Alessandro
Verra
Università Roma Tre
Forschungsseminar Algebraische Geometrie 📅
Institute
Head
Gavril Farkas
and
Bruno Klingler
Usual time
Wednesday from 13:15 to 14:45
Usual venue
Room: 3.007 John von Neumann-Haus
Number of talks
82
Comment
Currently past talks going back to 2010 are not included.
November 2024
October 2024
Jean Baptiste
Teyssier
Paris
Pedro
Montero
Universidad Técnica Valparaíso
Additive structures on quintic del Pezzo varieties
Abstract.
A classical problem of F. Hirzebruch concerns the classification of compactifications of affine space into smooth projective varieties with Picard rank one. It turns out that any such compactification must be a Fano manifold, i.e., it has an ample anti-canonical divisor. After reviewing some known results, I will focus on the specific case of equivariant compactifications of affine space (i.e., of the "vector group" \(\mathcal{G}_a^n\)), particularly in the case of del Pezzo varieties.<br>We will recall that del Pezzo varieties are a natural higher-dimensional generalization of classical del Pezzo surfaces. Over the field of complex numbers, these varieties were extensively studied by T. Fujita in the 1980s, who classified them by their degree.<br>I will present a result on the existence and uniqueness of "additive structures" on del Pezzo quintic varieties. Specifically, we determine when and how many distinct ways they can be obtained as equivariant compactifications of the commutative unipotent group. As an application, we obtain results on the k-forms of quintic del Pezzo varieties over an arbitrary field k of characteristic zero, as well as for singular quintic varieties. This is a joint work with Adrien Dubouloz and Takashi Kishimoto.
Irene
Spelta
HU Berlin
Decomposable G-curves and special subvarieties of the Torelli locus
Abstract.
As it is well known, the Torelli morphism \(j:\mathcal{M}_g\to\mathcal{A}_g\) sends an algebraic curve \(C\) to its Jacobian variety \(JC\). The (closure of the) image inside \(\mathcal{A}_g\) is the so-called Torelli locus. In this talk, we will discuss on the extrinsic geometry of this locus. In particular, we will consider certain special subvarieties coming from families of decomposable G-curves.
June 2024
Carl
Lian
Tufts University
Complete quasimaps to \(Bl_p(\mathbb{P}^2)\)
Abstract.
We consider the problem of counting curves \(C\) of fixed moduli in a target variety \(X\) passing through the maximal number of points ("Tevelev degrees"). A broad program for obtaining such a count is:<br>(i) Establish a Brill-Noether theorem for maps to \(X\).<br>(ii) Use (i) to construct a compact moduli space \(M\), generically parametrizing maps to \(X\), which witnesses (without excess intersections) the desired count.<br>(iii) Compute integrals on \(M\), e.g., by degeneration methods.<br>Typically, the moduli spaces coming from, e.g., Gromov-Witten theory, are not sufficient in step (ii). We review the case \(X=\mathbb{P}^r\), where this program has been carried out. Here, one may take \(M\) to be the moduli space of complete collineations (relative to the space of linear series on \(C\)), which is an iterated blow-up of a Quot scheme. We then report on work in progress with Alessio Cela on the case where \(X\) is a blow-up of \(\mathbb{P}^2\) at a point. Here, a Brill-Noether statement is given (more generally, for the blow-up of \(\mathbb{P}^r\) at any linear space) by a result of Farkas, and we construct a moduli space \(M\) of "complete quasimaps" to \(X\). Degenerations on this space are made possible by ideas from the previous calculation on \(\mathbb{P}^r\) and from Coskun's geometric Littlewood-Richardson rule. Our construction seems to hint toward a more general theory of complete quasimaps to other targets.
Ruijie
Yang
HU Berlin
Minimal exponent of a hypersurface
Abstract.
Recently, the minimal exponent of a hypersurface over complex numbers has been understood as a useful refined invariant of the log canonical threshold. It has found many new applications including deformation of Calabi-Yau 3-folds (Friedman-Laza), higher rational/du Bois singularities (Mustata-Popa) and geometric Schottky problem (Schnell-Yang). However, some basic properties of this invariant remain mysterious. In this talk I will discuss the conjecture of Mustata and Popa on birational characterization of the minimal exponent, which is the main obstruction for the computation in practice. I will explain the heuristic of the Mustata-Popa conjecture from Igusa's work on counting integer solutions of congruence equations and Igusa’s strong monodromy conjecture. Then I will discuss how several ideas from mixed Hodge modules and geometric representation theory can lead to a better understanding of the minimal exponents. This is based on two joint works with Christian Schnell and Dougal Davis, respectively.
Richard
Rimanyi
UNC Chapel Hill and Newton Institute Cambridge
3d mirror symmetry for characteristic classes
Abstract.
In this joint work with Tommaso Botta we study the elliptic characteristic classes called stable envelopes introduced by M. Aganagic and A. Okounkov. Stable envelopes measure singularities, they geometrize quantum group representations, and they can be interpreted as monodromy matrices of certain differential or difference equations. We prove that for a rich class of holomorphic symplectic varieties (called bow varieties) their elliptic stable envelopes display a duality inspired by mirror symmetry in d=3, N=4 quantum field theories. In the key step of our proof, we "resolve" large charge branes to a number of smaller charge branes. This phenomenon turns out to be the geometric counterpart of the algebraic fusion procedure. Along the way we discover more about the rich geometry of bow varieties, such as their Bruhat order and the elliptic Hall algebra structure on their stable envelopes.
May 2024
Marian
Aprodu
University of Bucharest
Resonance, syzygies, and rank-3 Ulrich bundles on the del Pezzo threefold \(V_5\)
Abstract.
This is a joint work with Yeongrak Kim. We investigate a geometric criterion for a smooth curve of genus 14 and degree 18 to be described as the zero locus of a section in an Ulrich bundle of rank 3 on a del Pezzo threefold \(V_5\). The main challenge is to read off the Pfaffian quadrics defining \(V_5\) from geometric properties of the curve. We find that this problem is related to the existence of a special rank-two vector bundle on the curve, with trivial resonance. From an explicit calculation of the Betti table, we also deduce the uniqueness of the del Pezzo threefold.
Alexander
Suciu
Northeastern University
Alexandru
Suciu
Northeastern University
The Milnor fibrations of hyperplane arrangements
Abstract.
To each multi-arrangement \((A,m)\), there is an associated Milnor fibration of the complement \(M=M(A)\). Although the Betti numbers of the Milnor fiber \(F=F(A,m)\) can be expressed in terms of the jump loci for rank 1 local systems on \(M\), explicit formulas are still lacking in full generality, even for \(b_1(F)\). After introducing these notions and explaining some of the known results, I will consider the "generic" case, in which \(b_1(F)\) is as small as possible. I will describe ways to extract information on the cohomology jump loci, the lower central series quotients, and the Chen ranks of the fundamental group of the Milnor fiber in this situation.
Nazim
Khelifa
IHES
Density criteria for typical Hodge loci and applications
Abstract.
After recalling the Zilber-Pink paradigm introduced in Hodge theory by Klingler and further developed by Baldi-Klingler-Ullmo, I will present joint work with David Urbanik giving sufficient conditions that ensure that the Hodge locus, i.e. the locus in the base of an integral polarized variation of Hodge structures where the fibers acquire non-generic Hodge tensors, is dense for the complex analytic topology in the base. I will then explain how to relate this result to classical results on Noether-Lefschetz loci. Finally, I will explain how the current knowledge of the Hodge locus can be used to revisit and improve classical bounds on the dimension of the image of period maps, studied among others by Carlson, Griffiths, Kasparian, Mayer and Toledo.
April 2024
Gregory
Pearlstein
Pisa
KSBA stable limits associated to quasi-homogeneous surface singularities
Abstract.
Smooth minimal surfaces of general type with \(K^2=1\), \(p_g=2\), and \(q=0\) constitute a fundamental example in the geography of algebraic surfaces. The moduli space of their canonical models admits a modular compactification \(M\) via the minimal model program. In previous work with Patricio Gallardo and Luca Schaffler we constructed eight new irreducible boundary divisors in \(M\) arising from unimodal singularities. In this talk, we will discuss extension of this work to quasi-homogeneous surface singularities.
February 2024
Diane
Maclagan
Warwick
Tropical vector bundles
Raluca
Vlad
Leipzig
Wonderful polytopes
Bernd
Sturmfels
Leipzig
Kinematics on \(\mathcal{M}_{0,n}\)
Annette
Werner
Frankfurt
Non-abelian p-adic Hodge theory
Rahul
Pandharipande
ETH Zürich
Tautological projections and the cohomology of the moduli space of abelian varieties
Abstract.
I will construct the projection operator on the Chow ring for the moduli of abelian varieties and compute many new examples (related to the geometry of the Lagrangian Grassmannian) elucidating the structure of this ring.
Andrei
Bud
Goethe Universität Frankfurt
Degenerations of Prym-Brill-Noether loci
Abstract.
I will describe the Prym-Brill-Noether loci for curves in the boundary of the moduli of Prym curves. As consequences of this, I prove the irreducibility of the Universal Prym-Brill-Noether locus and compute the class of the Prym-Brill-Noether divisor.
January 2024
Gian Pietro
Pirola
Università di Pavia
Second fundamental form on \(\mathcal{M}_g\) associated to the period map and its asymptotic lines
Abstract.
The aim is to study the second fundamental form associated with the image period map for curves. We present some computational improvements that allow classifying the asymptotic low-rank complex line with respect to the infinitesimal variation of the Hodge structure map and its relation to the Clifford index. This is a joint work with Elisabetta Colombo and Paola Frediani.
Andreas
Kretschmer
Magdeburg
Characteristic Numbers for Cubic Hypersurfaces
Abstract.
Given an \(N\)-dimensional family \(F\) of subvarieties of some projective space, the number of members of \(F\) tangent to \(N\) general linear spaces is called a characteristic number for \(F\). More general contact problems can often be reduced to the computation of these characteristic numbers. However, determining the characteristic numbers themselves can be very difficult, and very little is known even for the family of smooth degree \(d\) hypersurfaces of projective \(n\)-space as soon as both \(n\) and \(d\) are greater than 2. In this talk I will survey the construction of a so-called 1-complete variety of cubic hypersurfaces, generalizing Aluffi's variety of complete plane cubic curves to higher dimensions. This allows us to explicitly compute, for instance, the numbers of smooth cubic surfaces, threefolds and fourfolds tangent to 19, 34 and 55 lines, respectively.
Andreas
Kreschmer
Magdeburg
Characteristic Numbers for Cubic Hypersurfaces
Abstract.
Given an \(N\)-dimensional family \(F\) of subvarieties of some projective space, the number of members of \(F\) tangent to \(N\) general linear spaces is called a characteristic number for \(F\). More general contact problems can often be reduced to the computation of these characteristic numbers. However, determining the characteristic numbers themselves can be very difficult, and very little is known even for the family of smooth degree \(d\) hypersurfaces of projective \(n\)-space as soon as both \(n\) and \(d\) are greater than 2. In this talk I will survey the construction of a so-called 1-complete variety of cubic hypersurfaces, generalizing Aluffi's variety of complete plane cubic curves to higher dimensions. This allows us to explicitly compute, for instance, the numbers of smooth cubic surfaces, threefolds and fourfolds tangent to 19, 34 and 55 lines, respectively.
Thomas
Walpuski
HU Berlin
The Gopakumar–Vafa finiteness conjecture
Abstract.
The purpose of this talk is to illustrate an application of the powerful machinery of geometric measure theory to a conjecture in Gromov–Witten theory arising from physics. Very roughly speaking, the Gromov–Witten invariants of a symplectic manifold \((X,\omega)\) equipped with a tamed almost complex structure \(J\) are obtained by counting pseudo-holomorphic maps from mildly singular Riemann surfaces into \((X,J)\). It turns out that Gromov–Witten invariants are quite complicated (or “have a rich internal structure”). This is true especially for if \((X,\omega)\) is a symplectic Calabi–Yau 3–fold (that is: \(\mathrm{dim}X=6\), \(c_1(X,\omega) = 0\)). In 1998, using arguments from M–theory, Gopakumar and Vafa argued that there are integer BPS invariants of symplectic Calabi–Yau 3–folds. Unfortunately, they did not give a direct mathematical definition of their BPS invariants, but they predicted that they are related to the Gromov–Witten invariants by a transformation of the generating series. The Gopakumar–Vafa conjecture asserts that if one defines the BPS invariants indirectly through this procedure, then they satisfy an integrality and a (genus) finiteness condition. The integrality conjecture has been resolved by Ionel and Parker. A key innovation of their proof is the introduction of the cluster formalism: an ingenious device to side-step questions regarding multiple covers and super-rigidity. Their argument could not resolve the finiteness conjecture, however. The reason for this is that it relies on Gromov’s compactness theorem for pseudo-holomorphic maps which requires an a priori genus bound. It turns out, however, that Gromov’s compactness theorem can (and should!) be replaced with the work of Federer–Flemming, Allard, and De Lellis–Spadaro–Spolaor. This upgrade of Ionel and Parker’s cluster formalism proves both the integrality and finiteness conjecture. This talk is based on joint work with Eleny Ionel and Aleksander Doan.
December 2023
Hélène
Esnault
FU Berlin/Harvard/Copenhagen
Survey on some arithmetic properties of rigid local systems
Abstract.
A central conjecture of Simpson predicts that complex rigid local systems on a smooth complex variety come from geometry. In the last couple of years, we proved some arithmetic consequences of it: integrality (using the arithmetic Langlands program), F-isocrystal properties, crystallinity of the underlying p-adic representation (using the Cartier operator over the Witt vectors and the Higgs-de Rham flow) (for Shimura varieties of real rank at least 2, this is the corner piece of Pila-Shankar-Tsimerman's proof of the André-Oort conjecture), weak integrality of the character variety (using de Jong's conjecture proved with the geometric Langlands program) (yielding a new obstruction for a finitely presented group to be the topological fundamental group of a smooth complex variety). We'll survey some aspects of this (please ask if there is something on which you would like me to focus on). The talk is based mostly on joint work with Michael Groechenig, also, even if less, with Johan de Jong.
Simon
Felten
Columbia University
Global logarithmic deformation theory
Abstract.
A well-known problem in algebraic geometry is to construct smooth projective Calabi-Yau varieties \(X\). In the smoothing approach, one constructs first a degenerate (reducible) Calabi-Yau variety \(V\) by gluing pieces. Then one aims to find a family with special fiber \(V\) and smooth general fiber \(X\). In this talk, we see how infinitesimal logarithmic deformation theory solves the second step of this approach: the construction of a family out of a degenerate fiber \(V\). This is achieved via the logarithmic Bogomolov-Tian-Todorov theorem as well as its variant for pairs of a log Calabi-Yau space \(f_0: X_0 \to S_0\) and a line bundle \(\mathcal{L}_0\) on \(X_0\). Both theorems are a consequence of the abstract unobstructedness theorem for curved Batalin-Vilkovisky algebras.
Gergely
Berczi
Aarhus University
Geometry of the Hilbert scheme of points on manifolds, part II
Abstract.
While the Hilbert scheme of points on surfaces is well-explored and extensively studied, the Hilbert scheme of points on higher-dimensional manifolds proves to be exceptionally intricate, exhibiting wild and pathological characteristics. Our understanding of their topology and geometry is very limited. In these talks I will present recent results on various aspects of their geometry. I will discuss i) the distribution of torus fixed points within the components of the Hilbert scheme (work with Jonas Svendsen) ii) intersection theory of Hilbert schemes: a new formula for tautological integrals over geometric components, and applications in enumerative geometry iii) new link invariants of plane curve (and hypersurface) singularities coming from p-adic integration and Igusa zeta functions (work with Ilaria Rossinelli). This talk will be relatively independent from part I on 12th December at the Arithmetic Geometry Seminar
Igor
Burban
Universität Padelborn
Algebraic geometry of the torus model of the fractional quantum
Hall effect
Abstract.
The experimental discovery of the quantum Hall effect is widely considered to be a one of the major events in the condensed matter physics in the second half of the twentieth century. Both experimental and theoretical aspects of this phenomenon still continue to attract an enormous attention. In 1993 Keski-Vakkuri and Wen introduced a model for the quantum Hall effect based on multilayer two-dimensional electron systems satisfying quasi-periodic boundary conditions. Such a model is specified by a choice of a complex torus \(E\) and a symmetric positively definite matrix \(K\) of size \(g\) with integer coefficients. The space of the corresponding wave functions turns out to be \(d\)-dimensional, where \(d\) is the determinant of \(K\). I am going to explain a construction of a hermitian holomorphic bundle of rank \(d\) on the abelian variety \(A\) (which is the \(g\)-fold product of the torus \(E\) with itself), whose fibres can be identified with the space of wave function of Keski-Vakkuri and Wen. A rigorous construction of this "magnetic bundle" involves the technique of Fourier-Mukai transforms on abelian varieties. This bundle turns out to be simple and semi-homogeneous. Moreover, for special classes of the matrix \(K\), the canonical Chern-Weil connection of the magnetic bundle is shown to be projectively flat. This talk is based on a joint work with Semyon Klevtsov (arXiv:2309.04866).
November 2023
Giancarlo
Urzúa
Pontificia Universidad Católica de Chile
The birational geometry of Markov numbers
Abstract.
The projective plane is rigid. However, it may degenerate to surfaces with quotient singularities. After the work of Bădescu and Manetti, Hacking and Prokhorov 2010 classified these degenerations completely. They are \(\mathbb{Q}\)-Gorenstein partial smoothings of \(\mathbb{P}(a^2,b^2,c^2)\), where \(a,b,c\) satisfy the Markov equation \(x^2+y^2+z^2=3xyz\). Let us call them Markovian planes. They are part of a bigger picture of degenerations with Wahl singularities, where there is an explicit MMP whose final results are either \(K\) nef, smooth deformations of ruled surfaces, or Markovian planes. Although it is a final result of MMP, we can nevertheless run MMP on small modifications of Markovian planes to obtain new numerical/combinatorial data for Markov numbers via birational geometry. New connections with Markov conjecture (i.e. Frobenius Uniqueness Conjecture) are byproducts. This is joint work with Juan Pablo Zúñiga (Ph.D. student at UC Chile), the pre-print can be found here.
Frank-Olaf
Schreyer
Universität des Saarlandes
Tate resolutions of Gorenstein Rings and a construction from Clifford modules of complete intersection of two quadrics
Abstract.
The concept of MCM approximations of Auslander-Buchweitz is a beautiful concept which builds on Tate resolution. I will decribe the complexes explicitly in case for the case of the coordinate ring a complete intersection \((x_1,\ldots,x_n)\) as a module over a coordinate ring of a further complete intersection \((q_1,\ldots,q_r)\). In the second part I will explain how one can directly construct Tate resolution from a module over the Clifford algebra of a complete intersection of two quadrics \(X \subset \mathbb{P}^{2g+1}\) and their relation to Ulrich bundles on \(X\).
Alessandro
Verra
Università Roma Tre
From Enriques surfaces to the Artin-Mumford counterexample
Abstract.
The talk deals with the multiple relations between Enriques surfaces and rationality problems. Artin-Mumford's counterexample to Lueroth's problem is revisited: the role of Enriques surfaces, the family of Reye congruences is emphasized and the 2-torsion cohomology of the threefold is geometrically reconstructed from that of these surfaces. The same construction extends to higher dimensions.
Kristian
Ranestad
University of Oslo
Quaternary quartic forms and Gorenstein rings
Abstract.
The Betti tables of their apolar rings give rise to a stratification of the space of quartic forms. The strata may be characterized by possible power sum decompositions and liftings to Calabi-Yau 3-folds. I shall report on work with G. And M. Kapustka, H. Schenk, M. Stillman and B. Yuan.
October 2023
Rahul
Pandharipande
ETH Zürich
The Gromov-Witten/Donaldson-Thomas correspondence, Hilbert schemes of the affine plane and the moduli of abelian varieties
Abstract.
I will explain how these three directions of study are fundamentally linked.
July 2023
Rahul
Pandharipande
ETH Zürich
Beyond the tautological ring of the moduli of curves
Yuri
Tschinkel
New York University
Birational types
Abstract.
I will discuss joint work with Chambert-Loir, Kontsevich, and Kresch on new invariants in birational geometry.
June 2023
Eric
Larson
Brown University
Interpolation for Brill-Noether Curves
Abstract.
In this talk, we determine when there is a Brill-Noether curve of given degree and given genus that passes through a given number of general points in any projective space.
Michael
McBreen
The Chinese University of Hong Kong
Introduction to microlocal sheaves (Lecture 2 of the minicourse)
Abstract.
Given a manifold \(M\) and an open subset \(U\) of the cotangent bundle \(T^*M\), we define the category \(\mu Sh(U)\) of microlocal sheaves on \(U\), as well as the category \(\mu Sh_Z(U)\) of microlocal sheaves supported on any given subset \(Z\) of \(U\). We sketch the basic features of this category, and describe it as explicitly as possible using the constructions from Lecture 1.
Phil
Engel
University of Georgia
The non-abelian Hodge locus
Abstract.
Given a family of smooth projective varieties, one can consider the relative de Rham space, of flat vector bundles of rank \(n\) on the fibers. The flat vector bundles which underlie a polarized \(\mathbb{Z}\)-variation of Hodge structure form the "non-abelian Hodge locus". Simpson proved that this locus is analytic, and he conjectured it is algebraic. This would imply a conjecture of Deligne that only finitely many representations of the fundamental group of a fiber appear. I will discuss a proof of Deligne's and Simpson's conjectures, under the additional hypothesis that the \(\mathbb{Z}\)-zariski closure of monodromy is a cocompact arithmetic group. This is joint work with Salim Tayou.
Cécile
Gachet
HU Berlin
Smooth projective surfaces with infinitely many real forms
Abstract.
A common undergraduate exercise is to classify quadratic forms over the real and complex numbers. Its conclusion could be that the two non-isomorphic real conics \(x^2 + y^2 + z^2 = 0\) and \(x^2 + y^2 - z^2 = 0\) are isomorphic as complex curves. In fact, the corresponding complex curve is the rational line, and it admits only the afore-mentioned two non-isomorphic real forms. Although it is quite common to find complex projective varieties admitting several real forms, the first example of a variety with infinitely many non-isomorphic real forms can be found in a 2018 paper by Lesieutre. More examples of varieties with infinitely many real forms have been found later, for instance as rational surfaces and as surfaces birational to K3 surfaces, see the 2022 paper by Dinh, Oguiso, and Yu. This talk, reporting on joint work with Tien-Cuong Dinh, Hsueh-Yung Lin, Keiji Oguiso, Long Wang, and Xun Yu, completes the picture sketched by theses examples in the case of smooth projective surfaces. It features the following two results: First, if a smooth projective surface admits infinitely many real forms, then it is rational, or birational to a K3 surface and non-minimal, or birational to an Enriques surface and non-minimal. Second, there are surfaces obtained by blowing-up one point in an Enriques surface, which admit infinitely many non-isomorphic real forms. In this talk, I will explain the key ideas involved in the proofs of the two results, and try to give an idea of the construction used for the second one. Interestingly, we will encounter a fair share of group theory, group actions, and dynamics.
May 2023
Nicola
Tarasca
Virginia Commonwealth University
Coinvariants of vertex algebras and abelian varieties
Abstract.
Spaces of coinvariants have classically been constructed by assigning representations of affine Lie algebras, and more generally, vertex operator algebras, to pointed algebraic curves. Removing curves out of the picture, I will construct spaces of coinvariants at abelian varieties with respect to the action of an infinite-dimensional Lie algebra. I will show how these spaces globalize to twisted D-modules on moduli of abelian varieties, extending the classical picture from moduli of curves. This is based on my recent preprint arXiv:2301.13227.
Marian
Aprodu
University of Bucharest
Resonance and vector bundles
Abstract.
Resonance varieties are algebro-geometric objects that emerged from the geometric group theory. They are naturally associated to vector subspaces in second exterior powers and carry natural scheme structures that can be non-reduced. In algebraic geometry, they made an unexpected appearance in connection with syzygies of canonical curves. In this talk, based on a joint work with G. Farkas, C. Raicu and A. Suciu, I report on some recent results concerning the geometry of resonance schemes in the vector bundle case.
April 2023
Samouil
Molcho
ETH Zürich
Extending Brill-Noether classes to the boundary of moduli space of curves
Abstract.
Inside the Jacobian of the universal curve of the moduli space of genus \(g\), \(n\)-pointed curves lie the Brill-Noether loci, parametrizing pairs of curves with line bundles that have more than expected sections. Pulling the (virtual fundamental classes of the) Brill-Noether loci to \(M_{g,n}\) by any section of the universal Jacobian produces interesting cycles in the tautological ring, which play a key role in its structure as a ring. Pagani, Ricolfi and van Zelm have proposed an extension of these classes to the Deligne-Mumford compactification \(\bar{M}_{g,n}\), and conjecture that they are also in the tautological ring. In this talk, I will explain a natural refinement of these classes from the perspective of logarithmic geometry, which allows us to prove the PRvZ conjecture.
Ignacio
Barros
University of Antwerp
Moduli spaces of hyperkähler varieties
Abstract.
I will discuss general aspects of the geometry of moduli spaces of hyperkähler varieties and talk about the state of the art regarding their birational classification. In the second half, I will present recent results on their Kodaira dimension generalizing to higher dimension results of Gritsenko-Hulek-Sankaran in the surface and fourfold K3-type cases. The talk is based on joint work with Pietro Beri, Emma Brakkee, and Laure Flapan.
February 2023
Andrés
Rojas
HU Berlin
Cohomological rank functions on abelian surfaces via Bridgeland stability
Abstract.
In the context of abelian varieties, Z. Jiang and G. Pareschi have introduced interesting invariants called cohomological rank functions, associated to \(\mathbb{Q}\)-twisted (complexes of) coherent sheaves. We will show that, in the case of abelian surfaces, Bridgeland stability provides an alternative description of these functions. This helps to understand their general structure, and allows to compute geometrically meaningful examples. As a main application, we will give new results on the syzygies of abelian surfaces. This is a joint work with Martí Lahoz.
January 2023
Angel
Rios Ortiz
MPI Leipzig
Asymptotic base loci on Hyperkähler varieties
Abstract.
Hyperkähler varieties can be thought as higher dimensional analogues of K3 surfaces. As such, it is generally expected that properties that hold for K3 surfaces can be generalised (appropriately) for this class of varieties as well. In this talk I will discuss ongoing joint work with Francesco Denisi (Bologna University) that characterizes the so-called asymptotic base loci of a big divisor in a Hyperkähler variety in terms of rational curves on it.
Joshua
Lam
HU Berlin
Infinitely many rank two motivic local systems
Abstract.
A natural question in the study of motivic local systems is whether there are infinitely many such with fixed rank and determinant on a fixed proper curve, or a punctured curve if we further fix the conjugacy classes at the punctures. I'll discuss joint work with Daniel Litt, in which we answer this in the positive in the punctured case. More precisely, we construct all the rank two motivic local systems on P^1-4 points, with unipotent monodromy at three points, and "1/2-unipotent"-monodromy at the remaining point. Time permitting, I'll show how this implies several conjectures of Sun, Yang and Zuo coming from the theory of Higgs-de Rham flow.
Robert
Lazarsfeld
Stony Brook
Measures of association for algebraic varieties
Abstract.
I will discuss joint work with Oliver Martin addressing the following (vague) question: given two projective varieties \(X\) and \(Y\) of the same dimension, how far are \(X\) and \(Y\) from being birationally isomorphic?
Leonid
Monin
MPI Leipzig
Inversion of matrices, ML-degrees and the space of complete quadrics
Abstract.
What is the degree of the variety \(L^{-1}\) obtained as the closure of the set of inverses of matrices from a generic linear subspace \(L\) of symmetric matrices of size \(n\times n\)? Although this is an interesting geometric question in its own right, it is also motivated by algebraic statistics: the degree of \(L^{-1}\) is equal to the maximum likelihood degree (ML-degree) of a generic linear concentration model. In 2010, Sturmfels and Uhler computed the ML-degrees for \(\mathrm{dim}(L)\) less than 5 and conjectured that for the fixed dimension of \(L\) the ML-degree is a polynomial in \(n\). In my talk I will describe geometric methods to approach the computation of ML-degrees which in particular allow to prove the polynomiality conjecture. The talk is based on a joint works with Laurent Manivel, Mateusz Michalek, Tim Seynnaeve, Martin Vodicka, and Jaroslaw A. Wisniewski.
December 2022
Jieao
Song
HU Berlin
Geometry and moduli of Debarre-Voisin hyperkähler fourfolds
Abstract.
Debarre-Voisin varieties are one of the few known locally complete families of projective hyperkähler fourfolds, constructed inside the Grassmannian \(\mathrm{Gr}(6,10)\). Motivated by the case of varieties of lines on cubic fourfolds, we study their geometry as well as that of two associated Fano varieties, focusing on some special divisors in the moduli. This is a joint work with Vladimiro Benedetti.
November 2022
Laurentiu
Maxim
University of Wisconsin
On variants of the Singer-Hopf conjecture in complex geometry
Abstract.
The conjectures of Singer and Hopf predict the sign of the topological Euler characteristic of a closed aspherical manifold. In this talk I will discuss various generalizations (e.g., for singular spaces or Hodge enhancements) and partial results concerning the conjectures of Singer and Hopf in the context of Kähler geometry.
Alex
Suciu
Northeastern University
Topological invariants of groups and tropical geometry
Abstract.
There are several topological invariants that one may associate to a finitely generated group \(G\) -- the characteristic varieties, the resonance varieties, and the Bieri–Neumann–Strebel invariants -- that keep track of various finiteness properties of certain subgroups of \(G\). These invariants are interconnected in ways that makes them both more computable and more informative. I will describe in this talk one such connection, made possible by tropical geometry, and I will provide examples and applications pertaining to complex geometry and low-dimensional topology.
Omid
Amini
Ecole Polytechnique Paris
The tropical Hodge conjecture
Abstract.
The aim of the talk is to present the formulation of the Hodge conjecture for tropical varieties, and to explain a proof in the case the tropical variety is triangulable. This provides a partial answer to a question of Kontsevich. The proof uses Hodge theoretic properties of tropical varieties established in our companion works, which will be reviewed in the talk. These results generalize to the global setting the work of Adiprasito-Huh-Katz on combinatorial Hodge theory, by going in the local setting beyond the case of matroids, and provide answers to conjectures of Mikhalkin and Zharkov. Based on joint works with Matthieu Piquerez.
Irene
Spelta
University of Barcelona
Prym maps and generic Torelli theorems: the case of plane quintics.
Abstract.
The talk deals with Prym varieties and Prym maps. Prym varieties are polarized abelian varieties associated with finite morphisms between smooth curves. Prym maps are accordingly defined as maps from the moduli space of coverings to the moduli spaces of polarized abelian varieties. Once recalled the classical generic Torelli theorem for the Prym map of étale double coverings, we will move to the more recent results on the ramified Prym map \(P_{g,r}\) associated with ramified double coverings. For most of the values of \((g,r)\) a generic Torelli theorem holds and, furthermore, a global Torelli theorem holds when \(r\) is greater (or equal to) 6. At the same time, it is known that \(P_{g,2}\) and \(P_{g,4}\) have positive dimensional fibres when restricted to the locus of coverings of hyperelliptic curves. But this is not a characterization: the study of the differential \(d P_{g,r}\) shows that there are also other configurations to be considered. We will focus on the case of degree 2 coverings of plane quintics ramified in 2 points. We will show that the restriction of \(P_{g,r}\) here is generically injective. This is joint work with J.C. Naranjo.
October 2022
Andrea
Di Lorenzo
HU Berlin
Degenerations of twisted maps to algebraic stacks
Abstract.
Line bundles over curves, cyclic covers, elliptic surfaces: what these objects have in common is that they can all be regarded as maps from a family of curves to some moduli stack. Therefore, to have a controlled way to degenerate maps to algebraic stacks means having a controlled way to degenerate all the objects above, and more. How can this be obtained and made precise will be the main focus of this talk, which is based on a joint work with Giovanni Inchiostro.
Fabrizio
Catanese
Bayreuth
Manifolds with vanishing Chern classes and some questions by Severi/Baldassari
Abstract.
We give a negative answer to a question posed by Severi in 1951, whether the Abelian Varieties are the only manifolds with vanishing Chern classes. We exhibit Hyperelliptic Manifolds which are not Abelian varieties (nor complex tori) and whose Chern classes are zero not only in integral homology, but also in the Chow ring. We prove moreover the surprising result that Bagnera de Franchis manifolds ( quotients \(T/G\) where \(T\) is a torus and \(G\) is cyclic) have topologically trivial tangent bundle. Motivated by a more general question addressed by Mario Baldassarri in 1956, we discuss the Hyperelliptic Manifolds, the Pseudo- Abelian Varieties introduced by Roth, and we introduce a new notion, of Manifolds Isogenous to a \(k\)-Torus Product: the latter have the last \(k\) Chern classes trivial in rational homology and vanishing Chern numbers. We show that the latter class is the correct substitute for some incorrect assertions by Enriques, Dantoni, Roth and Baldassarri: in dimension 2 these are the surfaces with \(K_X\) nef and \(c_2(X) = 0\). A similar picture does not hold in higher dimension, unless we consider manifolds (isogenous to manifolds) whose tangent (resp. cotangent bundle) has a trivial summand. We survey old and new results on Kähler manifolds whose tangent (resp. cotangent bundle) has a trivial summand, and pose some open problems.
July 2022
Haoyang
Guo
Max Planck Institute for Mathematics
Prismatic approach to p-adic local systems
Abstract.
Let X be a smooth proper scheme over a p-adic field that admits a good reduction. Inspired by the de Rham comparison theorem in complex geometry, Grothendieck asked if there is a "mysterious functor", relating étale cohomology of the generic fiber and crystalline cohomology of the special fiber. The question was subsequently answered by Fontaine, Faltings and many others' work, which was one of the foundational results in the p-adic Hodge theory. In particular, this motivates the definition of a p-adic local system being crystalline, generalizing the representational property of the etale cohomology of X as above. In this talk, we will give an overview for crystalline representations and crystalline local systems. Building on the recent advance of Bhatt-Scholze, we then introduce the prismatic approach to crystalline local systems. This is a joint work with Emanuel Reinecke.
Xiaolei
Zhao
UCSB
Gushel-Mukai varieties, stability conditions, and moduli of stable objects
Abstract.
Gushel-Mukai varieties are smooth Fano varieties of dimension between 3 to 6, Picard rank 1, degree 10 and co-index 3. Depend on the parity of the dimension, their derived categories contain a so called Kuznetsov component, behaving similarly to an Enriques surface or a K3 surface. I will survey the construction of Bridgeland stability conditions on the Kuznetsov component, and some properties about moduli of stable objects on it. Applications to the geometry of Fano varieties will also be explained. This is based on joint work with Alex Perry and Laura Pertusi.
June 2022
Junliang
Shen
Yale
A tale of three moduli spaces of sheaves
Abstract.
I will discuss cohomological structures for three moduli spaces of sheaves: the moduli of vector bundles on a curve, the moduli of Higgs bundles on a curve, and the moduli of 1-dimensional torsion sheaves on \(\mathbb{P}^2\). These moduli spaces have been studied intensively from various perspectives. In recent years, enumerative geometry and string theory sheds new lights on the cohomological structure of these classical moduli spaces. In the talk I will discuss some results and conjectures in this direction; this concerns the \(\chi\)-independence phenomenon, tautological generators, and the \(P=W\) conjecture. Based on joint works with Davesh Maulik, Weite Pi, and an on-going project with Yakov Kononov and Weite Pi.
Carl
Lian
HU Berlin
Degenerations of complete collineations and geometric Tevelev degrees of \(\mathbb{P}^r\)
Abstract.
This is a report on work in progress. We will discuss a complete answer, in terms of Schubert calculus, to the problem of enumerating maps of degree \(d\) from a fixed general curve of genus \(g\) to \(\mathbb{P}^r\) satisfying incidence conditions at the appropriate number of marked points, that is, we compute the geometric Tevelev degrees of \(\mathbb{P}^r\). Previously, after the work of many people, the answers were known only when \(r = 1\), or when d is large compared to \(r\), \(g\); in the latter case, the answers agree with virtual counts in Gromov-Witten theory, but when \(d\) is small, the situation is considerably more subtle. The method proceeds by reduction to genus 0 via limit linear series, and then by an analysis of certain Schubert-type cycles on moduli spaces of complete collineations upon further degeneration.
Nicola
Tarasca
Virginia Commonwealth University
Incident varieties of algebraic curves and canonical divisors
Abstract.
The theory of canonical divisors on curves has witnessed an explosion of interest in recent years, motivated by recent developments in the study of limits of canonical divisors on nodal curves. Imposing conditions on canonical divisors allows one to construct natural geometric subvarieties of moduli spaces of pointed curves, called strata of canonical divisors. These strata are the projection on moduli spaces of curves of incidence varieties in the projectivized Hodge bundle. I will present a formula for the class of such incident varieties over the locus of pointed curves with rational tails. The formula is expressed as a linear combination of tautological classes indexed by decorated stable graphs, with coefficients enumerating appropriate weightings. I will conclude discussing applications to the study of relations in the tautological ring. Joint work with Iulia Gheorghita.
Ziquan
Zhuang
MIT
Finite generation and Kähler-Ricci soliton degenerations of Fano varieties
Abstract.
By the Hamilton-Tian conjecture on the limit behavior of Kähler-Ricci flows, every complex Fano manifold degenerates to a Fano variety that has a Kähler-Ricci soliton. In this talk, I'll discuss the algebro-geometric analogue of this statement and explain its connection to certain finite generation results in birational geometry. Based on joint work with Harold Blum, Yuchen Liu and Chenyang Xu.
May 2022
Philippe
Eyssidieux
Grenoble
\(L_2\) Cohomology for Hodge Modules
Abstract.
The talk will be based on arxiv:2203.06950[math.AG]. For an infinite Galois covering space of a compact Kähler manifold, we define \(L_p\) cohomology for \(1\le p<+\infty\) for various types of coefficients (perverse sheaves, coherent D-modules, Mixed Hodge Modules) and explain how to control it when \(p=2\). The formalism encompasses both my 2000 work on \(L_2\)-coherent cohomology (see also the independant and simultaneous work of Campana-Demailly) and Dingoyan's 2013 work on \(L_2\)-De Rham cohomology of an open subset. We describe a conjectural MHS on the reduced \(L_2\)-cohomology of a MHM and explain what can be proved with today's technology.
Reinier
Kramer
University of Alberta
The spin Gromov-Witten/Hurwitz correspondence
Abstract.
There are two important ways to calculate the number of maps between Riemann surfaces with given conditions: Hurwitz theory is over a hundred years old and uses the monodromy representation to transport the problem to symmetric groups, representation theory, and the Kadomstev-Petviashvili (KP) integrable hierarchy. Gromov-Witten theory for curves recasts the problem as intersection theory on the moduli space of such (stable) maps. By work of Okounkov-Pandharipande, there is a strict correspondence between the two. Both of these sides have an analogue with spin: this takes into account a bundle on the source which squares to the canonical bundle. On the Hurwitz side, this has relations to spin-symmetric or Sergeev groups, and the BKP hierarchy, while on the Gromov-Witten side, these invariants arise from localising the invariants of surfaces with smooth canonical divisor. I will explain that, at least for P^1, there is a correspondence between these two sides as well, which hinges on a spin analogue of the Ekedahl-Lando-Shapiro-Vainshtein formula and cosection localisation. This is joint work (partially in progress) with Alessandro Giacchetto, Danilo Lewański, and Adrien Sauvaget.
Andrei
Bud
HU Berlin
The Prym-Brill-Noether divisor
Abstract.
Understanding the birational geometry of the moduli space \(\mathcal{R}_g\) parametrizing Prym curves has been the subject of several papers, with great insight into this problem coming from the work of Farkas and Verra. Of particular importance for this study is finding divisors of small slope on the space \(\overline{\mathcal{R}}_g\). Drawing parallels with the situation on \(\overline{\mathcal{M}}_g\), we consider the Prym-Brill-Noether divisor and compute (some relevant coefficients of) its class. We will highlight the role of strongly Brill-Noether loci in understanding Prym-Brill-Noether loci. A consequence of our study is that the space \(\mathcal{R}_{14,2}\) parametrizing \(2\)-branched Prym curves of genus \(14\) is of general type.
Ruijie
Yang
HU Berlin
Singularities of theta divisors of hyperelliptic curves
Abstract.
It is a classical result that the theta divisor on the Jacobian variety associated to a smooth projective hyperelliptic curve has maximal dimension of singularities among indecomposable principally polarized abelian varieties. A version of the Schottky problem asks if this condition on the dimension of singularities characterizes hyperelliptic Jacobians. Motivated by this problem, it is natural to study the singularities of hyperelliptic theta divisors in more details to understand why they are special. In this talk, I will explain a natural and explicit embedded resolution of hyperelliptic theta divisors inside Jacobians by successively blowing up (proper transforms of) Brill-Noether subvarieties. A key observation is that we can use the geometry of Abel-Jacobi maps and secant varieties of rational normal curves to avoid the analysis of singularities in the blow-up process. From the point of view of singularities of pairs, we obtain all the essential information. If time permits, I will discuss how this resolution can be used to understand the mixed Hodge module structure on the vanishing cycle of a hyperelliptic theta divisor, using a global version of Esnault-Viehweg’s cyclic covering construction, limiting mixed Hodge structures on normal crossing divisors, and twisted D-modules. This is joint work (partially in progress) with Christian Schnell.
April 2022
Philippe
Eyssidieux
Grenoble
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.
March 2022
Michele
Pernice
Scuola Normale Superiore, Pisa
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.
February 2022
Rahul
Pandharipande
ETH Zürich
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.
Qianyu
Chen
Stony Brook
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.
Konstantin
Jakob
MIT
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.
Yichen
Qin
École Polytechnique
\(L\)-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.
January 2022
Raju
Krishnamoorthy
Wuppertal
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.
Andrés
Rojas
Uni Bonn
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.
Kien Huu
Nguyen
KU Leuven
Exponential sums modulo \(p^m\) for Deligne polynomials
Abstract.
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Luca
Battistella
Uni. Heidelberg
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.
Andres
Fernandez Herrero
Cornell
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.
Olivier
de Gaay Fortman
ENS Paris
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.
December 2021
Ruijie
Yang
HU Berlin
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.
Daniele
Agostini
MPI Leipzig
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.
Gavril
Farkas
HU Berlin
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.
November 2021
Scott
Mullane
HU Berlin
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.
Christian
Schnell
Stony Brook, visiting Bonn
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.
Rahul
Pandharipande
ETH Zürich
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.
October 2021
Alessandro
Verra
Università Roma Tre
A series of hyperkähler varieties and their Pfaffian counterparts