Algebra and geometry seminar
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Abstracts
2019/2020
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Paolo Cascini
(Imperial College)
Geometria birazionale
delle foliazioni su varietà di dimensione 3
Lo scopo del seminario è descrivere alcuni
risultati recenti sulla geometria birazionale delle foliazioni su
varietà complesse di dimensione 3, e in particolare sull'esistenza di un
modello minimale. In collaborazione con C. Spicer.
Frank Loray
(Rennes)
Neighborhoods of curves in complex surfaces
In this talk, we consider a
complete complex smooth curve C (compact Riemann surface) with an embedding
into a smooth complex surface S and we want to understand the structure of the
germ of surface (S,C) near the image of the curve. After surveying on old
results, we focus on the case of elliptic curve with trivial normal bundle.
With O. Thom and F. Touzet, we recently completed the
formal
classification of such
germs of neighborhood. In a work in progress, we investigate the analytic
classification, with F. Touzet and S. M. Voronin.
Alek Vainshtein
(Haifa)
Plethora of cluster structures on GLn
I will explain our recent
results in the study of multiple cluster structures in the rings of regular
functions on GLn, SLn
and Matn that are compatible with
Poisson-Lie and Poisson-homogeneous structures. All possible An type
Belavin-Drinfeld (BD) data will be subdivided into
oriented and non-oriented kinds. In the oriented case, one can single out BD
data satisfying a certain combinatorial condition called aperiodicity.
Our main result claims that
for any BD data of this kind there exists a regular cluster structure
compatible with the corresponding Poisson–Lie bracket. In fact, we extend
the aperiodicity condition to pairs of oriented BD data and prove a more
general result that establishes an existence of a regular cluster structure on SLn compatible with a Poisson bracket
homogeneous with respect to the right and left action of two copies of SLn
equipped with two different Poisson-Lie brackets. In all the cases, the
corresponding upper cluster algebra is isomorphic to the ring of regular
functions. If the aperiodicity condition is not satisfied, a compatible cluster
structure has to be replaced with a generalized cluster structure.
(Based on a joint work with
M. Gekhtman and M. Shapiro).
Jacopo Stoppa
(SISSA)
Deformed Hermitian Yang-Mills connections,
extended gauge group and scalar curvature
I will discuss joint work
in progress with Enrico Schlitzer (SISSA).
We introduce a new system
of partial differential equations that describe special metrics on pairs formed
by a Kähler manifold together with a line
bundle, thought of as an object in the derived category. These comprise the
deformed Hermitian Yang-Mills equations of Leung-Yau-Zaslow, and specialise to the Kähler Yang-Mills equations of Alvarez-Consul,
Garcia-Fernandez and Garcia-Prada in the large radius limit.
The first part of the talk
will be devoted to geometric motivation, in particular through a moment map
interpretation.
The second part will be
more analytic and will focus on abelian varieties, following ideas of
Feng-Szekelyhidi.
Jean-Benoît
Bost (Paris-Sud)
Transcendence proofs and theta invariants of
infinite-dimensional Euclidean lattices
I will explain how some
classical transcendence results, notably the theorem of Schneider-Lang, may be
given « natural proofs », formally similar to diverse classical algebraization proofs in analytic and formal geometry,
based on the consideration of infinite dimensional avatars of Euclidean
lattices and of their theta invariants.
Francesco Russo (Catania)
Razionalità di cubic fourfolds via Flop delle
Trisecanti e superfici K3 (non minimali) associate
Una formulazione recente della congettura di Kuznetsov
sulla razionalità dei cubic fourfolds specifica una quantità numerabile di
divisori di discriminante d “ammissibile" (nel senso di Hassett) nello spazio dei moduli la cui unione dovrebbe
descrivere esattamente il luogo dei cubic fourfolds razionali. I valori per cui la congettura risulta
finora provata sono: d=14 (Fano), d=26,38 (insieme a Staglianò).
Tramite la costruzione del Flop delle Trisecanti e la
teoria delle congruenze di curve secanti sveleremo il ruolo giocato dalle
superfici K3 (non minimali) in varie questioni di razionalità di fourfolds di Fano.
Presenteremo poi
l’applicazione di questi nuovi metodi alla dimostrazione del caso
d=42 della congettura e formuleremo alcune domande sui possibili sviluppi di
queste idee.
Presentazione basata su lavori con Giovanni Staglianò.
Piotr Pragacz
(Polish Academy of Sciences)
Flag bundles, Segre polynomials, and
push-forwards
We give Gysin formulas for
all flag bundles of types A, B, C,
D. The formulas (and also the
proofs) involve only the Segre classes of the original vector bundles and
characteristic classes of universal bundles. As an application we provide new determinantal formulas.
This is a joint work with
Lionel Darondeau.
Tommaso Pacini (Torino)
Extremal length in higher dimensions: a new
family of holomorphic invariants\
The notion of extremal
length, in the classical theory of one complex variable, provided a uniform
method for building conformal invariants. These invariants then found a wide
range of applications in the theory of conformal maps, Teichmüller
theory, systolic geometry, etc.
Using intuition from submanifold geometry, calibrations and Calabi-Yau
manifolds, we recently proposed a higher-dimensional analogue, cf. https://arxiv.org/abs/1904.07807
I will survey the
definition and some results, also trying to place this topic into a broader
geometric context.
Antonio J. Di Scala (Politecnico Torino)
Homogeneous Riemannian manifolds with
non-trivial nullity
The goal of the talk is to
explain the ideas and tools used to study the nullity distribution of the
curvature tensor of a Riemannian homogeneous space.
The main results are stated
in the paper https://arxiv.org/abs/1802.02642
written in collaboration with Carlos Olmos and Francisco Vittone.
Margherita Lelli Chiesa (Roma Tre)
Genus two curves on Abelian surfaces
Let (S,L) be a general (d1,d2)-polarized
abelian surfaces. The minimal geometric genus of any curve in the linear system
|L| is two and there are finitely many curves of such genus. In analogy with
Chen's results concerning rational curves on K3 surfaces, it is natural to ask
whether all such curves are nodal. In the seminar I will prove that this holds
true if and only if d2 is not divisible by 4. In the cases where d2
is a multiple of 4, I will construct curves in |L| having a triple, 4-tuple or
6-tuple point, and show that these are the only types of unnodal
singularities a genus 2 curve in |L| may acquire.
This is joint work with A.
L. Knutsen.
Federico Pellarin (St Etienne)
Famiglie analitiche di
forme modulari di Drinfeld
Le forme modulari di Drinfeld
possono essere viste come una variante delle forme modulari classiche. Esse
sono definite su un semipiano complesso 'di Drinfeld'
e prendono valori in campi algebricamente chiusi e completi di caratteristica
positiva. In questo seminario, dopo aver presentato gli aspetti elementari di
questa teoria, abborderemo l'esistenza di famiglie di forme modulari di Drinfeld che sono analitiche rispetto alla valutazione
all’infinito (di peso invariante e di livello variabile). In
compatibilità col tempo a disposizione, concluderemo con qualche
speculazione sull’esistenza di simili strutture nel caso complesso.
Daniel Labardini-Fragoso (Autónoma Mexico)
Algebraic and combinatorial decompositions of
Fuchsian groups
The discrete subgroups of PSL2(R)
are often called 'Fuchsian groups'. For Fuchsian groups G whose action on the hyperbolic plane H is free, the orbit space H/G has a canonical structure of Riemann surface with a
hyperbolic metric, whereas if the action of G is not free, then H/G has a structure of 'orbifold'.
In the former case, there is a direct and very clear relation between G and the fundamental group p1(H/G,x): a theorem of the theory of covering spaces states
that they are isomorphic. When the action of \Gamma is not free, the relation
between G and p1(H/G,x) is subtler. A 1968 theorem of Armstrong states that
there is a short exact sequence 1®E®G®p1(H/G,x)®1,
where E is the subgroup of G
generated by the elliptic elements. For G finitely generated, non-elementary and with at least one
parabolic element, I will present full algebraic and combinatorial
decompositions of G in
terms of p1(H/G,x) and a specific finitely generated subgroup of E, thus
improving Armstrong's theorem.
This talk is based on an
ongoing joint project with Sibylle Schroll and Yadira Valdivieso-Díaz
that aims at describing the bounded derived categories of skew-gentle algebras
in terms of curves on surfaces with orbifold points
of order 2.
Jorge Vitorio Pereira (IMPA)
Miyaoka's algebraicity criterion and variations
I will review some old and
new results/arguments on the algebraicity of leaves of foliations with
"positive" tangent sheaf.
Gaëtan Borot (MPI Bonn)
Topological recursion(s) for Masur-Veech
volumes
Statistics of the simple
length spectrum of bordered hyperbolic surfaces define functions on the moduli
space. Andersen, Orantin and myself showed recently that they satisfy a
recursion on the Euler characteristic, which implies a topological recursion
for their averages over the Weil-Petersson measure. This can be seen as a
generalization of Mirzakhani's identity and her proof of a topological
recursion for the Weil-Petersson voulmes.
We show how this result implies topological recursion (here taking the form of
Virasoro constraints) for the Weil-Petersson averages of the asymptotic growth
of the number of long curves. By invoking the relation between Weil-Petersson
measure on the Teichmuller space, Thurston measure on the space of measured
laminations, and Masur-Veech measure on the space of quadratic differentials,
this gives a recursion to compute polynomials Pg,n(L1,
..., Ln) whose constant term are the Masur-Veech volumes of the
principal stratum. This retrieves and generalizes a result of Delecroix et al.
obtained via different (combinatorial) methods.
If time permits, I will present a second topological recursion computing
different polynomials Rg,n(L1,
..., Ln) whose constant terms is the same Masur-Veech volume, and
which is a consequence of an intersection-theoretic approach of Chen, Moeller
and Sauvaget.
This is based on joint works with Jorgen Ellegaard Andersen, Severin
Charbonnier, Vincent Delecroix, Alessandro Giacchetto, Danilo Lewanski,
Campbell Wheeler.
Paolo Piazza (Roma “Sapienza”)
Varietà
singolari, generi superiori e operatori di Dirac
Il teorema di Atiyah-Singer
esprime l'indice dei tre operatori geometrici fondamentali, e cioè
l'operatore di segnatura, l'operatore spin-Dirac e
l'operatore di Riemann-Roch su varietà
complesse, in termini del genere L, del genere A-roof
e del genere di Todd. E` possibile generalizzare ognuno di questi numeri ad una
collezione di numeri, detti generi superiori, indicizzata dalle classi di coomologia del gruppo fondamentale della varietà in
esame. Risultati profondi di Kasparov, Connes, Moscovici
e molti altri connettono questi generi superiori, oggetti squisitamente
geometrici, ad opportune classi di K-Teoria associate ai relativi operatori e
questa connessione è cruciale dello studio delle proprietà di
stabilità dei generi superiori. Cosa di tutto questo può essere
esteso a varietà singolari? In questo seminario darò una risposta
(necessariamente parziale) a questa domanda; i risultati illustrati si basano
su lavori in collaborazione con Albin-Leichtnam-Mazzeo,
Botvinnik-Rosenberg, Bei.
Francesco Meazzini (Roma
Sapienza)
On the Kaledin-Lehn
formality conjecture
Kaledin and Lehn conjectured that (the homotopy class of) the DG-Lie algebra of derived endomorphisms of any polystable
sheaf on a K3 surface is formal. The relevance of the formality conjecture
relies on its consequences concerning the geometry of the moduli space of semistable sheaves on the K3. The conjecture was recently
proven after several contributions mainly due to Kaledin-Lehn
themselves, Zhang, Yoshioka, Arbarello-Saccà, Budur-Zhang.
In this talk I will present
a joint work with R. Bandiera and M. Manetti, where we introduce an alternative and more
algebraic approach to the problem, eventually proving that the formality conjecture
holds for polystable sheaves on any smooth minimal
projective surface of Kodaira dimension 0.
Daniele Agostini
(Humboldt Universität)
On the Schottky problem for genus five
Jacobians with a vanishing theta null
In this talk, I will
present a solution to the weak Schottky problem for
genus five Jacobians with a vanishing theta null, extending a result of Grushevsky and Salvati Manni in genus four. More precisely, I will show that if an
abelian fivefold has a vanishing theta null with a quadric tangent cone of rank
at most three, then it is in the Jacobian locus, up to extra irreducible
components.
This is joint work with Lynn Chua.
Laura Capuano (Politecnico di Torino)
Lang-Vojta
conjecture over function fields for surfaces dominating tori
The celebrated Lang-Vojta Conjecture
predicts degeneracy of integral (or more in general S-integral) points on
varieties of log general type defined over number fields. It admits a geometric
analogue over function fields, where stronger results have been obtained
applying a method developed by Corvaja and Zannier.
In this talk, we present a recent result for non-isotrivial surfaces over
function fields dominating a two-dimensional torus. This extends Corvaja and
Zannier result in the isotrivial case and it is based on an estimate about
greatest common divisors of polynomials evaluated at S-units.
This is a joint work with A. Turchet.
Giovanni Cerulli Irelli (Roma Sapienza)
On quiver Grassmannians
A quiver Grassmannian is a
projective variety which parametrizes the subrepresentations of a fixed
dimension of a quiver representation. It is easy to show that every projective
variety can be realized in this way. One hence restricts the study of such projective
varieties to relevant classes of quivers and of quiver representations.
In a joint work with F. Esposito, H. Franzen and M. Reineke we prove geometric
and cohomological properties of these varieties in the case of quivers of
finite and tame type and in the case of representations which are rigid, i.e.
whose Ext1 is zero. I will illustrate the results, give examples and
provide an idea of the techniques of proofs.
Gustavo Jasso (Bonn)
The symplectic
geometry of higher Auslander algebras
It is well known that the
partially wrapped Fukaya category of a marked disk is
equivalent to the perfect derived category of a Dynkin
quiver of type A.
In this talk I will present
a higher-dimensional generalisation of this
equivalence which reveals a deep connection between three a priori unrelated
subjects:
* Floer
theory of symmetric products of marked surfaces
* Higher Auslander-Reiten theory in the sense of Iyama
* Waldhausen
K-theory of differential graded categories
If time permits, as a first
application of the above relationship, I will outline a symplecto-geometric
proof of a recent result of Beckert concerning the
derived equivalence between higher Auslander algebras
of different dimensions.
This is a report on joint
work with Tobias Dyckerhoff and Yankı
Lekili.
Andrea Seppi
(Grenoble)
Examples of four-dimensional geometric
transition
Roughly speaking, a
geometric transition is a deformation of geometric structures on a manifold, by
“transitioning” between different geometries. Danciger
introduced a new such transition, which enables to deform from hyperbolic
structures to Anti-de Sitter structure, going through another type of real
projective structures called “half-pipe”, and provided conditions
for a compact 3-manifold to admit a geometric transition of this type. By
extending a construction of Kerckhoff and Storm, I
will describe examples of finite-volume geometric transition in dimension 4.
This is joint work with
Stefano Riolo.
Ralf Köhl
(Giessen)
The geometry of Kac-Moody
groups
Kac-Moody groups enjoy very many nice geometric
properties via their action on twin buildings, on generalized flag manifolds,
and on highest-weight modules endowed with an anisotropic form.
In this talk I want to
topological Kac-Moody groups by comparing their
properties to semisimple Lie groups. I will introduce
symmetric spaces for Kac-Moody groups, I will compute
the fundamental group of topological Kac-Moody
groups, and I will give evidence for Kostant
convexity.
Laura Pertusi (Milano statale)
Stability conditions and moduli spaces for Kuznetsov components of Gushel-Mukai
varieties
A generic Gushel-Mukai variety X is a quadric section of a linear
section of the Grassmannian Gr(2,5). Kuznetsov and Perry proved that the bounded derived
category of X has a semiorthogonal decomposition with
exceptional objects and a non-trivial subcategory Ku(X), known as the Kuznetsov component.
In this talk we will discuss the construction of stability conditions on Ku(X) and, consequently, on the bounded
derived category of X. As applications, for X of even-dimension, we will
construct locally complete families of hyperkaehler
manifolds from moduli spaces of stable objects in Ku(X) and we will determine when X has a homological associated K3
surface.
This is a joint work with
Alex Perry and Xiaolei Zhao..
Marco Boggi
(Belo Horizonte)
Automorphisms of procongruence mapping class
groups
In this talk, I will
discuss the automorphism group of the procongruence mapping class group and of
the associated procongruence curve and pants complexes. In analogy with a
classical result of Ivanov for mapping class groups, this allows to determine
the group of automorphisms of the arithmetic procongruence mapping class
group which satisfy a natural geometric condition. It is a nontrivial fact that
this condition holds in genus 0. Let M0,n be the moduli space
of n-labeled, genus 0 algebraic curves. It follows, in particular, that Out(π1ét(M0,n⊗ℚ))≅Σn,
for n≥5.
This talk is based on a joint work with Louis Funar and Pierre Lochak (cf.
arXiv:2004.04135).
Marco Maculan
(Paris Sorbonne)
Affine vs. Stein varieties in complex and
rigid geometry
Serre’s GAGA theorem states that, on a projective
complex variety, holomorphic objects (functions, vector bundle and their
sections, etc.) are algebraic. Without compactness hypothesis this is not true.
Yet, one may wonder whether a variety that can be embedded holomorphically
into an affine space, can be embedded therein algebraically. A classical example of Serre shows
that the answer is negative.
In an ongoing joint work
with J. Poineau, we investigate what happens when one
replaces the complex numbers by the p-adic ones.
Despite the formal similarities between the corresponding analytic theories,
the p-adic outcome is somewhat surprising.
Alexander Lytchak (Köln)
Structure of non-positively curved spaces
In the talk I would like to
discuss local geometric, analytic and topological structure of spaces with
upper curvature bounds and extendible geodesics.
The talk will be based on joint work with Koichi Nagano.
Michele D’Adderio (Université libre de Bruxelles)
Partial and global representations of finite
groups
The notions of partial
actions and partial representations have been extensively studied in several
algebraic contexts in the last 25 years.
In this talk we introduce these concepts and give a short overview of the
results known for finite groups. We will briefly show how this theory extends
naturally the classical global theory, in particular in the important case of
the symmetric group.
This is joint work with William Hautekiet, Paolo Saracco and Joost Vercruysse.
Cristiano Spotti (Aarhus Universitet)
On log Kähler-Einstein
metrics
In this talk I will discuss
examples and some geometric properties of KE metrics with cone angle
singularities along possibly singular (in general worse than normal crossing)
divisors.
10 June 2020 – Starting at 16:30!
Valentino Tosatti (Northwestern University)
Metric limits of Calabi-Yau
manifolds
I will discuss the problem
of understanding the behavior of degenerating families of Ricci-flat Kähler metrics on a Calabi-Yau
manifold, and what their possible metric limits are. I will explain what we
know in general about such metric limits, what techniques are used to approach
these questions, and what applications these results have.
Based on joint works with
Gross-Zhang and with Hein.
Karen Vogtmann
(University of Warwick)
Outer space for RAAGs
In recent years
right-angled Artin groups (RAAGs) have assumed a
prominent place in geometric group theory and related fields,
Now attention is also being
focused on their (outer) automorphism groups. The most well-understood examples of
outer automorphism groups of RAAGs are GL(n,Z)
and Out(Fn).
GL(n,Z) can be profitably studied via its
action on the symmetric space of marked lattices, and Out(Fn) via its action on Outer
space.
We propose an analogous
“outer space” generalizing both of these, on which the outer automorphism group of an arbitrary RAAG acts properly, and
we prove this space is contractible.
This is joint work with
Corey Bregman and Ruth Charney.
Bojko Bakalov (North
Carolina State University)
A vertex algebra construction of
representations of toroidal Lie algebras
Given a simple
finite-dimensional Lie algebra and an automorphism of
finite order, one can construct a twisted toroidal Lie algebra.
Similarly to twisted affine
Lie algebras, which are well-studied in the literature, we can create
representations of twisted toroidal Lie
algebras with the help of
vertex algebras. In this talk, I will discuss twisted modules of vertex
algebras and will show how representations of twisted toroidal Lie algebras can
be constructed
from such twisted modules.
Joint work with Samantha
Kirk.