Humboldt-Universität zu Berlin
Mathem.-Naturwissenschaftliche Fakultät
Institut für Mathematik

## Forschungsseminar "Algebraische Geometrie"

Wintersemester 2021/22

Time: Wednesday 13:15 - 14:45

Room: Online (Zoom) or see individual talks below for rooms.

TimeRoomSpeaker
25.10.2021 (Monday!), 13:00 2.006 Rudower Chaussee 25 Alessandro Verra (Universita Roma Tre)
Title: A series of hyperkähler varieties and their Pfaffian counterparts
Abstract:
03.11.2021 3.007 Rahul Pandharipande (ETH Zürich)
Title: Gromov-Witten theory of hypersurfaces
Abstract: I will explain how to think about the GW theory of hypersurfaces in projective space (and more generally complete intersections). The interesting new aspect is the control of the primitive cohomology. Full use of monodromy, degeneration, and nodal relative geometry, leads to an inductive solution. A consequence is that all GW cycles for hypersurfaces (and complete intersections) lie in the tautological ring of the moduli space of curves. Joint work with Argüz, Bousseau, and Zvonkine.
10.11.2021 3.007 Christian Schnell (Stony Brook, visiting Bonn)
Title: A new look at degenerating variations of Hodge structure
Abstract: In the early 1970s, Schmid published a detailed analysis of variations of Hodge structure on the punctured disk. All subsequent developments in Hodge theory (including Saito's theory of Hodge modules), and most applications of Hodge theory to questions about families of algebraic varieties, ultimately depend on Schmid's results. For that reason, I think one should try to understand this topic as well as possible. In the talk, I will present a new take on Schmid's work that greatly simplifies the existing proofs; works in the more general setting of complex variations of Hodge structure; and, most importantly, makes it much clearer what is going on.
17.11.2021 3.007 No Seminar
24.11.2021 3.007 Scott Mullane (HU Berlin)
Title: The birational geometry of the moduli space of pointed hyperelliptic curves
Abstract: The moduli space of pointed hyperelliptic curves is a seemingly simple object with perhaps unexpectedly interesting geometry. I will report on joint work with Ignacio Barros towards a full classification of both the Kodaira dimension and the structure of the effective cone of these moduli spaces.
01.12.2021 0.307 Schrödinger Zentrum Gavril Farkas (HU Berlin)
Title: The birational geometry of $$M_g$$ : new developments via non-abelian Brill-Noether theory and tropical geometry
Abstract: I will discuss how novel ideas from non-abelian Brill-Noether theory can be used to prove that the moduli space of genus 16 is uniruled and that the moduli space of Prym varieties of genus 13 is of general type. For the much studied question of determining the Kodaira dimension of moduli spaces, both these cases were long understood to be crucial in order to make further progress. I will also explain the use of tropical geometry in order to establish the Strong Maximal Rank Conjecture, necessary to carry out this program.
08.12.2021 0.307 SZ Daniele Agostini (MPI Leipzig)
Title: The Martens-Mumford Theorem and the Green-Lazarsfeld Secant Conjecture
Abstract: The Green-Lazarsfeld secant conjecture predicts that the syzygies of a curve of sufficiently high degree are controlled by its special secants. We prove this conjecture for all curves of Clifford index at least two and not bielliptic and for all line bundles of a certain degree. Our proof is based on a classic result of Martens and Mumford on Brill-Noether varieties and on a simple vanishing criterion that comes from the interpretation of syzygies through symmetric products of curves.
15.12.2021 0.307 SZ Ruijie Yang (HU Berlin)
Title: Higher multiplier ideals
Abstract: Multiplier ideals of $$\mathbb{Q}$$-divisors are important invariants of singularities, which have had lots of applications in algebraic geometry. In this talk, I will introduce a new series of ideals, arising from the global study of vanishing cycles for D-modules. They can be thought as higher multiplier ideals and capture more refined information. I will discuss some general properties and how they are related to the theory of Hodge ideals developed by Mustata and Popa. This is joint work in progress with Christian Schnell.
12.01.2022, 13:00 Zoom Olivier de Gaay Fortman (ENS Paris)
Title: The integral Hodge conjecture for one-cycles on Jacobians of curves
Abstract: In this talk I will report on joint work with Thorsten Beckmann. I will prove that the minimal cohomology class of a principally polarized complex abelian variety of dimension $$g$$ is algebraic if and only if all integral Hodge classes in degree $$2g-2$$ are algebraic. In particular, this proves the integral Hodge conjecture for one-cycles on the Jacobian of a smooth projective curve over the complex numbers. The idea is to lift the Fourier transform on rational Chow groups to a homomorphism between integral Chow groups. I shall explore such integral lifts of the Fourier transform for an abelian variety over any field, partially answering a question of Moonen-Polishchuk and Totaro. Another corollary is the integral Tate conjecture for one-cycles on the Jacobian of a smooth projective curve over the separable closure over a finitely generated field.
14.01.2022 (Friday), 15:30 Zoom Andres Fernandez Herrero (Cornell)
Title: Intrinsic construction of moduli spaces via affine grassmannians
Abstract: One of the classical examples of moduli spaces in algebraic geometry is the moduli of vector bundles on a smooth projective curve $$C$$. More precisely, there exists a quasiprojective variety that parametrizes stable vector bundles on $$C$$ with fixed numerical invariants. In order to further understand the geometry of this space, Mumford constructed a compactification by adding a boundary parametrizing semistable vector bundles. If the smooth curve $$C$$ is replaced by a higher dimensional projective variety $$X$$, then one can compactify the moduli problem by allowing vector bundles to degenerate to an object known as a "torsion-free sheaf". Gieseker and Maruyama constructed moduli spaces of semistable torsion-free sheaves on such a variety $$X$$. More generally, Simpson proved the existence of moduli spaces of semistable pure sheaves supported on smaller subvarieties of $$X$$. All of these constructions use geometric invariant theory (GIT).
In this talk I will explain an alternative GIT-free construction of the moduli space of semistable pure sheaves which is intrinsic to the moduli stack of coherent sheaves. Our main technical tools are the theory of Theta-stability introduced by Halpern-Leistner, and some recent techniques developed by Alper, Halpern-Leistner and Heinloth. In order to apply these results, one needs to prove some monotonicity conditions for a polynomial numerical invariant on the stack. We show monotonicity by defining a higher dimensional analogue of the affine Grassmannian for pure sheaves. If time allows, I will also explain applications of these ideas to some other moduli problems. This talk is based on joint work with Daniel Halpern-Leistner and Trevor Jones.
19.01.2022, 13:00 Zoom Luca Battistella (Uni. Heidelberg)
Title: Curve singularities, linear series, and stable maps: the case of genus two
Abstract: I will discuss the geometry of Gorenstein curve singularities, and how these can be leveraged to produce compactifications of moduli spaces of smooth curves, embedded or otherwise. I will focus mainly on the case of genus two, where results have been established by myself with F. Carocci.
19.01.2022, 14:30 Zoom Kien Huu Nguyen (KU Leuven)
Title: Exponential sums modulo $$p^m$$ for Deligne polynomials
Abstract: .pdf
26.01.2022, 13:00 Zoom Andrés Rojas (Uni Bonn)
Title: Chern degree functions and applications to abelian surfaces
Abstract: Given a smooth polarized surface, we will introduce Chern degree functions associated to any object of its derived category. These functions encode the behaviour of the object along the boundary of a certain region of Bridgeland stability conditions. We will discuss their extension to continuous real functions and the meaning of their differentiability at certain points. These functions turn out to be especially interesting for abelian surfaces, as they recover the cohomological rank functions defined by Jiang and Pareschi. In the final part we will apply this equivalence to give new results on the syzygies of abelian surfaces. This is a joint work with Martí Lahoz.
26.01.2022, 14:30 Zoom Raju Krishnamoorthy (Wuppertal)
Title: Rank 2 local systems and abelian varieties
Abstract: Motivated by work of Corlette-Simpson over the complex numbers, we conjecture that all rank 2 $$\ell$$-adic local systems with trivial determinant on a smooth variety over a finite field come from families of abelian varieties. We will survey partial results on a $$p$$-adic variant of this conjecture. Time permitting, we will provide indications of the proofs, which involve the work of Hironaka and Hartshorne on positivity, the answer to a question of Grothendieck on extending abelian schemes via their p-divisible groups, Drinfeld's first work on the Langlands correspondence for $$GL_2$$ over function fields, and the pigeonhole principle with infinitely many pigeons. This is joint with Ambrus Pál.
02.02.2022, 13:00 Zoom Yichen Qin (École Polytechnique)
Title: $$L$$-functions of Kloosterman sheaves
Abstract: Kloosterman sums are exponential sums over finite fields, appearing as traces of Frobenius on some $$\ell$$-adic local systems $$Kl_{n+1}$$ on $$G_m$$, called Kloosterman sheaves. Fresán, Sabbah, and Yu have constructed a family of motives attached to the symmetric powers of Kloosterman sheaves $$Sym^k Kl_{n+1}$$. They prove that the motivic $$L$$-functions of the motives attached $$Sym^k Kl_2$$ have meromorphic extensions to the complex plane and satisfy functional equations conjectured by Broadhurst and Roberts.
In this talk, I will present some results about the motivic $$L$$-functions of the motives attached to $$Sym^k Kl_3$$ and some related motives of dimension 2. In particular, several $$L$$-functions arise from modular forms, which can be determined by the information acquired from motives.
02.02.2022, 16:00 Zoom Konstantin Jakob (MIT)
Title: Geometric Langlands correspondence for Airy sheaves
Abstract: The Airy equation is a classic complex ordinary differential equation. An $$\ell$$-adic analogue of this ODE was defined and studied in-depth by N. Katz and collaborators. In this talk I will report on recent work with M. Kamgarpour and L. Yi on the geometric Langlands correspondence for generalizations of the Airy equation to reductive groups. I will explain the construction of a class of $$\ell$$-adic local systems that generalize the Airy equation in a suitable sense. Our approach follows the rigidity method in the geometric Langlands program first applied by J. Heinloth, B.-C. Ngô and Z. Yun in their construction of Kloosterman sheaves for reductive groups. Imposing suitable local conditions on automorphic sheaves leads to a rigid situation in which one can construct a Hecke eigensheaf on the moduli stack of G-bundles. The eigenvalue of this eigensheaf is the sought-after local system.
09.02.2022, 16:00 Zoom Qianyu Chen (Stony Brook)
Title: Limits of Hodge structures via D-modules
Abstract: It is well-known that each cohomology group of compact Kähler manifold carries a Hodge structure. If we consider a degeneration of compact Kähler manifolds over a disk then it is natural to ask how the Hodge structures of smooth fibers degenerate. When the degeneration only allows a reduced singular fiber with simple normal crossings (i.e. semistable), Steenbrink constructed the limit of Hodge structures algebraically. A consequence of the existence of the limit of Hodge structures is the local invariant cycle theorem: the cohomology classes invariant under monodromy action come from the cohomology classes of the total space. In this talk, I will try to explain a method using D-modules to construct the limit of Hodge structures even when the degeneration is not semistable.
24.02.2022 (Thursday) 13:15 J. von Neumann Haus Room 1.013 Rahul Pandharipande (ETH Zürich)
Title: Log Abel-Jacobi theory
Abstract: The cohomological study of the Abel-Jacobi map can be viewed at three level: the standard double ramification cycle, the universal double ramification cycle, and the logarithmic double ramification cycle. I will discuss the log DR cycle and its recent calculation (joint work with Holmes, Molcho, Pixton, Schmitt). The method involves the geometry of the universal Jacobian over the moduli space of curves, which I will also discuss.
25.03.2022 (Friday) 14:00 J. von Neumann Haus Room 1.114 Michele Pernice (Scuola Normale Superiore, Pisa)
Title: Results about the Chow ring of moduli of stable curves of genus three
Abstract: In this talk, we will discuss some results concerning the Chow ring of $$\overline{\mathcal{M}}_3$$, the moduli stack of stable curves of genus three. In particular, we will describe the main new idea, which consists of enlarging the moduli stack of stable curves by adding curves with worse singularities, like cusps and tacnodes. This will make the geometry of the stack uglier but in turn its Chow ring will be easier to compute. The Chow ring of $$\overline{\mathcal{M}}_3$$ can then be recovered by excising the locus of non-stable curves.
06.04.2022 (moved to 1.114) Philippe Eyssidieux (Grenoble)
Title: Reshetikhin-Turaev representations as Kähler local systems
Abstract: Joint work, partially in progress, with Louis Funar. In "Orbifold Kähler Groups related to Mapping Class groups", arXiv:2112.06726, we constructed certain orbifold compactifications of the moduli stack of stable pointed curves labelled by an integer $$p$$ such that the corresponding Reshetikhin-Turaev representation of the mapping class group descend to a representation of the orbifold fundamental group. I will explain the construction of that orbifold and why it is uniformizable. I will then report on a work in progress on the uniformization of these orbifolds. I will sketch a proof of the steiness of its universal covering $$p$$ odd large enough. An interesting new quantum topological consequence is that the image of the fundamental group of the smooth base of a non isotrivial complex algebraic family of smooth complete curves of genus greater than 2 by the Reshetikhin-Turaev representation is infinite (generalizing the Funar-Masbaum and the Koberda-Santharoubane-Funar-Lochak infiniteness theorems). If time allows, I will explain why the corresponding complex projective fundamental group satisfies the Toledo conjecture if $$p$$ is divisible enough.
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