VBAC 2014
Algebraic
Varieties:
Bundles, Topology, Physics
Titles and Abstracts
Francis
Brown
Relative
completion and automorphisms
Given
a discrete group with a homomorphism to the points of a reductive
group, one can attach a pro-algebraic group
called the
relative unipotent completion. It generalises the notion of
unipotent (Malcev) completion. Applied to
fundamental groups of
manifolds, the relative completion describes certain categories of
local systems. Hain showed that
in certain geometric
situations, the relative completion carries a mixed Hodge
structure. The associated periods are often
interesting
numbers, such as multiple zeta values, which are very prevalent in
high-energy physics. General conjectures
state that there
should be a motivic Galois group acting upon these numbers. I will
explain how to compute this action for
general relative
completions and give examples relating to moduli spaces of curves
of genus 0 (where the automorphism
group is the
Grothendieck-Teichmuller group)
and
genus 1 (whose periods are multiple versions of L-values of
modular
forms).
Fabrizio
Catanese
Vector
bundles on curves which are the direct image of the relative
dualizing sheaf
Slides
In
the first part of my talk I shall describe joint work with Michael
Dettweiler.
First, I shall provide details for a theorem
announced by
Fujita 34 years ago.
THM
1. If
one has a Kaehler family fibred over a curve B,
then the direct image V
of
the relative dualizing sheaf is the direct
sum of an ample
vector bundle A
and
of a unitary flat vector bundle W.
I
shall then describe a counterexample to a question raised by
Fujita 31 years ago.
Question:
Is
V
semi
ample?
In view of the previous theorem, the question is
whether the flat bundle W
corresponds
to a finite representation of the
fundamental group. While the
answer is yes (Deligne) if we have a summand of W
of
rank one,or if the base has genus at
most one, we show
examples, based on hypergeometric integrals, where we get a
representation of infinite order, hence
THM
2. There
are curve fibrations for which V
is
not semiample.
In the final part I shall relate the
question with Teichmueller space of higher dimensional varieties,
and the problem of
rigidification for varieties of general
type: are there isomorphisms which are isotopic to the identity,
or act trivially on
homology and the fundamental group?
Time
permitting I shall also report on work in progress with G.
Gromadzki.
Ron
Donagi
Moduli
of super Riemann surfaces
We
study various aspects of supergeometry, including obstruction,
Atiyah,
and super-Atiyah classes. This is applied to the
geometry of
the moduli space of super Riemann surfaces and to its
Deligne-Mumford compactification. For genus greater than
or
equal to 5, this moduli space is not projected (and in particular
it is not split): it cannot be holomorphically projected
to
its underlying reduced manifold. Physically, this means that
certain approaches to superstring perturbation theory that
are
very powerful in low orders have no close analog in higher orders.
Mathematically, it means that the moduli space of super
Riemann
surfaces cannot be constructed in an elementary way starting with
the moduli space of ordinary Riemann surfaces. It
has a life
of its own. (Joint work with E. Witten)
Brent
Doran
M0,n
and
non-linearizable actions
We
discuss a method to study effective cones of varieties and related
questions, illustrated by means of the moduli space of
n-pointed
rational curves. In that example, there is an “algebraic
uniformization" of the moduli space by affine space, endowed
with a non-linearizable solvable group action whose geometric
invariant theory quotient is M0,n;
this uniformization description may be thought of as a
homotopically natural replacement for the universal torsor. In
particular, an algebraic A1-bundle
that is not an algebraic line bundle plays a crucial role.
Daniel
Greb
Construction
and variation of moduli spaces of vector bundle on
higher-dimensional base manifolds
While
the variation of moduli spaces of H-slope/Gieseker-semistable
sheaves on surfaces under change of the ample
polarisation H
is
well-understood, research on the corresponding question in the
case of higher-dimensional base manifolds
revealed a number of
pathologies. After presenting these, I will discuss recent joint
work with Matei Toma (Nancy) and
Julius Ross (Cambridge) which
resolves some of these pathologies by looking at curves instead of
divisors, and by
embedding the moduli problem for sheaves into
a moduli problem for quiver representations.
Matthew
Kerr
Feynman
integrals and the K3
of
a K3
Slides
Feynman
graphs are pictures which display propagating and interacting
particles in the context of perturbative quantum
field theory.
Associated to each graph is an amplitude, which is computed by an
"Feynman integral" directly associated to
the graph.
There is a large and vibrant international industry centered
around the evaluation of these integrals.
The purpose of this
talk is to explain how some of these integrals are computed in the
context of algebraic geometry
(including Hodge theory and
K-theory) where they yield interesting "periods". I will
concentrate on an example from recent
joint work with Spencer
Bloch and Pierre Vanhove, which is evaluated in terms of elliptic
trilogarithms arising from the
descrption of the integral as a
"higher normal function" associated to a certain
algebraic K3
class
on a family of K3 surfaces.
Marc
Levine
Comparing
motivic and classical homotopy theories
Suslin-Voevodsky
proved in the early 90s that mod n
motivic
cohomology agrees
with mod n
singular
cohomology for varieties
over the complex numbers. We will
explain how this result can be extended to give an equivalence of
the classical stable
homotopy category with a full subcategory
of the motivic stable homotopy category over an algebraically
closed field of
characteristic zero. We will also explain how
one of the basic constructions in classical stable homotopy
theory, the
Adams-Novikov spectral sequence, is of motivic
origin, and how it is related to Voevodsky’s slice tower for the
motivic sphere
spectrum.
Eduard
Looijenga
Topology
and Hodge theory of the abelianization of a cofinite subgroup of
a mapping class group
According
to a conjecture of Nikolai Ivanov (that dates around 1991) the
first Betti number of a finite index subgroup of a
mapping
class group of a surface should vanish. In this talk (which is
about joint work with Marco Boggi) we describe an
elementary
topological property equivalent to this conjecture. We also
connect this to Deligne-Mumford compactifications
and describe
the underlying Hodge theoretic (or rather motivic)
aspects.
Gabriele
Mondello
On
the Dolbeault cohomological dimension of the moduli space of
Riemann surfaces
The
moduli space Mg
of
Riemann surfaces of genus g
is
(up to a finite étale cover) a complex manifold and so it makes
sense
to speak of its Dolbeault cohomological dimension (i.e.
the highest k
such
that H0,k(Mg,E)
does not vanish for some
holomorphic vector bundle E
on
Mg).
The conjecturally optimal bound is g-2,
which is verified for g=2,3,4,5.
I can prove that
such dimension is at most 2g-2.
The key point is to show that the Dolbeault cohomological
dimension of each stratum of
translation surfaces is at most g
(still non-optimal bound). In order to do that, I produce an
exhaustion function, whose
complex Hessian has controlled
index: in the construction of such a function basic geometric
properties of translation
surfaces come into play.
Jonathan
P. Pridham
Motives
and derived Tannaka duality
Given
a k-valued
cohomology functor on the derived category of motives, Ayoub
constructed a motivic Galois group as
a Hopf algebra in the
derived category of k,
satisfying a weak universal property. I will explain how to
strengthen this,
recovering a quotient of the derived category
of motives from the Galois group. The key tool is a generalisation
of
Tannaka duality to dg coalgebras and dg categories, via
derived Morita theory.
Andrei
Teleman
Compact
subspaces of moduli spaces of stable bundles over class VII
surfaces
Slides
We
explain our program - based on Donaldson theory - to prove
existence of curves on class VII surfaces, and the newest results
obtained using this program. We explain a duality principle: If
the moduli space M(K,0)
of polystable bundles with
determinant isomorphic to K
and
c2=0
on a class VII surface X
has
no compact subspace satisfying certain properties, then
X
will
have a cycle of curves. Using this principle and a recent
non-existence result for compact subspaces of Ms(K,0),
we prove the existence of a cycle for class VII surfaces with
b2≤3.
Karen
Vogtmann
Building
cocycles for the group of automorphisms of a free
group
Slides
The
rational cohomology of Out(Fn)
vanishes in high and low dimensions, but the middle range remains
quite mysterious.
In this talk I will summarize what we do
know and then show how tools from topology and representation
theory can be used to construct a large number of new cocycles.
Those within range of computer calculations have been shown to
give nontrivial cohomology classes, and it is conjectured that
essentially all of them should.
Katrin
Wendland
The
elliptic genus in conformal field theory
Slides
The
elliptic genus has long become a classical link between algebraic
geometry, topology and mathematical physics.
This talk will
review the definition of the elliptic genus and its interpretation
in quantum field theory. Moreover, more
recent developments
addressing the role of the elliptic genus in conformal quantum
field theory will be presented.
Kirsten
Wickelgren
Motivic
desuspension
Certain
problems such as classifying manifolds up to cobordism are stable
in the sense that they are solved in categories
where it is
possible to desuspend. Other problems, such as classifying
algebraic vector bundles on schemes, require
analogous
unstable information. The EHP sequence in algebraic topology is a
tool for turning stable information into unstable
information.
This talk will present an EHP sequence in A1
homotopy
theory of schemes. Work of A. Asok and J. Fasel gives a
relationship with splitting vector bundles on smooth affine
schemes. This is joint work with Ben Williams, and partially joint
work
with Asok and Fasel.
Christopher
Woodward
Gauged
Gromov-Witten theory
I
will talk about joint work with E. Gonzalez and S. Venugopalan on
gauged Gromov-Witten invariants, especially in the
toric case.
Connections between quantum cohomology and the minimal model
program will be discussed.
Ben
Davison
A
few ways to think about the Cohomological Hall algebra of Higgs
bundles
Slides
I
will introduce the Cohomological Hall algebra for the stack of all
finite-dimensional representations of the fundamental
group of
a closed genus g Riemann surface S.
A surprising recent theorem states that this algebra Ag
is
a free
supercommutative algebra. This leads naturally to a
recent conjecture, identifying the generators of this algebra with
the
cohomology of the space of twisted representations of the
fundamental group of S
--
strong evidence for this conjecture is
given by the E
polynomial calculations of Hausel and Villegas. The algebra
structure on Ag
respects
Hodge structures,
and so we obtain a dictionary relating
conjectures regarding the Hodge structure of twisted character
varieties to ones
about untwisted character varieties. On
the other hand, all of the above suggests a natural Hall algebra
structure on the
stack of semistable Higgs bundles, respecting
a perverse filtration, so that the famous P=W conjecture again
acquires a
natural twin conjecture in the untwisted world.
I'll finish by explaining how this works, and some recent progress
towards
proving these conjectures.
Will
Donovan
Geometry
of 3-folds, and noncommutative deformations
Slides
Rational
curves on complex 3-folds are a rich source of geometric interest,
and much of
their geometry is encoded in their
deformation theory. I will
describe recent joint work with Michael Wemyss which analyses the
noncommutative deformations of such curves. Under suitable
geometric assumptions, we associate an associative algebra of
deformations to each curve,
and use this to solve a number of
problems. Firstly, the algebra detects whether the curve can be
contracted to a point
without simultaneously contracting a
divisor. Secondly, the algebra allows a general construction of a
new Fourier-Mukai
autoequivalence for a 3-fold, which is
naturally related to the flop equivalence of Bridgeland and Chen.
In fact, to address
these problems in the appropriate level of
generality, we work with a deformation functor associated to a
finite set of rational
curves in a 3-fold: this functor
encodes information on the deformations of the structure sheaves
of the curves, as well as their
mutual extensions, via a
quiver algebra. We suggest how this approach will also yield
useful noncommutative structures on
more general moduli spaces
of sheaves.
Sara
Angela Filippini
Conformal
limits of irregular connections
We
sketch the construction of isomonodromic families of irregular
meromorphic connections ∇(Z) on IP1,
with values in the derivations of a class of infinite dimensional
Poisson algebras. Our main results concern the limits of the
families as we vary a scaling parameter R.
In the R
0
'conformal limit' we recover a semi-classical version of the
connections introduced by Bridgeland and Toledano Laredo (and so
the Joyce holomorphic generating functions for DT invariants). The
connections ∇(Z)
are a rough but rigorous approximation to the (mostly conjectural)
four-dimensional tt*-connections introduced by
Gaiotto-Moore-Neitzke. A precise comparison with these is
established in a basic example. Joint work with Mario
Garcia-Fernandez and Jacopo Stoppa.
George
Hitching
Lagrangian
subbundles of orthogonal vector bundles over a curve
Slides
The
r'th
Segre invariant of a vector bundle W
over
a curve measures the degree of a maximal subbundle of rank r
in
W.
The Segreinvariants define stratifications on moduli spaces of
such W,
which have been studied by Lange, Brambila-Paz, Russo, Teixidor i
Bigas and others. If the bundles W
have
orthogonal structure, the moduli spaces are also stratified by
isotropic Segre invariants, which measure the maximal degrees of
totally isotropic subbundles. When the isotropic subbundles are
Lagrangian (of maximal rank ⌊rk(W)/2⌋),
we construct parameter spaces which dominate the strata, showing
that the strata are irreducible and computing their dimensions.
The main ingredient in the proofs is to exploit the geometry of
secants in certain spaces of vector bundle extensions, in an
analogous way to a famous result of Lange and Narasimhan (Math.
Ann., 1983).
This
is joint work with Insong Choe (Konkuk, Seoul).
Marcos
Jardim
Boundary
of the moduli space of instanton bundles on IP3
Recent
results by Tikhomirov, Markushevish and Tikhomirov and by the
author and Verbitsky have answered old questions about the
geometry of the moduli space I(c)
of rank 2 instanton bundles of charge c
on
IP3:
we now know that this is an irreducible, rational, non-singular
affine variety of dimension 8c-3.
The next step is to study its compactification. Since every rank 2
instanton bundle on IP3
is
stable, I(c)
can be regarded as an open subset of the Gieseker--Maruyama scheme
M(2,0,c,0)
of semistable rank 2 torsion free sheaves on I(c)
with Chern classes c1=c3=0
and c2=c.
One can then consider the closure of I(c)
within M(2,0,c,0).
In this talk we show that the singular locus of non-locally free
rank 2 instanton sheaves on IP3
have
pure dimension 1. We then describe certain irreducible components
of the boundary of I(c)
with dimension 8c-4.
Such components consist of stable, non-locally free rank 2
instanton sheaves whose singular loci are rational curves. The
results presented are joint work with M. Gargate and with D.
Markushevich and A. Tikhomirov.
Michael
Lönne
Mapping
Class Groups of Trigonal Loci
In
a joint project with Michele Bolognesi we study the topology of
the stack Tg
of
smooth trigonal curves of genus g
over
the
complex numbers. We make use of a construction by
Bolognesi and Vistoli, that describes Tg
as
a quotient stack of the
complement of a discriminant. This
allows us to use braid monodromy techniques to give presentations
of the
orbifold fundamental group of Tg,
of its substrata with prescribed Maroni invariant and describe
their relation with the
mapping class group Mapg
of
Riemann surfaces of genus g.
Mario
Maican
The
homology of the moduli spaces of plane sheaves of multiplicity 4
and 5
Slides
We
study the canonical torus action on the moduli spaces of
semi-stable plane sheaves supported on quartic or quintic
curves.
As an application, we compute the Hodge numbers of these moduli
spaces and other invariants.
Alessia
Mandini
Polygons
and hyperpolygons: A journey to moduli
spaces of parabolic Higgs bundles
In
this talk I will describe two classes of spaces, the polygon and
hyperpolygon spaces, that arise respectively as Kähler and
hyperkähler reduction. In particular, I will illustrate how
the geometrical structure of the polygon space M(α)
and of the
hyperpolygon space X(α)
depends upon the data of n
real
positive numbers, which are the entries of the “length vector”.
Along the way we will prove that the hyperpolygon space X(α)
is isomorphic to (certain) moduli spaces of parabolic
Higgs
bundles and give some applications of this result. This is joint
work with Godinho and with Biswas, Florentino and
Godinho.
Shoetsu
Ogata
Projective
normality of
toric weak Fano 3-folds
Slides
Let
X be a nonsingular toric 3-folds with nef and big
anti-canonical divisor. We show that all ample line bundles on X
are
normally generated. As an application, we show that on a
Calabi-Yau hypersurface of a toric Fano 4-fold any ample line
bundles are normally generated.
Armando
Sánchez Argáez
Natural
decompositions in closed subspaces of the space of representations
of discrete groups
Let
X
be
a compact Riemann surface of genus g
≥ 2,
Π
=
π1
(X)
and U (n)
the unitary group. Let Hom(Π,
U (n))
be the set of
representations of Π
on
U (n);
it is very well known that it has a natural structure of real
algebraic variety. Let Ξ
be
an arbitrary
subgroup of Π,
we define the equivalence relation in Hom(Π,
U (n))
given by ρ1
∼Ξ
ρ2
if
and only if ρ1|Ξ
=
ρ2|Ξ.
Now, the partition
FΞ
defined
by ∼Ξ
determines
a decomposition by closed sets in Hom(Π,
U (n));
this partition is stable under the action by
conjugation, hence
defines in the moduli space UX(n)
of flat vector bundles a partition FΞ
by
real algebraic closed subsets .
Given an etale Galois cover f:
Y
−→ X
with
Galois group G
we
set ΓY
=
[f#(π1(Y
)),
f#(π1(Y
))]
the
commutator group. Let
J
=
{ΓY
⊂ Π|f:
Y
−→ X
is
an etale Galois cover}. This is a direct system of subgroups of Π
and
defines a direct system of partitions
{FΓY
}ΓY
∈J
.
The purpose of the talk is to study the properties of the direct
limit FJ
:=
limΓY
∈JFΓY.
In this case we have a special fiber, we set WYn=[1]
ΓY
the
quotient of the closed set {ρ
∈ RX
−→ U
(n)|ρ
∼ΓY1}.
We can prove that
Theorem
1. WYn
is
naturally isomorphic to the moduli space parametrizing stable flat
vector bundles of rank n
on
the smooth
variety P ic1(Y
)/G.
One
of the problems posed is the study of the limit variety W(n)
= lim WYn.
The conjecture is that W(n)
is a fractal type object in
UX(n).
Laura
Schaposnik
Real
slices of the moduli space of Higgs bundles
Through
the hyperkähler structure of the moduli space of Higgs bundles
for complex groups we shall construct three natural
involutions
whose fixed points in the moduli space give branes in the A-model
and B-model with respect to each of the
complex structures.
Then, by means of spectral data, we shall look at these branes and
heir topological invariants, and
propose what the dual branes
should be under Langlands duality.
Adeel
Khan Yusufzai
Derived
categories and motives of varieties
An
important invariant of a smooth projective variety is its bounded
derived category of coherent sheaves. In another
direction one
can consider the Chow motive of the variety, which captures its
cohomological information in a universal way.
An important
theorem of D. Orlov says that the derived category determines the
rational Chow motive up to Tate twists. We
explain how this
result follows immediately by working in the framework of
noncommutative algebraic geometry as
envisioned by M.
Kontsevich, together with the theory of noncommutative motives
recently developed by G. Tabuada.
Berlin, September 1-5, 2014