ABSTRACTS OF LECTURES
ARITHMETIC ALGEBRAIC GEOMETRY 2016
JANUARY 25 - 28, UNIVERSITY OF TOKYO
Shingo Taki (Tokai Univ.)
On uniqueness of K3 surfaces with non-symplectic automorphisms
Abstract. We treat that K3 surfaces with a non-symplectic automorphism whose
the value of Euler’s totient function of the order is 12 or more. It is know that these
moduli spaces are 0-dimensional. In this talk, we show whether these moduli spaces
are irreducible in some cases.
Hisanori Ohashi (Tokyo Univ. Sci.)
The automorphism groups of the Enriques surfaces covered by some
symmetric quartic K3 surfaces
Abstract. The automorphism group of an Enriques surface is a discrete, usually
infinite group. It is finite in special cases and they are completely classified by Kondo
and Nikulin. Some other cases of infinite groups were found by Barth and Peters
(principal congruence subgroups, or the infinite dihedral groups). In the talk, we
introduce a new case for which we can understand Aut(S) completely. The group is
not virtually abelian, but the generators are geometrically explicit. This is a joint
work with S. Mukai.
Shouhei Ma(Tokyo Inst. Tech.)
Finiteness of 2-reflective lattices of signature (2, n)
Abstract. A modular form for an even lattice of signature (2, n) is said to be
2-reflective if its zero divisor is set-theoretically contained in the Heegner divisor
defined by the (-2)-vectors. We prove that there are only finitely many even lattices
with n > 6 which admit such modular forms, which was conjectured by Gritsenko
and Nikulin. In particular, there is no such lattice in n > 26.
Kazuhiko Yamaki (Kyoto Univ.)
The geometric Bogomolov conjecture for curves
Abstract. The Bogomolov conjecture for curves is a problem concerning finiteness
of points with small canonical height on curves over number fields or function fields.
Over number fields, this conjecture was proved in 1998 by Ullmo. Recently, we have
proved the conjecture over function fields, namely, the geometric Bogomolov conjecture for curves. Our proof consists of several parts, and we use non-archimedean
geometry in one of the most important parts.
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Takashi Taniguchi (Kobe Univ.)
Average rank of elliptic curves - The work of Manjul Bhargava Abstract. In a joint work with Shankar, Bhargava showed that the average rank
of elliptic curves over the rational number field, when ordered by height, is less
than 1 (in fact, less than .885). Moreover, they determined the average size of 2,
3, 4 and 5-Selmer groups for those elliptic curves. Their proof is based on powerful
new method in the geometry of numbers, dealing with the integer orbits of certain
representations of algebraic groups. This talk is an introduction to the theory.
Lin Weng (Kyushu Univ.)
Arithmetic central extensions and tame reciprocity laws for arithmetic
surfaces
Abstract. About 50 years ago, Tate developed a theory of residues of r curves
using traces and adelic cohomologies. In 1989, Tate’s work was integrated with the
K2 central extensions by Arbarello-De Concini-Kac. Later, in 2003, Osipov, based
on Kapranov’s dimension theory, constructed dimension two central extensions and
hence established the reciprocity law for algebraic surfaces using Parshin’s adelic
theory.
In essence, Osipov’s construction may be viewed as K2 type theory of central
extensions developed by Brylinski-Deligne. However, for arithmetic surfaces, K2
theory does not work. Instead, we first develop a new theory of central extensions
based on Arakelov theory, then we use our adelic cohomology theory to establish
the reciprocity law for arithmetic surfaces.
Denis Osipov (Steklov Inst.)
Higher Contou-Carrere symbol and tangent space to Milnor K-groups
of rings
Abstract. The higher Contou-Carrere symbol is a universal deformation of the
higher-dimensional tame symbol. I will describe the various fundamental properties
of the higher Contou-Carrere symbol and give an explicit formula for this symbol
restricted to Q-algebras. These results are based on calculation of the tangent space
of Milnor K-theory of rings.
The talk is based on joint papers with S. Gorchinskiy arXiv:1505.03829 and
arXiv:1505.03780.
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Shu Kawaguchi (Doshisha Univ.)
Effective faithful tropicalizations associated to linear systems on curves
Abstract. For a smooth projective curve X of genus g ≥ 2, global sections of any
line bundle L with deg(L) ≥ 2g + 1 give an embedding of the curve into projective space. We consider an analogous statement for a Berkovich skeleton in nonarchimedean geometry, in which an embedding is replaced by a homeomorphim
onto its image in tropical projective space preserving the integral structures (called
a faithful tropicalization). Suppose that X is defined over an algebraically closed
field which is complete with respect to a non-trivial non- Archimedean value and
that X an denotes the analytification of X in the sense of Berkovich. Let Γ be a
skeleton (that is allowed to have ends) of X an . Under the assumption that X, Γ,
and L are defined over a discrete valued subfield, we show that, if deg(L) ≥ 5g − 4,
then global sections of L give a faithful tropicalization of Γ into tropical projective
space. This is a joint work with Kazuhiko Yamaki.
Yoshinori Gongyo (Univ. Tokyo)
Rational points on log Fano threefolds over a finite field
Abstract. We discuss the number of rational points on log Fano threefolds over
a finite field from the geometric side. In particular, we prove the W O-rationality
of klt threefolds and the rational chain connectedness of klt Fano threefolds over a
perfect field of characteristic p > 5. As a consequence, any klt Fano threefold over
a finite field has a rational point.
Hélène Esault (Freie Univ. Berlin)
Some aspects of the p-curvature conjecture
Abstract. We explain Grothendieck p-curvature conjecture, recall the main results so far by Katz, Chudnosvky, Andre-Hrushovski, and, by reducing the conjecture to smooth projective curves over number fields, show, under some assumption
stronger that the vanishing of the p-curvature, that Z-polarized variations of Hodge
structures satisfy the conjecture.
This is joint work with Mark Kisin.
Chen Jiang (Univ. Tokyo)
Birational boundedness of singular log Fano threefolds
Abstract. A normal projective variety X is said of ϵ-Fano type if there exists
a boundary B such that (X, B) is a log Fano pair with ϵ-klt singularities. The
boundedness of varieties of ϵ-Fano type, which is conjectured by Borisov-AlexeevBorisov, is one of the most interesting problems in birational geometry. In this talk,
we show that the family of 3-folds of ϵ-Fano type is birationally bounded.
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Adrian Langer (Univ. Warsaw)
Bogomolov’s inequality for Higgs bundles and its applications
Abstract. I will talk about Bogomolov’s inequality for semistable Higgs bundles.
In characteristic zero this inequality was proven by Carlos Simpson using analytic
methods. At the beginning I will give a quick algebraic proof of this inequality using
Ogus-Vologodsky’s non-abelian Hidge theory in positive characteristic. Then I will
sketch some ideas allowing to generalize the result to positive characteristic. It is
known that the Bogomolov-Miyaoka-Yau inequality fails in positive characteristic.
Nevertheless, as an application of the above results I will show that the BogomolovMiyaoka-Yau inequality holds for surfaces of general type that are liftable modulo
p2 (also in the logarithmic case). I will also some other applications of the obtained
results related to lifting of curves on rational surfaces.
Hiroshi Iritani (Kyoto Univ.)
Mirror symmetry for toric stacks
Abstract. I will describe a partially compactified Landau-Ginzburg mirror for
toric Deligne-Mumford stacks. I will also describe its application to functoriality of
quantum cohomology.
Thomas Peternell (Univ. Bayreuth)
The Miyaoka-Yau inequality for singular varieties and uniformization of
canonical models
Abstract. I will report on recent joint work with D.Greb, S. Kebekus and B. Taji
in terms on the Miyaoka-Yau inequality in terms of orbifold Chern classes for the
tangent sheaf of a complex projective variety of general type with klt singularities
and nef canonical divisor. If the variety has at worst terminal singularities and
equality holds, then the associated canonical model is the quotient of the ball by a
discrete group.
Matthias Schütt (Hannover Univ.)
Gorenstein Ql -cohomology fake projective planes
Abstract. Inspired by fake projective planes, this talk concerns normal projective
surfaces in odd characteristic whose etale cohomology equals that of P2 . I shall
concentrate on the case of rational double point singularities and numerically trivial
canonical bundle. The characteristic zero setting which has also been investigated
by Hwang-Keum-Ohashi is closely related to Enriques surfaces. In my talk I will lay
out the subtleties and novelties featuring in odd characteristic. Time permitting, I
will also discuss partial results in characteristic two.
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Dulip Piyaratne (Univ.Tokyo, IPMU)
Polarization and stability conditions on a derived equivalent abelian
variety
Abstract. In this talk I will explain how one can define a polarization on a derived
equivalent abelian variety by using Fourier-Mukai theory. Furthermore, we see how
such a realisation is connected with stability conditions on their derived categories.
In particular, I will discuss these ideas for abelian surfaces and abelian 3-folds in
detail.
Ken-Ichi Yoshikawa (Kyoto Univ.)
Analytic torsion for K3 surfaces with involution
Abstract. A holomorphic torsion invariant of K3 surfaces with involution was
introduced by the speaker in 2004. Its automorphy on the moduli space and its
explicit formula on some components of the moduli space have been known so far.
Very recently, in the joint work with Shouhei Ma, we determine the structure of
the invariant for all components of the moduli space of K3 surfaces with involution.
In our talk, we will explain the construction of the (twisted) holomorphic torsion
invariant and its explicit formula as automorphic function on the moduli space. If
time allows, we will explain its application to the determinant of period integrals of
plane curves of small degree.