International Open ChineseRussian conference
«Torus Actions: Topology, Geometry and Number Theory»
September 2  7, 2013
Khabarovsk, Russia
ABSTRACTS OF PLENARY TALKS

Ivan Arzhantsev, Torus quotient presentations and automorphisms of varieties.
Torus actions play a key role in modern algebraic and arithmetic geometry.
It is well known that toric varieties can be described in terms of fans of
rational polyhedral cones. A generalization of this description to actions
of arbitrary complexity is given in [1], see also [5]. This approach allows
to study geometric properties of varieties combinatorially.
An alternative technique is a canonical quotient presentation of a variety
by a (quasi)torus action. This is related to the theory of Cox rings [7],
[2] and universal torsors [9], and leads to the Galedual version of toric
geometry. Moreover, if cones correspond to local charts, the quotient
presentation realizes the variety globally.
We plan to give an overview of recent results in these directions and to
discuss applications to the study of automorphisms [3], [4], [6]. An
important combinatorial ingredient is the concept of Demazure's roots [8].
The work is partially supported by the Ministry of Education and Science of
Russian Federation, project 8214, and RFBR grant 120100704a.
References:
 Altmann K., Hausena J. Polyhedral divisors and algebraic torus actions
// Math. Ann. 334 (2006), no. 3, 557607.
 Arzhantsev I., Derenthal U., Hausen J., Laface A. Cox rings
// Cambridge University Press, to appear; arXiv:1003.4229; see also authors'
webpages.
 Arzhantsev I., Flenner H., Kaliman S., Kutzschebauch F., Zaidenberg M.
Flexible varieties and automorphism groups // Duke Math. J. 162 (2013),
no. 4, 767823.
 Arzhantsev I., Hausen J., Herppich E., Liendoa A. The automorphism
group of a variety with torus action of complexity one // arXiv:1202.4568.
 Arzhantsev I., Liendo A. Polyhedral divisors and $SL_2$actions on affine
$T$varieties // Michigan Math. J. 61 (2012), no. 4, 731762.
 Arzhantsev I., Perepechko A., Sub H. Infinite transitivity on universal
torsors // arXiv:1302.2309.
 Cox D.A. The homogeneous coordinate ring of a toric variety //
J. a Alg. Geometry 4 (1995), no. 1, 1750.
 Demazure M. Sousgroupes algebriques de rang maximum du groupe de Cremona
// Ann. Sci. Ecole Norm. Sup. (4) 3 (1970), 507588.
 Skorobogatova A.N. Torsors and rational points // Cambridge Tracts
in Mathematics 144, Cambridge University Press, Cambridge, 2001.
 Yaroslav Bazaikin,
Complete Riemannian Metrics with Holonomy Group $G_2$
on Deformations of Cones over $S^3\times S^3$.
We consider general class of $G_2$structure on cone over the space $M=S^3\times S^3$.
This structure corresponds to asymptotically conic metric which can be written as
$$
d\bar{s}^2=dt^2+\sum_{i=1}^3 A_i(t)^2 \left(\eta_i+\tilde{\eta_i}\right)^2
+\sum_{i=1}^3 B_i(t)^2 \left( \eta_i\tilde{\eta_i}\right)^2,
$$
where $\eta_i, \tilde{\eta_i}$ is the standard coframe of $1$forms on $S^3\times S^3$,
whereas the functions $A_i(t), B_i(t)$ define a deformation of the cone singularity.
Special initial conditions to functions $A_i(t), B_i(t)$ guarantee that the metric can be
extended to complete Riemannian metric on deformation of standard cone over $M$.
We investigate system of nonlinear ODE's obtained on this way and describe new solutions
which correspond to new $G_2$ holonomy metrics.
 Victor Buchstaber,
Problems and results in toric topology.
 Victor Bykovskii, Elliptic systems of sequences
and functions.
 Haibao Duan, Parallel tangency for immersions
into Euclidean spaces.
Let $M$ and $N$ be two smoothly immersed manifolds of codimension 2 in
2ndimensional Euclidean space. We study the following enumerative problem:
find the number of the pairs $(x,y)\in M\times N$ of points so that $Mx$
is parallel to $Ny$, where $Mx$ is the tangent space to $M$ at $x$.
 Fuquan Fang, Reflection groups, nonnegative
curvature and Tits geometry.
A reflection in a euclidean space (sphere) is one of the fundamental
notions of symmetry of geometric figures. It plays a central role in
Killing and Cartan's work on Lie algebra in 19th century. Reflections groups
on a hyperbolic space is important in hyperbolic geometry, and the first
example goes back to F. Klein and Poincare. In this talk I will present
 A complete classification of reflection groups and the equivariant
structures of complete non negatively curved manifolds.
 A complete classification of positively curved polar manifolds
of cohomogeneity at least 2, which is achieved partially based on Tits geometry.
(joint works with Karsten Grove and G. Thorbergsson).
 Alexandr Gaifullin, Coxeter groups, small covers,
and realisation of cycles.
The following classical question is due to Steenrod (late 1940s) and is
usually called the problem on realisation of cycles:
Given a topological space and an integral homology class of it, can we
realise this homology class as a continuous image of the fundamental class
of an oriented closed smooth manifold?
The classical approach to this problem is based on Thom's transversality
theorem and tools of algebraic topology. It was developed in 1950s1970s
by Thom, Milnor, Novikov, Buchstaber, and Sullivan.
Since 2007 the author has developed a new constructive approach to
Steenrod's problem based on an explicit construction for a manifold that
realises a multiple of the given homology class, see [1], [2]. This
construction involves rightangular Coxeter groups and special convex
polytopes called permutahedra. Using this approach the author managed to
prove that in every dimension $n$ there exists a manifold $M^n$ possessing
the following
Universal Realisation of Cycles (URC) property:
For each topological space $X$ and each homology class
$z\in H_n(X;\mathbb{Z})$, there is a finitesheeted covering $\widehat{M}^n$
of $M^n$, and a continuous mapping $f\colon\widehat M^n\to X$ such that
$f_*[\widehat{M}^n]=qz$ for some $q\ne 0$.
The talk will be devoted to the problem of studying of the class of all
URCmanifolds, i.e., manifolds that satisfy the URCcondition. We shall
find many examples of URCmanifolds among small covers of simple convex
polytopes. In particular, in dimension $4$ we shall present an example of
a hyperbolic (i.e. possessing a Riemannian metric of constant negative
sectional curvature) URCmanifold, namely, the universal Abelian cover of
the regular 120cell.
We shall also discuss the relation of URCmanifolds to simplicial volume.
Most of these results are published in [3].
References:
 A.A. Gaifullin. Explicit construction of manifolds realising prescribed
homology classes // Russian Math. Surveys, V. 62, Issue 6, 2007.
 A.A. Gaifullin. The manifold of isospectral symmetric tridiagonal
matrices and realization of cycles by aspherical manifolds // Proceedings
of the Steklov Institute of Mathematics, V. 263, 2008.
 A.A. Gaifullin. Universal realisators for homology classes // Geometry
& Topology, V. 17, Issue 3, 2013.
 Hiroaki Ishida, Complex manifolds with maximal torus
actions and several similarities between them and toric varieties.
When a compact torus $G$ acts on a connected smooth manifold $M$ effectively,
one can see that $\dim G+\dim G_x \leq \dim M$ for any $x \in M$. For example,
if $M$ has a $G$fixed point then $2\dim G \leq \dim M$ and if the $G$action
is locally free (that is, the isotropy subgroup is finite at each point) then
$\dim G \leq \dim M$. Motivated by this fact, we say that an effective action
of a compact torus $G$ on a connected smooth manifold $M$ is \emph{maximal}
if there exists a point $x \in M$ such that $\dim G+\dim G_x = \dim M$.
In [3], Panov and Ustinovsky have shown that the momentangle manifold of a
starshaped sphere admits a complex structure invariant under the natural
torus action if it is even dimensional. It is easy to see that the complex
structures on momentangle manifolds constructed by Panov and Ustinovsky are
invariant under the natural torus actions. So, the momentangle manifolds are
typical examples of complex manifolds with maximal torus actions. In [2],
it has been shown that if $M$ is a compact complex manifold of complex
dimension $n$ with an $(S^1)^n$action having a fixed point, then $M$ is
equivariantly biholomorphic to a nonsingular complete toric variety. In [1],
it has been shown that compact complex manifolds with maximal torus action
can be described in terms of fans and complex vector (sub)spaces, like toric
varieties.
I would like to explain this story and discuss their similar properties to
toric varieties. Especially, we focus on their plurigenera and Lie algebras
of infinitesimal automorphisms.
References:
 H. Ishida, Complex manifolds with maximal torus actions, preprint,
available at arXiv:1302.0633.
 H. Ishida and Y. Karshon, Completely integrable torus actions on
complex manifolds with fixed points, to appear in Math. Res. Lett.,
available at arXiv:1203.0789.
 T. Panov and Y. Ustinovsky, Complexanalytic structures on momentangle
manifolds, Mosc. Math. J. 12 (2012), no. 1, 149172.
 Askold Khovanskii,
Universal Grobner basis and toric compactifications.
Let $A\subset\Bbb Z^n$ be the set of exponents of a finite set $\chi_A$ of characters of the
torus $(C^*)^n$. An algebraic variety $X\subset\Bbb (C^*)^n$ is called {\it $A$variety} if it
can be definite by a system $P_1=\dots=P_k=0$ where all Laurent polynomials $P_i$ are linear
combinations of the characters from $\chi_A$ (we do not assume that $k=n\dim X$). A toric
compactification $M\supset (C^*)^n$ is called {\it $A$toric compactification} if for any
$A$variety $X\subset (C^*)^n$ the closure $\overline X\subset M$ does not intersect any orbit
$O\subset M$ satisfying the inequality $\dim O+\dim X<n$.
{\bf Theorem 1.}{\it For any finite set $A\subset\Bbb Z^n$ there exists a smooth $A$toric
compactification of $(C^*)^n$.}
Our proof of the theorem 1 relays on the Grobner basis technique. Consider an ideal $I$ in the ring
$\Bbb C[\bold x]$ of polynomials in $n$ variables. A finite subset $G\subset I$ is call an {\it
universal Grobner basis} of the ideal $I$ if for any Grobner ordering $\prec$ of the semigroup
$\Bbb Z^n_{\geq 0}$ a Grobner basis of $I$ with respect to the ordering $\prec$ can be chosen from
the subset $G\subset I$.
{\bf Theorem 2.}{\it For any ideal $I\subset \Bbb C[\bold x]$ there exists a universal Grobner
basis $G$ in $I$.}
Theorem 2 is known. In the talk I will present its proof (a simplified version of the proof from
[1]). I also will sketch of a proof of the theorem 1.
The ring of conditions for an $n$dimensional reductive group $G$ is a version of the Chow ring
for $G$ (see [2]). In a work in progress [3] we found two very different models for the ring of
conditions for $(C^*)^n$. One of the models deals with Tropical Geometry, another one with Mixed
Volumes. An isomorphism between those models has nontrivial geometrical corollaries. We discover
the theorem 1 and rediscover the theorem 2 thinking about the ring of conditions for $(C^*)^n$.
References:
 B.Ya. Kazarnovskii, A.G. Khovanskii. Universal Groebner basis // Proceedings of the
International Conference on Polynomial Computer Algebra, Saint Peterburg. 2011. 6569.
 C. De Concini. Equivariant Embeddings of Homogeneous Spaces // Proceedings of the International
Congress of Mathematicians Berkeley, California, USA, 1986. 369377.
 B.Ya. Kazarnovskii, A.G. Khovanskii. The ring of condition for $(C^*)^n$, tropical geometry and
mixed volumes // Preprint in preparation, 2013, 41 pp.
 Zhi Lu, Generalized configuration spaces.
We consider two kinds of generalized configuration spacesorbit configuration
space and graphic configuration space. We first investigate the orbit
configuration spaces of some equivariant closed manifolds over simple
convex polytopes in toric topology, such as small covers, quasitoric
manifolds and (real) momentangle manifolds; especially for the cases of
small covers and quasitoric manifolds. These kinds of orbit configuration
spaces are all nonfree and noncompact, but still built via simple convex
polytopes. Our purpose is to explore the essential link between topology
and geometry of these kinds of orbit configuration spaces and combinatorics
of simple convex polytopes.
We obtain an explicit formula of Euler characteristic for orbit configuration
spaces of small covers and quasitoric manifolds in terms of the $h$vector
of a simple convex polytope. As a byproduct of our method, we also obtain a
formula of Euler characteristic for the classical configuration space, which
generalizes the F\'elixThomas formula.
In addition, we also study the homotopy type of such orbit configuration
spaces. In particular, we determine an equivariant strong deformation
retract of the orbit configuration space of 2 distinct orbitpoints in a
small cover or a quasitoric manifold, which turns out that we are able to
further study the algebraic topology of such an orbit configuration space by
using the MayerVietoris spectral sequence. Second, we also study the
graphic configuration space, and will talk about some results.
(Joint work with Junda Chen and Jie Wu).
 Dmitry Millionshchikov, Cohomology of Nilmanifolds:
Computation and Applications.
 Andrey Mironov, Minimal Lagrangian tori in CP2.
We associate a periodic twodimensional Schrodinger operator to every
Lagrangian torus in CP2 and define the spectral curve of a torus as the
Floquet spectrum on this operator on the zero energy level. In this event
minimal Lagrangian tori correspond to potential operators. We show that
the NovikovVeselov hierarchy of equations induces integrable deformations
of a minimal Lagrangian torus in CP2 preserving the spectral curve.
 Jianzhong Pan, Rational homotopy of manifolds.
 Taras Panov, Homotopy theory of momentangle complexes.
 Andrey Raigorodskii, Problems of splitting
of polyhedrons into parts of smaller diameter.
 Dong Youp Suh, Torus cobordism of lens spaces.
Let $T^n$ be the real $n$torus group. We show that any $3$dimensional lens
space $L(p;q)$ is $T^2$equivariantly cobordant to zero. We also give some
sufficient conditions for higher dimensional lens spaces
$L(p; q_1, \ldots, q_n)$ to be $T^{n+1}$equivariantly cobordant to zero
using toric topological arguments.
 Xuezhi Zhao, The free degree of a space.
The free degree of a topological space $X$ is the minimum positive
integer $n$ such that any selfhomeomorphism on $X$ has a periodic
point with period less or equal to $n$. We shall talk about the role
of this concept and some methods to consider the varies of free degrees.