## Algebraic Geometry Seminar

Date, time & place Tuesday 15:30 - 17:00 122Room #122 (Graduate School of Math. Sci. Bldg.)

Seminar information archive

### 2014/02/12

14:00-17:30   Room #122 (Graduate School of Math. Sci. Bldg.)
Shin-ichi Matsumura (Kagoshima University) 14:00-15:30
An injectivity theorem with multiplier ideal sheaves of singular metrics with transcendental singularities (ENGLISH)
[ Abstract ]
In this talk, I give an injectivity theorem with multiplier ideal sheaves of singular metrics.
This theorem is a powerful generalization of various injectivity and vanishing theorems.
The proof is based on a combination of the theory of harmonic integrals and the L^2-method for the \\dbar-equation.
To treat transcendental singularities, after regularizing a given singular metric, we study the asymptotic behavior of the harmonic forms with respect to a family of the regularized metrics.
Moreover we obtain L^2-estimates of solutions of the \\dbar-equation, by using the \\check{C}ech complex.
As an application, we obtain a Nadel type vanishing theorem.
Junyan Cao (KIAS) 16:00-17:30
Ohsawa-Takegoshi extension theorem for K\\"ahler manifolds (ENGLISH)
[ Abstract ]
In this talk, we first prove a version of the Ohsawa-Takegoshi
extension theorem valid for on arbitrary K\\"ahler manifolds, and for
holomorphic line bundles equipped with possibly singular metrics. As an
application, we generalise Berndtsson and Paun 's result about the
pseudo-effectivity of the relative canonical bundles to arbitrary
compact K\\"ahler families.

### 2014/02/03

15:30-17:00   Room #122 (Graduate School of Math. Sci. Bldg.)
Kento Fujita (RIMS)
Classification of log del Pezzo surfaces of index three (JAPANESE)
[ Abstract ]
Log del Pezzo surfaces constitute an interesting class of rational surfaces and naturally appear in the minimal model program. I will describe an algorithm to classify all the log del Pezzo surfaces of fixed (Q-Gorenstein) index $a$. Especially, I will focus on the case that $a$ is equal to three. This is joint work with Kazunori Yasutake.

### 2014/01/22

15:00-16:30   Room #122 (Graduate School of Math. Sci. Bldg.)
Thomas Ducat (University of Warwick)
Divisorial Extractions from Singular Curves in Smooth 3-Folds (ENGLISH)
[ Abstract ]
Consider a singular curve C contained in a smooth 3-fold X.
Assuming the existence of a Du Val general elephant S containing C,
I give a normal form for the equations of C in X and an outline of how to
construct a divisorial extraction from this curve. If the general S is
Du Val of type D_{2k}, E_6 or E_7 then I can give some explicit
conditions for the existence of a terminal extraction. A treatment of
the D_{2k+1} case should be possible by similar means.

### 2014/01/20

15:30-17:00   Room #126 (Graduate School of Math. Sci. Bldg.)
Taro Sano (University of Warwick)
Deforming elephants of Q-Fano 3-folds (ENGLISH)
[ Abstract ]
Shokurov and Reid proved that a Fano 3-fold with canonical
Gorenstein singularities has a Du Val elephant, that is,
a member of the anticanonical linear system with only Du Val singularities.
The classification of Fano 3-folds is based on this fact.
However, for a Fano 3-fold with non-Gorenstein terminal singularities,
the anticanonical system does not contain such a member in general.
Alt{\\i}nok--Brown--Reid conjectured that, if the anticanonical system is non-empty,
a Q-Fano 3-fold can be deformed to that with a Du Val elephant.
In this talk, I will explain how to deform an elephant with isolated
singularities to a Du Val elephant.

### 2013/12/09

15:30-17:00   Room #122 (Graduate School of Math. Sci. Bldg.)
On birationally tririgid Q-Fano threefolds (JAPANESE)
[ Abstract ]
I will talk about birational geometry of Q-Fano threefolds. A Mori
fiber space birational to a given Q-Fano threefold is called a birational Mori fiber structure of the threefold. The existence of Q-Fano threefolds with a unique birational Mori fiber structure (resp. with two birational Mori fiber structures) is known. In this talk I will give an example of Q-Fano threefolds with three birational Mori fiber structures and also discuss about the behavior of birational Mori fiber structures in a family.

### 2013/11/25

15:30-17:00   Room #122 (Graduate School of Math. Sci. Bldg.)
Takayuki Koike (The University of Tokyo)
Minimal singular metrics of some line bundles with infinitely generated section rings (JAPANESE)
[ Abstract ]
We consider Hermitian metrics of pseudo-effective line bundles on smooth
projective varieties defined over $\\mathbb{C}$.
Especially we are interested in (possibly singular) Hermitian metrics
with semi-positive curvatures when the section rings are not finitely generated.
We study where and how minimal singular metrics, special Hermitian
metrics with semi-positive curvatures, diverges in the following two situations;
a line bundle admitting no Zariski decomposition even after any
modifications (Nakayama example)
and a nef line bundle $L$ on $X$ satisfying $D \\subset |mL|$ and $|mL-D| = \\emptyset$ for some divisor $D \\subset X$ and for all $m \\geq 1$ (
Zariski example).

### 2013/11/18

15:30-17:00   Room #122 (Graduate School of Math. Sci. Bldg.)
Florin Ambro (IMAR)
An injectivity theorem (ENGLISH)
[ Abstract ]
I will discuss a generalization of the injectivity theorem of Esnault-Viehweg, and an
application to the problem of lifting sections from the non-log canonical locus of a log variety.

### 2013/11/11

15:30-17:00   Room #122 (Graduate School of Math. Sci. Bldg.)
Sung Rak Choi (POSTECH)
Geography via the base loci (ENGLISH)
[ Abstract ]
The geography of log model refers to the decomposition of the set of effective adjoint divisors into the cells defined by the resulting models that are obtained by the log minimal model program.
We will describe the geography in terms of the asymptotic base loci and Zariski decompositions of divisors.
As an application, we give a partial answer to a question of B. Totaro concerning the structure of partially ample cones.

### 2013/10/28

15:30-17:00   Room #122 (Graduate School of Math. Sci. Bldg.)
Chen Jiang (University of Tokyo)
Weak Borisov-Alexeev-Borisov conjecture for 3-fold Mori Fiber spaces (ENGLISH)
[ Abstract ]
We investigate $\\epsilon$-klt log Fano 3-folds with some Mori fiber space structure, more precisely, with a del Pezzo fibration structure, or a conic bundle structure over projective plane. We give a bound for the log anti-canonical volume of such pair. The method is constructing non-klt centers and using connectedness lemma. This result is related to birational boundedness of log Fano varieties.

### 2013/07/22

15:30-17:00   Room #122 (Graduate School of Math. Sci. Bldg.)
Equivariant degenerations of spherical modules (ENGLISH)
[ Abstract ]
Given a reductive algebraic group G and an invariant
Hilbert function h, Alexeev and Brion have defined
a moduli scheme M which parametrizes affine G-schemes X
with the property that the coordinate ring of X decomposes,
as G-module, according to the function h. The talk will
be about joint work with Bart Van Steirteghem (New York)
which studies the moduli scheme M under some additional
assumptions.

### 2013/04/22

16:30-18:00   Room #118 (Graduate School of Math. Sci. Bldg.)
Professor Igor Reider (Universite d'Angers / RIMS)
Kodaira-Spencer classes, geometry of surfaces of general type and Torelli
theorem (ENGLISH)
[ Abstract ]
In this talk I will explain a geometric interpretation of Kodaira-Spencer classes and apply
it to the study of the differential of the period map of weight 2 Hodge structures for surfaces
of general type.
My approach is based on interpreting Kodaira-Spencer classes as higher rank bundles and
then studing their stability. This naturally leads to two parts:
1) unstable case
2) stable case.
I will give a geometric characterization of the first case and show how to relate the second
case to a special family of vector bundles giving rise to a family of rational curves. This family
of rational curves is used to recover the surface in question.

### 2013/01/15

15:30-17:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Jungkai Alfred Chen (National Taiwan University)
Three Dimensional Birational Geoemtry--updates and problems (ENGLISH)
[ Abstract ]
In this talk I will talk about some recent results on
biratioanl classification and biratioanl geoemtry of threefolds.

Given a threefold of general type, we improved our previous result by
showing that $Vol \\ge 1/1680$ and $|mK_X|$ is biratioanl for $m \\ge 61$.
Compare with the worst known example that $X_{46} \\subset \\mathbb{P}(4,5,6,7,23)$, one also knows that there are only finiteley
many singularities type
for threefolds of general type with $1/1680 \\le Vol \\le 1/420$. It is
then intereting to study threefolds of general type with given basket
of singularities and with given fiber structure.
Concerning threefolds with intermediate Kodaira dimension, we
considered the effective Iitaka fibration. For this purpose, it is
interesting to study threefolds with $\\kappa=1$ with given basket of
singularities and abelian fibration.

For explicit birational geoemtry, we will show our result that each
biratioanl map in minimal model program can be factored into a
sequence of following maps (or its inverse)
1. a divisorial contraction to a point of index r with discrepancy 1/r.
2. a blowup along a smooth curve
3. a flop

### 2012/12/13

10:40-12:10   Room #118 (Graduate School of Math. Sci. Bldg.)
Jean-Paul Brasselet (CNRS (Luminy))
The asymptotic variety of polynomial maps (ENGLISH)
[ Abstract ]
The asymptotic variety, or set of non-properness has been intensively studied by Zbigniew Jelonek. In a recent paper, Anna and Guillaume Valette associate to a polynomial map $F: {\\mathbb C}^n \\to {\\mathbb C}^n$ a singular variety $N_F$ and relate properness property of $F$ to the vanishing of some intersection homology groups of $N_F$. I will explain how stratifications of the asymptotic variety of $F$ play an important role in the story and how recently, one of my students, Nguyen Thi Bich Thuy, found a nice way to exhibit such a suitable stratification.

### 2012/12/10

15:30-17:00   Room #122 (Graduate School of Math. Sci. Bldg.)
Kotaro Kawatani (Nagoya University)
A hyperbolic metric and stability conditions on K3 surfaces with $¥rho=1$ (JAPANESE)
[ Abstract ]
We introduce a hyperbolic metric on the (normalized) space of stability conditions on projective K3 surfaces $X$ with Picard rank 1. Furthermore we demonstrate how this hyperbolic metric is helpful for us by discussing two or three topics.

### 2012/11/26

15:30-17:00   Room #122 (Graduate School of Math. Sci. Bldg.)
Toshiyuki Katsura (Hosei University)
A configuration of rational curves on the superspecial K3 surface (JAPANESE)

### 2012/11/19

15:30-17:00   Room #122 (Graduate School of Math. Sci. Bldg.)
Yukinobu Toda (IPMU)
Stability conditions and birational geometry (JAPANESE)
[ Abstract ]
I propose a conjecture which claims that MMP for a smooth projective variety is realized as a variation of Bridgeland moduli spaces of semistable objects in the derived category of coherent sheaves. I will discuss the surface case and extremal contractions for 3-folds. In the former case, the conjecture is completely solved. In the latter case, I will construct the perverse t-structure associated to the extremal contraction, and construct a candidate of the desired stability condition as a double tilting of the perverse heart.

### 2012/11/12

15:30-17:00   Room #122 (Graduate School of Math. Sci. Bldg.)
Kazunori Yasutake (Kyushu University)
On Fano fourfolds with nef vector bundles $Λ^2T_X$ (JAPANESE)
[ Abstract ]
By using results about extremal contractions on smooth fourfolds, we give a classification of fano fourfolds whose the second exterior power of tangent bundles are numerically effective.

### 2012/11/05

15:30-17:00   Room #122 (Graduate School of Math. Sci. Bldg.)
Shouhei Ma (Nagoya University)
The rationality of the moduli spaces of trigonal curves (JAPANESE)

### 2012/10/29

15:30-17:00   Room #122 (Graduate School of Math. Sci. Bldg.)
Kento Fujita (RIMS)
The Mukai conjecture for log Fano manifolds (JAPANESE)
[ Abstract ]
The concept of log Fano manifolds is one of the most natural generalization of the concept of Fano manifolds. We will give some structure theorems of log Fano manifolds. For example, we will show that the Mukai conjecture for Fano manifolds implies the `log Mukai conjecture' for log Fano manifolds.

### 2012/10/15

15:30-17:00   Room #122 (Graduate School of Math. Sci. Bldg.)
Yoshinori Gongyo (University of Tokyo)
On the moduli b-divisors of lc-trivial fibrations (JAPANESE)
[ Abstract ]
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.

### 2012/10/01

13:30-15:00   Room #122 (Graduate School of Math. Sci. Bldg.)
Robert Laterveer (CNRS, IRMA, Université de Strasbourg)
Weak Lefschetz for divisors (ENGLISH)
[ Abstract ]
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.

### 2012/10/01

15:30-17:00   Room #122 (Graduate School of Math. Sci. Bldg.)
Ryo Ohkawa (RIMS, Kyoto University)
Frobenius morphisms and derived categories on two dimensional toric Deligne--Mumford stacks (JAPANESE)
[ Abstract ]
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.

### 2012/07/30

15:30-17:00   Room #122 (Graduate School of Math. Sci. Bldg.)
Gianluca Pacienza (Université de Strasbourg)
Log Bend-and-Break on Deligne-Mumford stacks (ENGLISH)
[ Abstract ]
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.

### 2012/07/23

15:30-17:00   Room #122 (Graduate School of Math. Sci. Bldg.)
Shinnosuke Okawa (University of Tokyo)
Derived category of smooth proper Deligne-Mumford stack with p_g>0 (JAPANESE)
[ Abstract ]
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)

### 2012/06/25

15:30-17:00   Room #122 (Graduate School of Math. Sci. Bldg.)
Keiji Oguiso (Osaka University)
Automorphism groups of Calabi-Yau manifolds of Picard number two (JAPANESE)
[ Abstract ]
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.