## Algebraic Geometry Seminar

Seminar information archive ～07/21｜Next seminar｜Future seminars 07/22～

Date, time & place | Friday 13:30 - 15:00 ハイブリッド開催/117Room #ハイブリッド開催/117 (Graduate School of Math. Sci. Bldg.) |
---|---|

Organizer(s) | GONGYO Yoshinori, NAKAMURA Yusuke, TANAKA Hiromu |

**Seminar information archive**

### 2016/06/27

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Generic vanishing and birational geometry in char p>0 (ENGLISH)

http://www.math.utah.edu/~hacon/

**Christopher Hacon**(University of Utah)Generic vanishing and birational geometry in char p>0 (ENGLISH)

[ Abstract ]

Many precise results on the birational geometry of irregular varieties have been obtained by combining the generic vanishing theorems of Green and Lazarsfeld with the Fourier-Mukai transform. In this talk we will discuss the failure of the generic vanishing theorems of Green and Lazarsfeld in positive characteristic. We will then explain a different approach to generic vanishing based on the theory of F-singularities that leads to concrete applications in birational geometry in positive characteristics

[ Reference URL ]Many precise results on the birational geometry of irregular varieties have been obtained by combining the generic vanishing theorems of Green and Lazarsfeld with the Fourier-Mukai transform. In this talk we will discuss the failure of the generic vanishing theorems of Green and Lazarsfeld in positive characteristic. We will then explain a different approach to generic vanishing based on the theory of F-singularities that leads to concrete applications in birational geometry in positive characteristics

http://www.math.utah.edu/~hacon/

### 2016/06/20

16:30-18:00 Room #123 (Graduate School of Math. Sci. Bldg.)

BUILDING BLOCKS OF POLARIZED ENDOMORPHISMS OF NORMAL PROJECTIVE VARIETIES (English)

http://www.math.nus.edu.sg/~matzdq/

**De-Qi Zhang**(National University of Singapore)BUILDING BLOCKS OF POLARIZED ENDOMORPHISMS OF NORMAL PROJECTIVE VARIETIES (English)

[ Abstract ]

An endomorphism f of a normal projective variety X is polarized if f∗H ∼ qH for some ample Cartier divisor H and integer q > 1.

We first assert that a suitable maximal rationally connected fibration (MRC) can be made f-equivariant using a construction of N. Nakayama, that f descends to a polarized endomorphism of the base Y of this MRC and that this Y is a Q-abelian variety (quasi- ́etale quotient of an abelian variety). Next we show that we can run the minimal model program (MMP) f-equivariantly for mildly singular X and reach either a Q-abelian variety or a Fano variety of Picard number one.

As a consequence, the building blocks of polarized endomorphisms are those of Q- abelian varieties and those of Fano varieties of Picard number one.

Along the way, we show that f always descends to a polarized endomorphism of the Albanese variety Alb(X) of X, and that a power of f acts as a scalar on the Neron-Severi group of X (modulo torsion) when X is smooth and rationally connected.

Partial answers about X being of Calabi-Yau type or Fano type are also given with an extra primitivity assumption on f which seems necessary by an example.

This is a joint work with S. Meng.

[ Reference URL ]An endomorphism f of a normal projective variety X is polarized if f∗H ∼ qH for some ample Cartier divisor H and integer q > 1.

We first assert that a suitable maximal rationally connected fibration (MRC) can be made f-equivariant using a construction of N. Nakayama, that f descends to a polarized endomorphism of the base Y of this MRC and that this Y is a Q-abelian variety (quasi- ́etale quotient of an abelian variety). Next we show that we can run the minimal model program (MMP) f-equivariantly for mildly singular X and reach either a Q-abelian variety or a Fano variety of Picard number one.

As a consequence, the building blocks of polarized endomorphisms are those of Q- abelian varieties and those of Fano varieties of Picard number one.

Along the way, we show that f always descends to a polarized endomorphism of the Albanese variety Alb(X) of X, and that a power of f acts as a scalar on the Neron-Severi group of X (modulo torsion) when X is smooth and rationally connected.

Partial answers about X being of Calabi-Yau type or Fano type are also given with an extra primitivity assumption on f which seems necessary by an example.

This is a joint work with S. Meng.

http://www.math.nus.edu.sg/~matzdq/

### 2016/06/20

14:45-16:15 Room #123 (Graduate School of Math. Sci. Bldg.)

Fujita's freeness conjecture for 5-fold (English)

**Zhixian Zhu**(KIAS)Fujita's freeness conjecture for 5-fold (English)

[ Abstract ]

Let X be a smooth projective variety of dimension n and L any ample line bundle. Fujita conjectured that the adjoint line bundle O(K_X+mL) is globally generated for any m greater or equal to dimX+1. By studying the singularity of pairs, we prove Fujita's freeness conjecture for smooth 5-folds.

Let X be a smooth projective variety of dimension n and L any ample line bundle. Fujita conjectured that the adjoint line bundle O(K_X+mL) is globally generated for any m greater or equal to dimX+1. By studying the singularity of pairs, we prove Fujita's freeness conjecture for smooth 5-folds.

### 2016/05/31

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

A Characterization of Symplectic Grassmannians (JAPANESE)

**Kiwamu Watanabe**(Saitama University)A Characterization of Symplectic Grassmannians (JAPANESE)

[ Abstract ]

In the series of their works, J. M. Hwang and N. Mok have been developing the theory of Varieties of Minimal Rational Tangents (VMRT for short). In this direction, the results of Mok and J. Hong-Hwang allow us to recognize a homogeneous Fano manifold X of Picard number one by looking at its VMRT at a general point. This characterization works for all rational homogeneous manifolds of Picard number one whenever the VMRT is rational homogeneous, which is always the case except for the short root cases; namely for symplectic Grassmannians, and for two varieties of type F*4*.

In this talk we show that, if we impose that the VMRT is the expected one at every point of the variety, we may still characterize symplectic Grassmannians. This is a joint work with G. Occhetta and L. E. Sola Conde (arXiv:1604.06867).

In the series of their works, J. M. Hwang and N. Mok have been developing the theory of Varieties of Minimal Rational Tangents (VMRT for short). In this direction, the results of Mok and J. Hong-Hwang allow us to recognize a homogeneous Fano manifold X of Picard number one by looking at its VMRT at a general point. This characterization works for all rational homogeneous manifolds of Picard number one whenever the VMRT is rational homogeneous, which is always the case except for the short root cases; namely for symplectic Grassmannians, and for two varieties of type F*4*.

In this talk we show that, if we impose that the VMRT is the expected one at every point of the variety, we may still characterize symplectic Grassmannians. This is a joint work with G. Occhetta and L. E. Sola Conde (arXiv:1604.06867).

### 2016/05/24

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

ON PSEUDO KOBAYASHI HYPERBOLICITY OF SUBVARIETIES OF ABELIAN VARIETIES

(tba)

**Katsutoshi Yamanoi**(Osaka University)ON PSEUDO KOBAYASHI HYPERBOLICITY OF SUBVARIETIES OF ABELIAN VARIETIES

(tba)

[ Abstract ]

We prove that the Kobayashi pseudo distance of a closed subvariety X of an abelian variety A is a true distance outside the special set Sp(X) of X, where Sp(X) is the union of all positive dimensional translated abelian subvarieties of A which are contained in X. More strongly, we prove that a closed subvariety X of an abelian variety is taut modulo Sp(X); Every sequence fn : D → X of holomorphic mappings from the unit disc D admits a subsequence which converges locally uniformly, unless the image fn(K) of a fixed compact set K of D eventually gets arbitrarily close to Sp(X) as n gets larger. These generalize a classical theorem on algebraic degeneracy of entire holomorphic curves in irregular varieties.

We prove that the Kobayashi pseudo distance of a closed subvariety X of an abelian variety A is a true distance outside the special set Sp(X) of X, where Sp(X) is the union of all positive dimensional translated abelian subvarieties of A which are contained in X. More strongly, we prove that a closed subvariety X of an abelian variety is taut modulo Sp(X); Every sequence fn : D → X of holomorphic mappings from the unit disc D admits a subsequence which converges locally uniformly, unless the image fn(K) of a fixed compact set K of D eventually gets arbitrarily close to Sp(X) as n gets larger. These generalize a classical theorem on algebraic degeneracy of entire holomorphic curves in irregular varieties.

### 2016/05/17

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

On dual defects of toric varieties (TBA)

https://sites.google.com/site/atsushiito221/

**Atsushi Ito**(Dep. of Math. Kyoto Univ. )On dual defects of toric varieties (TBA)

[ Abstract ]

For a projective variety embedded in a projective space,

we can define the dual variety in the dual projective space.

By dimension count, the codimension of the dual variety is expected to be one,

but it can be greater than one for some varieties.

For a smooth toric variety, it is known that the codimension of the dual variety is greater than one

if and only if the toric variety is a suitable projective bundle over some toric variety.

In this talk, I will explain a generalization of this result to toric varieties without the assumption of singularities.

This is a joint work with Katsuhisa Furukawa.

[ Reference URL ]For a projective variety embedded in a projective space,

we can define the dual variety in the dual projective space.

By dimension count, the codimension of the dual variety is expected to be one,

but it can be greater than one for some varieties.

For a smooth toric variety, it is known that the codimension of the dual variety is greater than one

if and only if the toric variety is a suitable projective bundle over some toric variety.

In this talk, I will explain a generalization of this result to toric varieties without the assumption of singularities.

This is a joint work with Katsuhisa Furukawa.

https://sites.google.com/site/atsushiito221/

### 2016/04/26

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

A gentle introduction to K-stability and its recent development (Japanese)

https://sites.google.com/site/yujiodaka2013/

**Yuji Odaka**(Dept. of Math., Kyoto U.)A gentle introduction to K-stability and its recent development (Japanese)

[ Abstract ]

K安定性とは複素代数多様体上の「標準的な」ケーラー計量の存在問題に端を発する，代数幾何的な概念です．二木先生や満渕先生等の先駆的な仕事に感化されて導入され，特に近年ホットに研究され始めている一方，未だその大半はより微分幾何的な研究者の方々や背景の中でなされているように講演者には感じられます．

代数幾何的にもどのように面白いか，どういった意義があるかに私見で軽く触れた上で，その基礎付けをより拡張した枠組みで説明しつつ，最先端でどのようなことが問題になっているかをいくらか（私の力量と時間の許す限り）解説しつつ，文献をご紹介できればと思っています

[ Reference URL ]K安定性とは複素代数多様体上の「標準的な」ケーラー計量の存在問題に端を発する，代数幾何的な概念です．二木先生や満渕先生等の先駆的な仕事に感化されて導入され，特に近年ホットに研究され始めている一方，未だその大半はより微分幾何的な研究者の方々や背景の中でなされているように講演者には感じられます．

代数幾何的にもどのように面白いか，どういった意義があるかに私見で軽く触れた上で，その基礎付けをより拡張した枠組みで説明しつつ，最先端でどのようなことが問題になっているかをいくらか（私の力量と時間の許す限り）解説しつつ，文献をご紹介できればと思っています

https://sites.google.com/site/yujiodaka2013/

### 2016/04/19

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Isomorphic quartic K3 surfaces and Cremona transformations (JAPANESE)

**Keiji Oguiso**(University of Tokyo)Isomorphic quartic K3 surfaces and Cremona transformations (JAPANESE)

[ Abstract ]

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.

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.

### 2016/04/11

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Gysin maps, duality and Schubert classes (English)

https://www.impan.pl/~pragacz/main.htm

**Piotr Pragacz**(Institute of Mathematics, Polish Academy of Sciences )Gysin maps, duality and Schubert classes (English)

[ Abstract ]

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.

[ Reference URL ]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.

https://www.impan.pl/~pragacz/main.htm

### 2015/12/17

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Polarization and stability on a derived equivalent abelian variety (English)

http://db.ipmu.jp/member/personal/3989en.html

**Dulip Piyaratne**(IPMU)Polarization and stability on a derived equivalent abelian variety (English)

[ 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 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.

[ Reference URL ]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.

http://db.ipmu.jp/member/personal/3989en.html

### 2015/12/14

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Extending Hori-Vafa toric mirror symmetry via SYZ and modular forms (English)

**Atsushi Kanazawa**(Harvard)Extending Hori-Vafa toric mirror symmetry via SYZ and modular forms (English)

[ Abstract ]

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.

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.

### 2015/12/07

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Flops and spherical functors (English)

**Alexey Bondal**(IPMU)Flops and spherical functors (English)

[ Abstract ]

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.

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.

### 2015/11/30

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Interesting surfaces which are coverings of a rational surface branched over few lines (English)

**Fabrizio Catanese**(Universität Bayreuth)Interesting surfaces which are coverings of a rational surface branched over few lines (English)

[ Abstract ]

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.

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.

### 2015/11/16

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Counting curves on surface in Calabi-Yau threefolds and the proof of S-duality modularity conjecture (English)

**Artan Sheshmani**(IPMU/ Ohio State University)Counting curves on surface in Calabi-Yau threefolds and the proof of S-duality modularity conjecture (English)

[ Abstract ]

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.

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.

### 2015/11/09

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

3-dimensional McKay correspondence (English)

**Yukari Ito**(Nagoya University)3-dimensional McKay correspondence (English)

[ Abstract ]

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.

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.

### 2015/11/05

15:30-17:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Compact moduli of marked noncommutative del Pezzo surfaces via quivers (English)

**Shinnosuke Okawa**(Osaka University)Compact moduli of marked noncommutative del Pezzo surfaces via quivers (English)

[ Abstract ]

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.

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.

### 2015/10/26

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Asymptotic syzygies and the gonality conjecture (English)

**Lawrence Ein**(University of Illinois at Chicago)Asymptotic syzygies and the gonality conjecture (English)

[ Abstract ]

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.

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.

### 2015/10/05

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Weighted Compactifications of Configuration Spaces (English)

**Evangelos Routis**(IPMU)Weighted Compactifications of Configuration Spaces (English)

[ Abstract ]

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.

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.

### 2015/06/29

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Twisted cubics and cubic fourfolds (English)

**Manfred Lehn**(Mainz/RIMS)Twisted cubics and cubic fourfolds (English)

[ Abstract ]

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.

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.

### 2015/06/22

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Rational cohomology tori

(English)

http://webusers.imj-prg.fr/~marti.lahoz/

**Martí Lahoz**(Institut de Mathématiques de Jussieu )Rational cohomology tori

(English)

[ Abstract ]

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.

[ Reference URL ]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.

http://webusers.imj-prg.fr/~marti.lahoz/

### 2015/06/15

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Boundedness of the KSBA functor of

SLC models (English)

http://www.math.utah.edu/~hacon/

**Christopher Hacon**(University of Utah/RIMS)Boundedness of the KSBA functor of

SLC models (English)

[ Abstract ]

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

[ Reference URL ]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

http://www.math.utah.edu/~hacon/

### 2015/06/01

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Rank 2 weak Fano bundles on cubic 3-folds (日本語)

**Daizo Ishikawa**(Waseda University)Rank 2 weak Fano bundles on cubic 3-folds (日本語)

[ Abstract ]

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.

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.

### 2015/05/25

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Good reduction of K3 surfaces (日本語 or English)

https://www.ms.u-tokyo.ac.jp/~ymatsu/index_j.html

**Yuya Matsumoto**(University of Tokyo)Good reduction of K3 surfaces (日本語 or English)

[ Abstract ]

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.

[ Reference URL ]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.

https://www.ms.u-tokyo.ac.jp/~ymatsu/index_j.html

### 2015/05/18

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Twists and braids for general 3-fold flops (English)

http://db.ipmu.jp/member/personal/4007en.html

**Will Donovan**(IPMU)Twists and braids for general 3-fold flops (English)

[ Abstract ]

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.

[ Reference URL ]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.

http://db.ipmu.jp/member/personal/4007en.html

### 2015/05/11

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Deformations of weak Fano varieties (日本語 or English)

https://sites.google.com/site/tarosano222/

**Taro Sano**(Kyoto University)Deformations of weak Fano varieties (日本語 or English)

[ Abstract ]

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.

[ Reference URL ]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.

https://sites.google.com/site/tarosano222/