Purdue Algebraic Geometry Seminar
The Algebraic Geometry Seminar takes place on Wednesdays 2:30–3:30PM. Pre-talks (which graduate students are particularly encouraged to attend) take place Wednesdays 1:45–2:15PM. This semester the seminar will take place in a hybrid format with some talks being presented online via Zoom and some in person. The in person talks will take place in STON 215. Please see below for the most up to date information. Here is the Zoom link for the talks:
https://purdue-edu.zoom.us/j/97786675764?pwd=ZUxkQzFsck5IaFBBZnhiOVNTZys4UT09
January 10: Takumi Murayama (Purdue University)
Location: HAMP 2118 1:45–3:30PM
Title: "Projective manifolds with ample tangent bundles" by Shigefumi Mori (Classical Papers/Preprint Seminar)
Abstract: Let X be a complex projective manifold. By definition, X is can be embedded into a projective space CP^n for some n. A natural question is: When is X isomorphic to CP^n? Frankel and Hartshorne conjectured that X is isomorphic to CP^n if and only if the tangent bundle of X is ample, i.e. has everywhere positive holomorphic bisectional curvature.
In this talk, I will present Mori's proof of this conjecture. The key idea is what is now known as Mori's "bend and break" technique, which paved the way for modern birational geometry and the minimal model program. Surprisingly, Mori's technique requires positive characteristic techniques, even if one starts over the complex numbers.
The first half of the talk will be introductory and will only assume background from MA 595. In the second half of the talk, I will explain Mori's bend and break technique and use it to sketch Mori's characterization of projective space. This is the first talk in the "Classical Papers/Preprint Seminar" series of the Algebraic Geometry Seminar.
January 31: François Greer (Michigan State University)
Location: STON 215 2:30–3:30PM
Title: Compactifying special cycles
Abstract: A classical theorem of Borcherds states that the cohomology classes of Hodge loci in a moduli space of K3-type Hodge structures form the coefficients of a modular form. We investigate how well this theorem survives upon passing to a smooth compactification. As an application, we sketch a counterexample to the Severi Problem for rational surfaces.
February 7: Morgan Opie (UCLA)
Note: This will be a special joint Algebraic Geometry/Topology Seminar.
Location: STON 215 2:30–3:30PM
Title: Enumerating stably trivial topological vector bundles with higher real K-theories
Abstract: The zeroeth complex topological K-theory of a space encodes complex vector bundles up to stabilization. Since complex topological K-theory is highly computable, this is a great place to start when asking questions about topological vector bundles. But, in general, there are many non-equivalent vector bundles with the same K-theory class. Bridging the gap between K-theory and actual bundle theory is challenging, even for the simplest CW complexes.
Building on work of Hu, we use Weiss-theoretic techniques in tandem with a little chromatic homotopy theory to translate vector bundle enumeration questions to tractable stable homotopy theory computations. Our main result is to compute lower bounds for the number of stably trivial rank complex rank r topological vector bundles on complex projective n-space, for infinitely many n and r. The talk will include a gentle discussion of the tools involved. This is joint work with Hood Chatham and Yang Hu.
February 14: Xiaojiang Cheng (Washington University in St. Louis)
Location: STON 215 2:30–3:30PM
Title: Hodge Classes in the Cohomology of Local Systems
Abstract: Let S be an arithmetic quotient of a Hermitian symmetric domain and X/S be a family of varieties over S. One interesting problem is to find the Hodge classes of X, and if possible, to prove the Hodge conjecture for X. Using techniques from automorphic forms, we studied the Hodge conjecture for certain families of varieties over arithmetic quotients of balls and the Siegel domain of degree two. As a byproduct, we derived formulas for Hodge numbers in terms of automorphic forms.
February 21: David Stapleton (University of Michigan)
Location: STON 215 2:30–3:30PM
Title: Complexes of stable birational invariants
Abstract: Degenerating algebraic varieties has been an important tool to study birational geometry in the past 10 years. There are many ways to understand the geometric fiber of a degeneration using the special fiber: e.g. (1) the dual complex, (2) the decomposition of the diagonal, and (3) the motivic volume. In this talk we introduce a chain complex that we attach to such a degeneration that is (A) functorial, and (B) a stable birational invariant of the geometric fiber. This invariant lives somewhere between (1), (2), and (3). As an application, we show that A1-connectedness specializes in smooth projective families. This is joint work with James Hotchkiss.
February 28: Greg Pearlstein (Universita di Pisa)
Location: Zoom (11:30-12:30 AM)
Title: Infinitesimal Torelli and rigidity results for a remarkable class of elliptic surfaces
Abstract: I will discuss joint work with Chris Peters which extends rigidity results of Arakalov, Faltings and Peters to period maps arising from families of complex algebraic varieties which are non-necessarily proper or smooth. Inspired by recent work with P. Gallardo, L. Schaffler, Z. Zhang, I will discuss two classes of elliptic surfaces which can be presented as hypersurfaces in weighted projective spaces which have a unique canonical curve. In each case, we will show that infinitesimal Torelli fails for H^2 of the compact surface, but is restored when one considers the period map for the complement of the canonical curve.
March 6: Salim Tayou (Harvard University)
Location: STON 215 2:30–3:30PM
Title: The non-abelian Hodge locus
Abstract: Classical finiteness results of Arakelov and Parshin state that a fixed quasi-projective curve can only carry finitely many non-isotrivial families of smooth projective curves of fixed genus g. These results have been generalized by Faltings and Deligne for polarized variations of Hodge structure of arbitrary weight. In this talk, I will explain a further generalization which only depends on the topology of the base and not the algebraic structure, giving thus a partial answer to a question asked by Deligne. I will then explain an application proving the algebraicity of the non-abelian Hodge locus, partially solving a conjecture of Simpson. The results in this talk are joint work with Philip Engel.
March 20: Laure Flapan (Michigan State University)
Location: STON 215 2:30–3:30PM
Title: Modular Forms and Divisors on Orthogonal Modular Varieties
Abstract: We describe how the vanishing of modular forms along certain divisors can be used to prove general-type results for orthogonal modular varieties, in particular new general-type results for moduli spaces of polarized Hyperkähler manifolds of $K3^{[n]}$ type. These techniques also yield an approach to providing a negative answer to the question of whether the effective cone of such an orthogonal modular variety, particularly in the case of moduli spaces of quasi-polarized K3 surfaces, is generated by irreducible components of Noether-Lefschetz divisors. This is joint work with Ignacio Barros, Pietro Beri, and Emma Brakkee.
March 27: Tong Zhou (UC Berkeley)
Location: STON 215 2:30–3:30PM
Title: Some Results on Microlocal Aspects of ℓ-adic Sheaves
Abstract: The analogy between D-modules and ℓ-adic sheaves suggests that there could be a microlocal ℓ-adic sheaf theory. In this talk, I will expound on this analogy, and discuss the ℓ-adic analogues of two classical results: 1) the stability of vanishing cycles, which classically states that vanishing cycles form a local system with respect to the variation of transverse test functions; 2) the invariance of characteristic cycles under the Fourier transform for monodromic sheaves.
April 3: Carlos Alberto Agrinsoni and Ruipeng Zou (Purdue University)
Location: STON 215 1:30–3:30PM
Title: "Théorème de Lefschetz et critères de dégénérescence de suites spectrales" by Pierre Deligne (Classical Papers/Preprint Seminar)
Abstract: In this presentation we will review part of Pierre Deligne’s article “Théorème de Lefschetz et critères de dégénérescence de suites spectrales”. In the first part of this talk, we will introduce the notion of cohomological functors, derived categories, and spectral sequences. In the second part of this talk, we will give equivalent conditions for a spectral sequence to degenerate. Finally, we will apply Deligne's results on sheaf cohomology.
April 10: Mert Akdenizli (Purdue University)
Location: STON 215 2:30–3:30PM
Title: A counterexample to the Hodge conjecture for Kähler manifolds. (Based on the paper "The Hodge conjecture for cubic fourfolds" by Steven Zucker) (Classical Papers/Preprint Seminar)
Abstract: The aim of this talk is to present a counterexample to the Hodge conjecture for Kähler manifolds, as given in Zucker's paper. Firstly, I will talk about Kähler manifolds, Hodge decomposition, and then state the conjecture in its true form. Then I will show a counterexample for the conjecture in the Kähler setting.
April 17: Nicolas Diaz-Wahl (Purdue University)
Location: STON 215 1:30–3:30PM
Title: Tate's isogeny theorem and Tate's conjecture for abelian varieties over finite fields.
Abstract: The goal of the talk will be to prove Tate's conjecture for abelian varieties over finite fields. I will reduce this claim to Tate's isogeny theorem and prove Tate's isogeny theorem using deep finiteness theorems in arithmetic geometry. The first half will introduce the results, and the second half will prove them.
April 24: Deepam Patel (Purdue University)
Location: STON 215 2:30–3:30PM
Title: Local monodromy of constructible sheaves
Abstract: Let X be a complex algebraic variety, and X → D a proper morphism to a small disk which is smooth away from the origin. In this setting, the higher direct images of the constant sheaf form a local system on the punctured disk, and the Local Monodromy Theorem (due to Brieskorn-Grothendieck-Griffiths-Landsman) asserts that the eigenvalues of local monodromy are roots of unity. In this talk, we will discuss generalizations of this result to the setting of arbitrary morphisms between complex algebraic varieties, and with coefficients in arbitrary constructible sheaves. If there is time, I'll discuss applications to variation of monodromy in abelian covers, applications to the monodromy of alexander modules, and analogs in char. p.
This is based on joint work with Madhav Nori.
May 1: June Weiland (Purdue University)
Location: (Postponed to the Fall semester)
Title:
Abstract: