代数幾何学セミナー

開催情報 火曜日　10:30～11:30 or 12:00　数理科学研究科棟(駒場) ハイブリッド開催/002号室 權業 善範・中村 勇哉・田中公

2012年10月15日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室

On the moduli b-divisors of lc-trivial fibrations (JAPANESE)
[ 講演概要 ]
Roughly speaking, by using the semi-stable minimal model program, we prove that the moduli part of an lc-trivial fibration coincides with that of a klt-trivial fibration induced by adjunction after taking a suitable generically finite cover. As an application, we obtain that the moduli part of an lc-trivial fibration is b-nef and abundant by Ambro's result on klt-trivial fibrations. Moreover I may explain some applications of canonical bundle formulas. These are joint works with Osamu Fujino.

2012年10月01日(月)

13:30-15:00   数理科学研究科棟(駒場) 122号室
Robert Laterveer 氏 (CNRS, IRMA, Université de Strasbourg)
Weak Lefschetz for divisors (ENGLISH)
[ 講演概要 ]
Let $X$ be a complex projective variety (possibly singular), and $Y\\subset X$ a generic hyperplane section. We prove several weak Lefschetz results concerning the restriction $A^1(X)_{\\qq}\\to A^1(Y)_{\\qq}$, where $A^1$ denotes Fulton--MacPherson's operational Chow cohomology group. In addition, we reprove (and slightly extend) a weak Lefschetz result concerning the Chow group of Weil divisors first proven by Ravindra and Srinivas. As an application of these weak Lefschetz results, we can say something about when the natural map from the Picard group to $A^1$ is an isomorphism.

2012年10月01日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室

Frobenius morphisms and derived categories on two dimensional toric Deligne--Mumford stacks (JAPANESE)
[ 講演概要 ]
For a toric Deligne-Mumford (DM) stack over the complex number field, we can consider a certain generalization of the Frobenius endomorphism. For such an endomorphism of a two-dimensional toric DM stack, we show that the push-forward of the structure sheaf generates the bounded derived category of coherent sheaves on the stack. This is joint work with Hokuto Uehara.

2012年07月30日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
Gianluca Pacienza 氏 (Université de Strasbourg)
Log Bend-and-Break on Deligne-Mumford stacks (ENGLISH)
[ 講演概要 ]
We prove a logarithmic Bend-and-Break lemma on a LCI Deligne-Mumford stacks with projective moduli space and integral boundary divisor. As a by-product we obtain a logarithmic version of the Miyaoka-Mori numerical criterion of uniruledness for DM stacks (under additional conditions on the boundary and on the non-schematic locus) and a Cone Theorem for Deligne-Mumford stacks with boundary. These results hold on an algebraically closed field of any characteristic. This is joint work with Michael McQuillan.

2012年07月23日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室

Derived category of smooth proper Deligne-Mumford stack with p_g>0 (JAPANESE)
[ 講演概要 ]
Semiorthogonal decomposition (SOD) of the derived category of coherent sheaves reflects interesting geometry of varieties (more generally stacks), such as minimal model program. We show that the global sections of the canonical line bundle (if exists) give restrictions on the possible form of SODs. As a special case, we see that the global generation of the canonical line bundle implies the non-existence of SODs. (joint work with Kotaro Kawatani)

2012年06月25日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室

Automorphism groups of Calabi-Yau manifolds of Picard number two (JAPANESE)
[ 講演概要 ]
We prove that the automorphism group of an odd dimensional Calabi-Yau manifold of Picard number two is always a finite group. This makes a sharp contrast to the automorphism groups of K3 surfaces and hyperk\\"ahler manifolds and birational automorphism groups, as I shall explain. We also clarify the relation between finiteness of the automorphism group (resp. birational automorphism group) and the rationality of the nef cone (resp. movable cone) for a hyperk\\"ahler manifold of Picard number two. We will also discuss a similar conjectual relation for a Calabi-Yau threefold of Picard number two, together with exsistence of rational curve, expected by the cone conjecture.

2012年06月18日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室

アルバネーゼ次元最大の複素射影多様体の特殊集合について (JAPANESE)
[ 講演概要 ]
アルバネーゼ次元が最大の複素射影多様体の中に含まれる代数的あるいは超越的な複

2012年06月14日(木)

13:30-15:00   数理科学研究科棟(駒場) 122号室
Christian Schnell 氏 (IPMU)
Vanishing theorems for perverse sheaves on abelian varieties (ENGLISH)
[ 講演概要 ]
I will describe a few results, due to Kraemer-Weissauer and myself, about perverse sheaves on complex abelian varieties; they are natural generalizations of the generic vanishing theorem of Green-Lazarsfeld.

2012年06月04日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室

Smooth P1-fibrations and Campana-Peternell conjecture (ENGLISH)
[ 講演概要 ]
We give a complete classification of smooth P1-fibrations
over projective manifolds of Picard number 1 each of which admit another
smooth morphism of relative dimension one.
Furthermore, we consider relations of the result with Campana-Peternell conjecture
on Fano manifolds with nef tangent bundle.

2012年05月28日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
Mihnea Popa 氏 (University of Illinois at Chicago)
Generic vanishing and linearity via Hodge modules (ENGLISH)
[ 講演概要 ]
I will explain joint work with Christian Schnell, in which we extend the fundamental results of generic vanishing theory (for instance for the canonical bundle of a smooth projective variety) to bundles of holomorphic forms and to rank one local systems, where parts of the theory have eluded previous efforts. To achiever this, we bring all of the old and new results under the same roof by enlarging the scope of generic vanishing theory to the study of filtered D-modules associated to mixed Hodge modules. Besides Saito's vanishing and direct image theorems for Hodge modules, an important input is the Laumon-Rothstein Fourier transform for bundles with integrable connection.

2012年05月21日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室

Characterizations of projective spaces and hyperquadrics
(JAPANESE)
[ 講演概要 ]
After Mori's works on Hartshorne's conjecture, many results to
characterize projective spaces and hyperquadrics in terms of
positivity properties of the tangent bundle have been provided.
Kov\\'acs' conjecture states that smooth complex projective
varieties are projective spaces or hyperquadrics if the $p$-th
exterior product of their tangent bundle contains the $p$-th
exterior product of an ample vector bundle. This conjecture is
the generalization of many preceding results. In this talk, I will
explain the idea of the proof of Kov\\'acs' conjecture for varieties
with Picard number one by using a method of slope-stabilities
of sheaves.

2012年05月07日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室

Algebro-geometric characterization of Cayley polytopes (JAPANESE)
[ 講演概要 ]
A lattice polytope is called a Cayley polytope if it is "small" in some
sense.
In this talk, I will explain an algebro-geometric characterization of
Cayley polytopes
by considering whether or not the corresponding polarized toric
varieties are covered by lines, planes, etc.

We can apply this characterization to the study of Seshadri constants,
which are invariants measuring the positivity of ample line bundles.
That is, we can obtain an explicit description of a polarized toric
variety whose Seshadri constant is one.

2012年04月23日(月)

17:10-18:40   数理科学研究科棟(駒場) 122号室
いつもと時間が違います。

Motivic integration and wild group actions (JAPANESE)
[ 講演概要 ]
The cohomological McKay correspondence proved by Batyrev is the equality of an orbifold invariant
and a stringy invariant. The former is an invariant of a smooth variety with a finite group action and the latter is
an invariant of its quotient variety. Denef and Loeser gave an alternative proof of it which uses the motivic integration theory developped by themselves.
Then I pushed forward with their study by generalizing the motivic integration to
Deligne-Mumford stacks and reformulating the cohomological McKay correspondence from the viewpoint of
the birational geometry of stacks.
However all of these are about tame group actions (the order of a group is not divisible by the characteristic of the base field),
and the wild (= not tame) case has remained unexplored.
In this talk, I will explain my attempt to examine the simplest situation of the wild case. Namely linear actions of a cyclic group
of order equal to the characteristic of the base field are treated. A remarkable new phenomenon is that the space of generalized
arcs is a fibration over an infinite dimensional space with infinite dimensional fibers, where the base space is the space of
Artin-Schreier extensions of $k((t))$, the field of Laurent series.

2012年04月16日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室

Toric degenerations of minuscule Schubert varieties and mirror symmetry (JAPANESE)
[ 講演概要 ]
Minuscule Schubert varieties admit the flat degenerations to projective
Hibi toric varieties, whose combinatorial structure is explicitly
described by finite posets. In this talk, I will explain these toric
degenerations and discuss the mirror symmetry for complete intersection
Calabi-Yau varieties in Gorenstein minuscule Schubert varieties.

2012年04月09日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室

On mirror symmetry for weighted Calabi-Yau hypersurfaces (JAPANESE)
[ 講演概要 ]
In the talk, I will discuss relation between homological mirror symmetry for weighted projective spaces, their Calabi-Yau hypersurfaces, and weighted homogeneous singularities.
If the time permits, I will also discuss an application to monodromy of hypergeometric functions.

2012年01月30日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室

On varieties of globally F-regular type (JAPANESE)
[ 講演概要 ]
I will talk about recent topics on varieties of globally F-regular type.

2012年01月23日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室

Sigularities and Kodaira dimension of the moduli of stable sheaves on Enriques surfaces (JAPANESE)
[ 講演概要 ]
We shall estimate singularities of moduli of stable sheaves on Enriques/hyper-elliptic surfaces via the Kuranishi theory, consider when its singularities are canonical, and calculate its Kodaira dimension.

2012年01月16日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室

Mihai Paun 氏 (Institut Élie Cartan and KIAS)
TBA(中止になりました) (JAPANESE)

2011年12月26日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室

Birational unboundedness of Q-Fano varieties and rationally connected strict Mori fiber spaces (JAPANESE)
[ 講演概要 ]
It has been known that suitably restricted classes of Q-Fano varieties are bounded. I will talk about the birational unboundedness of (log terminal) Q-Fano varieties with Picard number one and of rationally connected strict Mori fiber spaces in each dimension $¥geq 3$. I will explain the idea of the proof which will be done after passing to a positive characteristic.

2011年12月12日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室

Robert Laterveer 氏 (CNRS, IRMA, Université de Strasbourg)
A version of Barth's theorem for singular varieties (中止になりました) (JAPANESE)

2011年12月05日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室

Obstructions to deforming curves on a uniruled 3-fold (JAPANESE)
[ 講演概要 ]
In this talk, I review some results from a joint work with Mukai:
1. a generalization of Mumford's example of a non-reduced component of the Hilbert scheme, and
2. a sufficient condition for a first order deformation of a curve on a uniruled 3-fold to be obstructed.
As a sequel of the study, we will discuss some obstructed deformations of degenerate curves on a higher dimensional scroll.

2011年12月01日(木)

16:30-18:00   数理科学研究科棟(駒場) 002号室
Dmitry Kaledin 氏 (Steklov Mathematics Institute/ KIAS)
Cyclic K-theory (ENGLISH)
[ 講演概要 ]
Cyclic K-theory is a variant of algebraic K-theory introduced by Goodwillie 25 years ago and more-or-less forgotten by now. I will try to convince the audience that cyclic K-theory is actually quite useful -- in particular, it can be effectively computed for varieties over a finite field.

2011年11月28日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室

Comparison with Gieseker stability and slope stability via Bridgeland's stability (JAPANESE)
[ 講演概要 ]
In this talk we compare two classical notions of stability (Gieseker stability and slope stability) for sheaves on K3 surfaces by using stability conditions which was introduced by Bridgeland. As a consequence of this work, we give a classification of 2 dimensional moduli spaces of sheaves on K3 surface depending on the rank of the sheaves.

2011年11月14日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室

On projective manifolds swept out by cubic varieties (JAPANESE)
[ 講演概要 ]
The structures of embedded complex projective manifolds swept out by varieties with preassigned properties have been studied by several authors. In this talk, we study structures of embedded projective manifolds swept out by cubic varieties.

2011年11月07日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室

Okounkov bodies and Seshadri constants (JAPANESE)
[ 講演概要 ]
Okounkov bodies, which are convex bodies associated to big line bundles, have rich information of the line bundles. On the other hand, Seshadri constants are invariants which measure the positivities of line bundles. In this talk, I will explain a relation between Okounkov bodies and Seshadri constants.