ISAAC 2015
Session 5: Complex Geometry
Titles and Abstracts
Nikolai
Beck
Principal
bundles with local decorations
A
local
decoration
on
a principal bundle over a smooth projective curve is a point in an
associated projective bundle. When the
associated bundle is
constructed from a generalized flag variety, a bundle with a local
decoration is just a parabolic principal bundle.
When the
associated bundle is constructed using the wonderful
compactification of the structure group, one obtains a principal
bundle
with a level structure. In this talk, we present a
unified approach to defining stability conditions for these
objects and to the construction of
the moduli space of
semistable objects. For parabolic principal bundles and vector
bundles with a level structure this procedure leads to the known
stability conditions.
Andrei
Bogatyrev
Period
mappings: combinatorial approach
->
Slides
<-
We
study a period mapping from the moduli space of real hyperelliptic
curves to the eucledian space. Such mappings arise in various
applications, in particular in approximation theory and
electrical filters design. We introduce a piecewise-linear
structure (decomposition
into polyhedra) of the moduli space
described in terms of planar graphs. This approach allows e.g. to
reconstruct the global topology of
low dimensional fibers of
the period mapping .
Cinzia
Casagrande
Parabolic
vector bundles on IP1
and
the variety of linear spaces contained in two odd-dimensional
quadrics
Let
g≥2
be an integer, set n:=2g-2,
and let Z⊂IPn+2
be
a smooth complete intersection of two quadrics. Let us consider
the variety G
of
(g-2)-planes
in IPn+2
contained
in Z;;
it is
well-known that G
is
a smooth, n-dimensional
Fano variety with b2(G)=n+4.
We will talk about
the relation between G,
the blow-up X
of
IPn
in
n+3
general points, and the moduli space N
of
stable parabolic vector bundles,
of rank 2 and degree 0 on IP1
with
n+3
marked points (with weights {½,...,½}). The relation among
X
and
N
has
been established
by Bauer, who showed that there is a (
finite)
sequence of flips N
-->X.
Concerning G,
we will explain the following.
Theorem
1. Let
p1,...,pn+3
∈IP1
be
distinct points; assume that pj
=
(λj:1)
with
λj
∈k.
Consider IPn+2
with
homogeneous coordinates (x1:⋯:xn+3),
and let Q1
and
Q2
denote
the following quadrics:
Q1 : ∑xij2=0; Q2 : ∑λjxij2=0:
Then
the moduli space N
of
stable parabolic vector bundles of rank 2 and degree 0 on
(IP1;p1,...,pn+3
)
is isomorphic to the variety
G
of
(g-2)-planes
in IPn+2
contained
in Q1\Q2
.
Summing-up,
we have three di fferent descriptions for the same Fano variety:
an embedded description in a grassmannian, a modular
description
via parabolic vector bundles on IP1,
and a birational description as the unique Fano small modication
of the blow-up of IPn
in
n+3
points.
Victoria
Hoskins
Algebraic
symplectic varieties via non-reductive geometric invariant theory
We
give a new construction of algebraic symplectic varieties by
taking a non-reductive algebraic symplectic reduction of the
cotangent lift of an action of the additive group on an affine
space. The quotient is taken using techniques of geometric
invariant
theory (GIT) for non-reductive groups developed by
Doran and Kirwan. For a linear action of the additive group on an
affine space
over the complex numbers, the non-reductive GIT
quotient is isomorphic to a reductive affine GIT quotient;
however, we show that
the corresponding non-reductive and
reductive algebraic symplectic reductions are not isomorphic, but
rather birationally
symplectomorphic.
Jun-Muk
Hwang
Webs
of algebraic curves
A
family of algebraic curves covering a projective variety X
is
called a web of curves on X
if
it has only finitely many members through
a general point of
X.
A
web of curves on X
induces
a web-structure, in the sense of local differential geometry, in a
neighborhood of a
general point of X.
We will discuss the relation between the local differential
geometry of the web-structure and the global algebraic
geometry
of X.
Priska
Jahnke
Submanifolds
with splitting tangent sequence
A
theorem of Van de Ven states that a projective submanifold of
complex projective space whose holomorphic tangent bundle
sequence splits holomorphically is necessarily a linear
subspace. Note that the sequence always splits differentiably but
in general not
holomorphically.
We are interested in
generalizations to the case when the ambient space is a
homogeneous manifold different from projective space:
quadrics,
Grassmannians or abelian manifolds, for example. Split
submanifolds are closely related to totally geodesic submanifolds
.
Wenfei
Liu
Automorphisms
of surfaces of general type acting trivially on cohomology
→
Slides
←
In
studying the automorphism group of a compact complex manifold it
is very natural to look at its induced action on the cohomology
ring. While it is well known that, for a curve of genus
greater than one, this action on cohomology is faithful, there are
indeed surfaces
of general type with (non-trivial)
automorphisms acting trivially on the whole cohomology ring. In
this talk I will give an optimal bound
on the group of such
automorphisms for irregular surfaces. This is joint work with J
Cai and L Zhang
Ngaiming
Mok
Analytic
continuation of germs of submanifolds on uniruled projective
manifolds inheriting geometric substructures
In
the late 1990s, Hwang and Mok introduced a geometric theory of
uniruled projective manifolds X
modeled
on their varieties of
minimal rational tangents (VMRTs), which
are the collections CxIPTx(X)
of projectivizations of vectors tangent to minimal rational
curves passing through a general point x∈X.
A central result was the Cartan-Fubini extension principle
established by Hwang-Mok in
2001, according to which a germ of
VMRT-preserving biholomorphism
f:
(X;x0)(Y;y0)
between two Fano manifolds of Picard number 1
extends
necessarily to a biholomorphism F:XY
whenever
X
and
Y
are
of Picard number 1, their VMRTs at general points are of positive
dimension, and their Gauss maps are generically finite. In
2010 Hong-Mok extended Cartan-Fubini extension to the
non-equidimensional
case under a certain relative
nondegeneracy condition on second fundamental forms. This leads to
characterization results of standard
embeddings between
certain rational homogeneous spaces G/P
of
Picard number 1. In 2015, in a joint work with Y. Zhang, we
considered
the problem of analytic continuation of germs of
submanifolds (S;x0)(X;x0)
of uniruled projective manifolds (X;K)
of Picard number 1 under
the assumption that S
inherits
a sub-VMRT structure defined by intersections of VMRTs with
projectivized tangent spaces. We established a
principle of
analytic continuation of subvarieties, viz., SZ
for
some projective subvariety ZX,dim(Z)
= dim(X),
by constructing a universal
family of chains of rational
curves by an analytic process and proving its algebraicity by
establishing a Thullen-type extension theorem on
paramentrized
families of sub-VMRT structures along chains of rational curves,
under a new nondegeneracy condition on second
fundamental
forms, a bracket generating condition on distributions spanned by
sub-VMRTs, and the assumption that members of (X;K)
are rational curves of degree 1.
Daniel
Sage
Minimal
K-types for flat G-bundles, moduli spaces, and isomonodromy
->
Slides
<-
In
this talk, I describe a new approach to the study of flat
G-bundles
on curves (for complex reductive G)
using methods of
representation theory. This approach is based
on a geometric version of the Moy-Prasad theory of minimal K-types
(or fundamental
strata) for representations of p-adic
groups. In the geometric theory, one associates a fundamental
stratum - data involving appropriate
filtrations on loop
algebras - to a formal flat G-bundle.
Intuitively, this stratum plays the role of the “leading term"
of the flat G-bundle
and can be used to define its slope, an
invariant measuring the degree of irregularity of the connection.
I will explain how these ideas
can be used to generalize work
of Jimbo, Miwa, and Ueno and Boalch to meromorphic connections on
P1
with
irregular singularities
that are not necessarily formally
diagonalizable. In particular, the isomonodromy equations for such
connections can be exhibited as
an explicit integrable system
on a suitable Poisson moduli space of connections; this moduli
space is defined automorphically in terms
of manifolds
encoding local data. This is joint work with C. Bremer.
Yum-Tong
Siu (Pleanary Talk)
The
past, present, and future of the theory of multiplier ideal
sheaves
Since
the late nineteen seventies multiplier ideal sheaves have been a
very powerful tool in the PDE approach to the theory of
several
complex variables, in Kaehler geometry, and in
algebraic geometry. This talk will discuss the background of the
theory of multiplier ideal
sheaves, recent results, and open
problems.
Marco
Spinaci
Maximal
holomorphic representations of Kähler groups
In
this talk we introduce the notion of Toledo invariant τ
for
representations of the fundamental group of a higher dimensional
Kähler
manifold X
into
a non-compact Lie group of Hermitian type G.
A variant of this definition has been given by Burger and Iozzi
using
bounded cohomology techniques, in the setting of complex
hyperbolic lattices. In this setting, a version of the Milnor-Wood
inequality
is available, bounding the maximal possible value
of |τ|.
It is conjectured that maximal representations (that is, the ones
that realize
the equality in the Milnor-Wood inequality) are
very rigid, essentially reducing to a “diagonal" embedding.
Using a theorem of Royden,
the above Milnor-Wood inequality
easily generalizes to the Kähler setup if one makes the further
assumption of the existence of a
holomorphic equivariant map
from the universal cover X~
of
X
to
the (Kähler) symmetric space associated to G.
We classify the maximal
ones among these “holomorphic"
representations, that turn out to be rigid. In particular, it
follows that, to prove the rigidity conjecture
for maximal
representations of cocompact complex hyperbolic lattices, it is
enough to prove that every maximal representation can be
deformed
to one admitting a holomorphic equivariant map. The proof is based
on Higgs bundles techniques; we will overview how the
concepts
introduced in this talk can be effectively restated in Higgs
bundles terms in order to give easy proofs of some
statements.
Sheng-Li
Tan
Chern
numbers and algebraic integrability of a holomorphic foliation
The
birational classification of holomorphic foliations F
on
compact complex surfaces is almost completed by using the Kodaira
dimensions. In order to get the biregular classification, we
introduce the Chern numbers c12(F),
c2(F)
and χ(F)
for a holomorphic foliation
F,
which are nonnegative rational numbers satisfying Noether's
equality c12(F)+c2(F)=12χ(F).
These Chern numbers are birational invariants,
and c12(F)=0
iff F
is
not of general type. If the foliation F
is
algebraically integrable, then these invariants are exactly the
modular Chern
numbers of the family of curves defined by the
rational first integral. These kinds of global invariants of
differential equations were desired by
Poincaré
and
Painlevé
in
the 19th century. As an application, we will give positive answers
to the problems of Poincaré
and
Painlevé
on
the
algebraic integrability of some foliations with small
slopes s(F)=
c12(F)/χ(F).
We will also discuss the behavior of the pluricanonical systems of
foliations of general type.
Malte
Wandel
Induced
automorphisms on hyperkähler manifolds
In
this talk I will give an overview over recent developments in the
study of automorphisms of hyperkaehler manifolds. In particular, I
will
introduce the notion of induced automorphisms. This class
of automorphisms is strongly related to moduli spaces of sheaves
and has
proven to be a very useful tool to construct and
classify automorphisms of hyperkaehler manifolds. In the case of
manifolds of K3[2]-type,
the
classification statement will be made more precise. For the use
and the study of induced automorphisms it is of utmost importance
that we understand the kernel of the so-called cohomological
representation, which is a measure how much the geometry of our
manifolds is determined by their second integral cohomology.
Thus I will present two recent results concerning this kernel in
the cases of
O'Grady's sporadic example.
Graeme
Wilkin
Morse
theory on singular spaces
I
will describe how to construct a Morse theory for a certain class
of functions on singular spaces with applications to questions
about the
topology of symplectic quotients of varieties.
Macao, August 3-8, 2015
5