri, and Tsimerman.
TBA
Title: Higher direct images of dualizing sheaves III.
Abstract: In a series of extremely influential papers that appeared 40 years ago, Kollár showed several surprising properties of higher direct images of dualizing sheaves via projective morphisms. In this talk I will report on recent joint work with Kollár that should be considered a sequel to his original papers (hence the counter in the title). We show that for a flat morphism between varieties with rational singularities even stronger statements are true. In particular, Kollár's splitting theorem from 40 years ago holds for the higher direct images of the relative dualizing sheaf and this has several interesting consequences. For instance it implies that the higher direct images of the structure sheaf are locally free. As an application in a seemingly unrelated direction this leads to the statement that the identity component of the relative Picard scheme of a flat morphism between varieties with rational singularities is a smooth algebraic group scheme.
TBA
Title: Algebraic geometry of hyper-Kähler varieties via hyperholomorphic bundles
Abstract: Hyper-Kähler varieties are higher-dimensional analogues of K3 surfaces. Addressing many fundamental questions about hyper-Kähler varieties requires constructions of specific algebro-geometric objects. For example, resolving the Bondal–Orlov D-equivalence conjecture involves constructing suitable Fourier–Mukai kernels; approaching the period–index problem requires constructing appropriate vector bundles; attacking the Orlov conjecture for motives requires constructing certain algebraic cycles. In this talk, I will explain how Markman’s hyperholomorphic bundles, which relied heavily on transcendental techniques (metric, twistor lines, etc) of Verbistsky, can be used to produce the desired algebro-geometric constructions in all the directions mentioned above. This is based on joint work with James Hotchkiss, Davesh Maulik, Qizheng Yin, and Ruxuan Zhang.
TBA