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Convex and Algebraic Geometry
Organized by: Klaus Altmann (1), Victor V. Batyrev (2) and Bernard Teissier (3)
(1) Fachbereich Mathematik und Informatik, Freie Universität Berlin, Arnimallee 3, 14195, BERLIN, GERMANY(2) Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076, TÜBINGEN, GERMANY
(3) Equipe 'Geometrie et dynamique', Institut Mathématique de Jussieu, 175 rue du Chevaleret, F-75251, PARIS CEDEX, FRANCE
The workshop \emph{Convex and Algebraic Geometry} was organized by
Klaus Altmann (Berlin), Victor Batyrev (T\"ubingen), and Bernard
Teissier (Paris). Both title subjects meet primarily in the theory of
toric varieties. These constitute the part of algebraic geometry where
all maps are given by monomials in suitable coordinates, and all
equations are binomial. The combinatorics of the exponents of
monomials and binomials is sufficient to embed the geometry of lattice
polytopes in algebraic geometry. Thus, toric geometry and its several
generalizations provide a kind of section from polyhedral into
algebraic geometry. While this reflects only a thin slice of
algebraic geometry, it is general enough to display many important
phenomena, techniques, and methods. It serves as a wonderful testing
ground for general theories such as the celebrated mirror symmetry in
its different flavours. In particular, much of the popularity of toric
geometry originates in mathematical physics.
The meeting was attended by almost 50 participants
from many European countries, Canada, the USA, and Japan.
The program consisted of talks by 23 speakers, among them
many young researchers. Most subjects fit more or less into the following
main areas:
\topic{Derived categories, quivers, and (homological) mirror symmetry}
{Bondal, Craw, Horja, Maclagan, Perling, Siebert, Ueda}
One of the major discussions during the meeting concerned the
existence of strongly exceptional sequences on toric varieties which
consist of line bundles. A full
exceptional sequence provides a kind of ``basis'' for the derived
category. While Hille and Perling presented an example that does not
carry such a sequence of full length, Bondal suggested a method to
link this question to sheaves on the dual real torus that are
constructible with respect to a certain stratification.
In general, one expects to gain exceptional sequences from the
universal bundles on moduli spaces. Using this method, Craw constructs
those sequences on smooth toric Fano threefolds. In this context,
Maclagan and Ueda consider the case of three-dimensional abelian
quotient singularities. Ueda investigates the Fukaya category of the
corresponding potential on the dual torus explicitly.
Using mirror symmetry, Horja establishes a connection between the
orbifold $K$-theory of toric Deligne-Mumford stacks and solutions to
GKZ-hypergeometric $D$-modules.
\topic{Degenerations and deformations}
{Brown, Hausen, Siebert, S\"uss, Vollmert}
Gross and Siebert have developed a program to understand mirror
symmetry as the duality of certain degeneration data. The special
fibers split into toric components, and the degeneration is encoded in
a topological manifold $B$ with an affine and a polytopal structure.
Duality is now inherited from discrete geometry, and the topology of
$B$ reflects the topology of the general fiber. In particular, if $B$
is a ($\Q$-homology) $\PP^n_\C$, then this construction might lead to
(compact) Hyperk\"ahler varieties.
Considering, in a special case, a certain contraction of the total
space of these families leads to a description of torus actions on
algebraic varieties via divisors on their Chow quotients. These
divisors carry polytopes or even polyhedral complexes as their
coefficients, compare the talks of Hausen, S\"uss, and Vollmert. In a
similar setting, but with an explicit manipulation of Pfaffians, Brown
and Reid construct smoothings of certain non-isolated singularities
giving rise to four-dimensional flips.
\topic{Tropical geometry and Welschinger invariants}
{Itenberg, Shustin, Siebert}
The most rigorous degeneration of a variety is the tropical one. Here,
everything takes place over the so-called tropical semiring, and one
ends up with piecewise linear spaces. In fact, Siebert's degeneration
data mentioned above correspond to these objects.
Itenberg and Shustin use this approach to calculate the Welschinger
invariants, which are a kind of real version of Gromov-Witten
invariants. Along the lines of the method of Gathmann and Markwig,
there is a recursive formula for theses invariants. In the case of del
Pezzo surfaces, it turns out that both invariants are
(log-)asymptotically equivalent.
\topic{Commutative algebra, GKZ-systems, and polytopes}
{Bruns, Haase, Hering, Hor\-ja, Miller, Pasquier, Stienstra}
A generalization of toric varieties in a different direction from the
torus actions mentioned above is given by the notion of spherical
varieties. Pasquier considers horospherical Fano varieties and comes
up with an adapted notion of (generalized, coloured) reflexive
polytopes. Bruns, Haase, and Hering deal with ordinary polytopes and
their relations to syzygies of toric varieties.
For an integral matrix $A$ one obtains a semigroup algebra $\C[\N A]$
(leading to the usual affine toric variety) and a GKZ-hypergeometric
system of differential equations. The latter depends on a parameter
$\beta$, and Miller has reported on a result that relates the set of
$\beta$ where the rank of the system jumps to the set of those
multidegrees where the semigroup algebra $\C[\N A]$ carries local
cohomology. In particular, the Cohen-Macaulay property is equivalent
to the constant rank condition, answering an old question of
Sturmfels.
One of the nighttime discussions gave rise to the suggestion to not
include normality in the definition of a toric variety, thus
overcoming the cumbersome term of a ``not necessarily normal toric
variety''.
The workshop was closed on Friday night by an informal piano recital
by Benjamin Nill and Milena Hering featuring Strawinsky, Liszt, and
Chopin.
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