ALGEBRAIC GEOMETRY SEMINAR
Time: Wednesday, 3:30-4:30 pm
Location: MATH 731
September 5th: Kenji Matsuki (Purdue)
Title: A Kawamata-Viehweg type formulation of the (logarithmic) Akizuki- Nakano Vanishing Theorem.
September 12th: Kenji Matsuki (Purdue)
Title: A Kawamata-Viehweg type formulation of the (logarithmic) Akizuki- Nakano Vanishing Theorem.Abstract: Continuation of last week's talk.
September 19th: Donu Arapura (Purdue)
Title: Chow groups and cycle maps.Abstract: This talk is entirely expository. I'll review basic facts about Chow groups and cycle maps to usual/etale (co)homology.
September 25th: Takeshi Saito (Tokyo)
Title: Characteristic cycle of an étale sheaf and its functoriality (Lecture 1)Abstract: For an étale sheaf on a smooth algebraic variety over a perfect field of characteristic p > 0, the characteristic cycle is defined as a cycle supported on the singular support, which is defined by Beilinson as a closed conical subset of the cotangent bundle. The characteristic cycles are characterized by and satisfy some functorial properties. I will explain basic definitions and properties of the singular support and characteristic cycle.
Title: Characteristic cycle of an étale sheaf and its functoriality (Lecture 2)Abstract: This will be a continuation of the first lecture.
Title: Characteristic cycle of an étale sheaf and its functoriality (Lecture 3)Abstract: This will be a continuation of the second lecture.
October 3rd: No Seminar
October 10th: Ziwen Zhu (University of Utah)
Title: Higher Codimensional Alpha Invariants and Characterization of Projective Spaces.
Abstract: Recent work of Kento Fujita, Yuji Odaka and Chen Jiang shows that among K-semistable Fano manifolds, the projective space can becharacterized in terms of either the alpha invariant or the volume. In this talk, I will generalize the definition of alpha invariant to arbitrary codimension, and show that we can characterize projective spaces among all K-semistable Fano manifolds in terms of higher codimensional alpha invariants. This result also demonstrates the relation between alpha invariants and volumes in the characterization problem of projective spaces among K-semistable Fano manifolds.
Title: Intersection pairing of cycles and biextensionsAbstract: We study the intersection of two cycles on a variety in a situation similar to the Bloch-Beilinson height pairing and Arakelov theory. The main question we answer is as follows. For a fixed cycle, find conditions on it, such that whenever we pair it with two rationally equivalent cycles (with possible multiple equivalences), we get the same equivalence between the images of the pairing. This question relates to the question of a biextension, associated to a paring of cycles. In particular, we show that Bloch's biextension of homologically trivial cycles cannot be extended to a biextension of numerically trivial cycles. As part of the proof we give an explicit expression of the Suslin-Voevodsky's isomorphism.
November 7th: Donu Arapura (Purdue University)
Title: Cohomology of local systems on the moduli space of abelian varietiesAbstract: In this expository talk I want to explain the Faltings-Chai method for computing the cohomology of local systems on the above moduli spaces. The answer is expressed in terms of the cohomology of certain vector bundles over toroidal compactifications. It has the advantage over some alternative methods of giving some information on the mixed Hodge structure.
November 14th: Michael Groechenig (University of Toronto)
Title: De Rham epsilon lines and the epsilon connectionAbstract: De Rham epsilon lines for holonomic D-modules on curves were introduced by Deligne and Beilinson-Bloch-Esnault. This formalism includes a product formula, expressing the determinant of cohomology of a holonomic D-module as a tensor product of the epsilon lines computed with respect to a non-zero rational 1-form. Patel generalised the theory of de Rham epsilon factors to arbitrary dimensions. A curious feature of BBE’s 1-dimensional theory, is the epsilon connection which appears when studying the variation of the epsilon lines on the space of non-zero 1-forms. In this talk I will explain how properties of algebraic K-theory yield a conjectural candidate for the epsilon connection in arbitrary dimensions.
December 5th: David Hansen (University of Notre Dame)