VBAC 2014

Algebraic Varieties:
Bundles, Topology, Physics







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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 K
3 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 2
g-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 b23.

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 R0 '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 Map
g 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 g2, 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 ρ1Ξ 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 U
X(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
= {ΓYΠ|f: Y −→ X is an etale Galois cover}. This is a direct system of subgroups of Π and defines a direct system of partitions
{F
ΓY }ΓYJ . The purpose of the talk is to study the properties of the direct limit FJ := limΓYJFΓY. 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 ic
1(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