## 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

### 2016年04月19日(火)

15:30-17:00   数理科学研究科棟(駒場) 122号室

Isomorphic quartic K3 surfaces and Cremona transformations (JAPANESE)
[ 講演概要 ]
We show that

(i) there is a pair of smooth complex quartic K3 surfaces such that they are isomorphic as abstract varieties but not Cremona equivalent.

(ii) there is a pair of smooth complex quartic K3 surfaces such that they are Cemona equivalent but not projectively equivalent.

These two results are much inspired by e-mails from Professors Tuyen Truong and J\'anos Koll\'ar.

### 2016年04月11日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
Piotr Pragacz 氏 (Institute of Mathematics, Polish Academy of Sciences )
Gysin maps, duality and Schubert classes
(English)
[ 講演概要 ]
We establish a Gysin formula for Schubert bundles
and a strong version of the duality theorem in Schubert calculus
on Grassmann bundles. We then combine them to compute the fundamental
classes of Schubert bundles in Grassmann bundles, which yields a new
proof of the Giambelli formula for vector bundles. This is a joint
work with Lionel Darondeau.
[ 参考URL ]
https://www.impan.pl/~pragacz/main.htm

### 2015年12月17日(木)

15:30-17:00   数理科学研究科棟(駒場) 122号室
このセミナーは講演者が急病のためキャンセルになりました。This seminer is canceled due to the speaker 's sick.
Dulip Piyaratne 氏 (IPMU)
Polarization and stability on a derived equivalent abelian variety (English)
[ 講演概要 ]
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 realisations is connected with stability conditions on their derived categories. Then I will discuss these ideas for abelian surfaces and abelian 3-folds in detail.
[ 参考URL ]
http://db.ipmu.jp/member/personal/3989en.html

### 2015年12月14日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室

Extending Hori-Vafa toric mirror symmetry via SYZ and modular forms (English)
[ 講演概要 ]
In this talk, I will introduce partial compactification for a class of toric Calabi-Yau manifolds. A fundamental question is how the Hori-Vafa toric mirror symmetry extends to this new class of Calabi-Yau manifolds. The answer leads us to a new connection between SYZ mirror symmetry and modular forms. If time permits, I will also discuss higher dimensional analogues of the Yau-Zaslow formula (for an elliptic K3 surface) in terms of Siegel modular forms. This talk is based on a joint work with Siu-Cheong Lau.

### 2015年12月07日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
Alexey Bondal 氏 (IPMU)
Flops and spherical functors (English)
[ 講演概要 ]
I will describe various functors on derived categories of coherent sheaves
related to flops and relations between these functors. A categorical
version of deformation theory of systems of objects in abelian categories
will be outlined and its relation to flop spherical functors will be
presented.

### 2015年11月30日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
Fabrizio Catanese 氏 (Universität Bayreuth)
Interesting surfaces which are coverings of a rational surface branched over few lines (English)
[ 講演概要 ]
Surfaces which are covers of the plane branched over 5 or 6 lines have provided answers to long standing questions, for instance the BCD surfaces for Fujita's question on semiampleness of VHS (Dettweiler-Cat); and examples of ball quotients (Hirzebruch), automorphisms acting trivially on integral cohomology (Cat-Gromadtzki), canonical maps with high degree or image-degree (Pardini, Bauer-Cat). I shall speak especially about the above Abelian coverings of the plane, the geometry of the del Pezzo surface of degree 5, the rigidity of BCD surfaces, and a criterion for a fibred surface to be a projective classifying space.

### 2015年11月16日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
Artan Sheshmani 氏 (IPMU/ Ohio State University)
Counting curves on surface in Calabi-Yau threefolds and the proof of S-duality modularity conjecture (English)
[ 講演概要 ]
I will talk about recent joint works with Amin Gholampour, Richard Thomas and Yukinobu Toda, on an algebraic-geometric proof of the S-duality conjecture in superstring theory, made formerly by physicists Gaiotto, Strominger, Yin, regarding the modularity of DT invariants of sheaves supported on hyperplane sections of the quintic Calabi-Yau threefold. Our strategy is to first use degeneration and localization techniques to reduce the threefold theory to a certain intersection theory over the relative Hilbert scheme of points on surfaces and then prove modularity; More precisely, together with Gholampour we have proven that the generating series, associated to the top intersection numbers of the Hilbert scheme of points, relative to an effective divisor, on a smooth quasi-projective surface is a modular form. This is a generalization of the result of Okounkov-Carlsson, where they used representation theory and the machinery of vertex operators to prove this statement for absolute Hilbert schemes. These intersection numbers eventually, together with the generating series of Noether-Lefschetz numbers as I will explain, will provide the ingredients to achieve a complete algebraic-geometric proof of S-duality modularity conjecture.

### 2015年11月09日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室

3-dimensional McKay correspondence (English)
[ 講演概要 ]
The original McKay correspondence is a relation between group theory of a finite subgroup G of SL(2,C) and geometry of the minimal resolution of the quotient singularity by G, and was generalized several ways. In particular, 3-dimensional generalization was extended to derived categorical eqivalence and the G-Hilbert scheme was useful to explain the correspondence. However, most results hold only for abelian subgroups. In this talk, I would like to introduce an iterated G-Hilbert scheme and show more geometrical McKay correspondence for non-abelian subgroups.

### 2015年11月05日(木)

15:30-17:00   数理科学研究科棟(駒場) 126号室
いつもと部屋と曜日が違います。The day of the week and room are different from usual.

Compact moduli of marked noncommutative del Pezzo surfaces via quivers (English)
[ 講演概要 ]
I will introduce certain GIT construction via quivers of compactified moduli spaces of marked noncommutative del Pezzo surfaces. For projective plane, quadric surface, and those of degree 3, 2, 1, we obtain projective toric varieties of dimension 2, 3, 8, 9, 10, respectively. Then I will discuss relations with deformation theory of abelian categories, blow-up of noncommutative projective planes, and three-block exceptional collections due to Karpov and Nogin. This talk is based on joint works in progress with Tarig Abdelgadir and Kazushi Ueda.

### 2015年10月26日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
Lawrence Ein 氏 (University of Illinois at Chicago)
Asymptotic syzygies and the gonality conjecture (English)
[ 講演概要 ]
We'll discuss my joint work with Lazarsfeld on the gonality conjecture about the syzygies of a smooth projective curve when it is embedded into the projective space by the complete linear system of a sufficiently very ample line bundles. We'll also discuss some results about the asymptotic syzygies f higher dimensional varieties.

### 2015年10月05日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
Evangelos Routis 氏 (IPMU)
Weighted Compactifications of Configuration Spaces (English)
[ 講演概要 ]
In the early 90's, Fulton and MacPherson provided a natural and beautiful way of compactifying the configuration space F(X,n) of n distinct labeled points on an arbitrary nonsingular variety. In this talk, I will present an alternate compactification of F(X,n), which generalizes the work of Fulton and MacPherson and is parallel to Hassett's weighted generalization of the moduli space of n-pointed stable curves. After discussing its main properties, I will give a presentation of its intersection ring and as an application, I will describe the intersection ring of Hassett's spaces in genus 0. Finally, as time permits, I will discuss some additional moduli problems associated with weighted compactifications.

### 2015年06月29日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
Manfred Lehn 氏 (Mainz/RIMS)
Twisted cubics and cubic fourfolds (English)
[ 講演概要 ]
The moduli scheme of generalised twisted cubics on a smooth
cubic fourfold Y non containing a plane is smooth projective of
dimension 10 and admits a contraction to an 8-dimensional
holomorphic symplectic manifold Z(Y). The latter is shown to be
birational to the Hilbert scheme of four points on a K3 surface if
Y is of Pfaffian type. This is a report on joint work with C. Lehn,
C. Sorger and D. van Straten and with N. Addington.

### 2015年06月22日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
Martí Lahoz 氏 (Institut de Mathématiques de Jussieu )
Rational cohomology tori
(English)
[ 講演概要 ]
Complex tori can be topologically characterised among compact Kähler
manifolds by their integral cohomology ring. I will discuss the
structure of compact Kähler manifolds whose rational cohomology ring is
isomorphic to the rational cohomology ring of a torus and give some
examples. This is joint work with Olivier Debarre and Zhi Jiang.
[ 参考URL ]
http://webusers.imj-prg.fr/~marti.lahoz/

### 2015年06月15日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
Christopher Hacon 氏 (University of Utah/RIMS)
Boundedness of the KSBA functor of
SLC models (English)
[ 講演概要 ]
Let $X$ be a canonically polarized smooth $n$-dimensional projective variety over $\mathbb C$ (so that $\omega _X$ is ample), then it is well-known that a fixed multiple of the canonical line bundle defines an embedding of $X$ in projective space. It then follows easily that if we fix certain invariants of $X$, then $X$ belongs to finitely many deformation types. Since canonical models are rarely smooth, it is important to generalize this result to canonically polarized $n$-dimensional projectivevarieties with canonical singularities. Moreover, since these varieties specialize to non-normal varieties it is also important to generalize this result to semi-log canonical pairs. In this talk we will explain a strong version of the above result that applies to semi-log canonical pairs.This is joint work with C. Xu and J. McKernan
[ 参考URL ]
http://www.math.utah.edu/~hacon/

### 2015年06月01日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室

Rank 2 weak Fano bundles on cubic 3-folds (日本語)
[ 講演概要 ]
A vector bundle on a projective variety is called weak Fano if its
projectivization is a weak Fano manifold. This is a　generalization of
Fano bundles.
In this talk, we will obtain a classification of rank 2 weak Fano
bundles on a nonsingular　cubic hypersurface in a projective 4-space.
Specifically, we will show that there exist rank 2 indecomposable weak
Fano bundles on it.

### 2015年05月25日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室

Good reduction of K3 surfaces (日本語 or English)
[ 講演概要 ]
We consider degeneration of K3 surfaces over a 1-dimensional base scheme
of mixed characteristic (e.g. Spec of the p-adic integers).
Under the assumption of potential semistable reduction, we first prove
that a trivial monodromy action on the l-adic etale cohomology group
implies potential good reduction, where potential means that we allow a
finite base extension.
Moreover we show that a finite etale base change suffices.
The proof for the first part involves a mixed characteristic
3-dimensional MMP (Kawamata) and the classification of semistable
degeneration of K3 surfaces (Kulikov, Persson--Pinkham, Nakkajima).
For the second part, we consider flops and descent arguments. This is a joint work with Christian Liedtke.
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~ymatsu/index_j.html

### 2015年05月18日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
Will Donovan 氏 (IPMU)
Twists and braids for general 3-fold flops (English)
[ 講演概要 ]
When a 3-fold contains a floppable rational curve, a theorem of Bridgeland provides a derived equivalence between the 3-fold and its flop. I will discuss recent joint work with Michael Wemyss, showing that these flop functors satisfy Coxeter-type braid relations. Using this result, we construct an action of a braid-type group on the derived category of the 3-fold. This group arises from the topology of a certain simplicial hyperplane arrangement, determined by the local geometry of the curve. I will give examples and explain key elements in the construction, including the noncommutative deformations of curves introduced in our previous work.
[ 参考URL ]
http://db.ipmu.jp/member/personal/4007en.html

### 2015年05月11日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室

Deformations of weak Fano varieties (日本語 or English)
[ 講演概要 ]
A smooth projective variety often has obstructed deformations.
Nevertheless, important varieties such as Fano varieties and
Calabi-Yau varieties have unobstructed deformations.
In this talk, I explain about unobstructedness of deformations of weak
Fano varieties, in particular a weak Q-Fano 3-fold.
I also present several examples to show delicateness of this unobstructedness.
[ 参考URL ]

### 2015年04月27日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室

Lagrangian embeddings of cubic fourfolds containing a plane (日本語)
[ 講演概要 ]
4次元3次超曲面は、複素シンプレクティック多様体の構成、有理性やK3曲面との関係などという観点から研究されている。1985年BeauvilleとDonagiは、4次元3次超曲面上の直線のなすFanoスキームがK3曲面上の2点のHilbertスキームと変形同値な複素シンプレクティック多様体であることを示した。2013年Lehnらは、平面を含まない4次元3次超曲面は8次元複素シンプレクティックにラグランジュ部分多様体として埋め込めることを示した。この8次元複素シンプレクティック多様体は4次元3次超曲面上のねじれ3次曲線全体を考えることにより得られる。

### 2015年04月20日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室

Fano 5-folds with nef tangent bundles (日本語)
[ 講演概要 ]
Campana と Peternell は, ネフな接束をもつ Fano 多様体は有理等質多様体で
あると予想した.

とが示されている.

によって場合分けすることができて, 趙・宮岡・Shepherd-Barron, 宮岡, Hwang,
Mok らの結果から, 5 次元の場合には, 擬指数が 4 であるときを除けば有理等

その結果として 5 次元のときには Campana と Peternell の予想が正しいこと
が従う.

### 2015年04月13日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
Frédéric Campana 氏 (Université de Lorraine)
An orbifold version of Miyaoka's semi-positivity theorem and applications (English)
[ 講演概要 ]
This orbifold' version of Miyaoka's theorem says that if (X,D)
is a projective log-canonical pair with K_X+D pseudo-effective,
then its 'cotangent' sheaf $¥Omega^1(X,D)$ is generically semi-positive.
The definitions will be given. The original proof of Miyaoka, which
mixes
char 0 and char p>0 arguments could not be adapted. Our proof is in char
0 only.

A first consequence is when (X,D) is log-smooth with reduced boudary D,
in which case the cotangent sheaf is the classical Log-cotangent sheaf:
if some tensor power of $¥omega^1_X(log(D))$ contains a 'big' line
bundle, then K_X+D is 'big' too. This implies, together with work of
Viehweg-Zuo,
the hyperbolicity conjecture' of Shafarevich-Viehweg.

The preceding is joint work with Mihai Paun.

A second application (joint work with E. Amerik) shows that if D is a
non-uniruled smooth divisor in aprojective hyperkaehler manifold with
symplectic form s,
then its characteristic foliation is algebraic only if X is a K3 surface.
This was shown previously bt Hwang-Viehweg assuming D to be of general
type. This result has some further consequences.

### 2015年01月26日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
Jungkai Chen 氏 (National Taiwan University)
Positivity in varieties of maximal Albanese dimension (ENGLISH)
[ 講演概要 ]
Given a variety of maximal Albanese dimension, it is known that the holomorphic Euler characteristic is non-negative. It is an interesting question to characterize varieties with vanishing Euler characteristic.

In our previous work (jointly with Debarre and Jiang), we prove that Ein-Lazarsgfeld's example is essentially the only variety of maximal Albanese and Kodaira dimension with vanishing Euler characteristic in dimension three. In the recent joint work with Jiang, we prove a decomposition theorem for the push-forward of canonical sheaf. As a consequence, we are able to generalized our previous characterization. The purpose of this talk is give a survey of these two works.

### 2015年01月19日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室

Crepant resolutions of Slodowy slice in nilpotent orbit closure in sl_N(C) (JAPANESE)
[ 講演概要 ]
Nilpotent orbit closures and their intersections with Slodowy slices are typical examples of symplectic varieties. It is known that every crepant resolution of a nilpotent orbit closure is obtained as a Springer resolution. In this talk, we show that every crepant resolution of a Slodowy slice in nilpotent orbit closure in sl_N(C) is obtained as the restriction of a Springer resolution and explain how to count the number of crepant resolutions. The proof of the main results is based on the fact that Slodowy slices can be described as quiver varieties.

### 2014年12月15日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室

A characterization of ordinary abelian varieties in positive characteristic (JAPANESE)
[ 講演概要 ]
This is joint work with Hiromu Tanaka. In this talk, we study F^e_*O_X on a projective variety over the algebraic closed field of positive characteristic. For an ordinary abelian variety X, F^e_*O_X is decomposed into line bundles for every positive integer e. Conversely, if a smooth projective variety X satisfies this property and its Kodaira dimension is non-negative, then X is an ordinary abelian variety.

### 2014年12月01日(月)

15:30-17:00   数理科学研究科棟(駒場) 122号室
Malte Wandel 氏 (RIMS)
Induced Automorphisms on Hyperkaehler Manifolds (ENGLISH)
[ 講演概要 ]
in this talk I want to report on a joint project with Giovanni Mongardi (Milano). We study automorphisms of hyperkaehler manifolds. All known deformation classes of these manifolds contain moduli spaces of stable sheaves on surfaces. If the underlying surface admits a non-trivial automorphism, it is often possible to transfer this automorphism to a moduli space of sheaves. In this way we obtain a big class of interesting examples of automorphisms of hyperkaehler manifolds. I will present a criterion to 'detect' automorphisms in this class and discuss several applications for the classification of automorphisms of manifolds of K3^[n]- and kummer n-type. If time permits I will try to talk about generalisations to O'Grady's sporadic examples.