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

**Seminar information archive**

### 2007/10/16

10:00-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Homogical methods in Non-commutative Geometry

**Dmitry KALEDIN**(Steklov研究所, 東大数理)Homogical methods in Non-commutative Geometry

[ Abstract ]

Of all the approaches to non-commutative geometry, probably the most promising is the homological one, developed by Keller, Kontsevich, Toen and others, where non-commutative eometry is understood as "geometry of triangulated categories". Examples of "geometric" triangulated categories come from representation theory, symplectic geometry (Fukaya category) and algebraic geometry (the derived category of coherent sheaves on an algebraic variety and

various generalizations). Non-commutative point of view is expected to be helpful even in traditional questions of algebraic geometry such as the termination of flips.

We plan to give an introduction to the subject, with emphasis on homological methods (such as e.g. Hodge theory which, as it turns out, can be mostly formulated in the non-commutative setting).

No knowledge of non-commutative geometry whatsoever is assumed. However, familiarity with basic homological algebra and algebraic geometry will be helpful.

Of all the approaches to non-commutative geometry, probably the most promising is the homological one, developed by Keller, Kontsevich, Toen and others, where non-commutative eometry is understood as "geometry of triangulated categories". Examples of "geometric" triangulated categories come from representation theory, symplectic geometry (Fukaya category) and algebraic geometry (the derived category of coherent sheaves on an algebraic variety and

various generalizations). Non-commutative point of view is expected to be helpful even in traditional questions of algebraic geometry such as the termination of flips.

We plan to give an introduction to the subject, with emphasis on homological methods (such as e.g. Hodge theory which, as it turns out, can be mostly formulated in the non-commutative setting).

No knowledge of non-commutative geometry whatsoever is assumed. However, familiarity with basic homological algebra and algebraic geometry will be helpful.

### 2007/10/10

16:30-17:30 Room #117 (Graduate School of Math. Sci. Bldg.)

Abel-Jacobi Maps Associated to Algebraic Cycles, I.

**James Lewis**(University of Alberta)Abel-Jacobi Maps Associated to Algebraic Cycles, I.

[ Abstract ]

This talk concerns the Bloch cycle class map from the higher Chow groups to Deligne cohomology of a projective algebraic manifold. We provide an explicit formula for this map in terms of polylogarithmic type currents.

This talk concerns the Bloch cycle class map from the higher Chow groups to Deligne cohomology of a projective algebraic manifold. We provide an explicit formula for this map in terms of polylogarithmic type currents.

### 2007/10/10

15:00-16:00 Room #122 (Graduate School of Math. Sci. Bldg.)

p-adic Hodge theory in the non-commutative setting

**Dmitry Kaledin**(Steklov Institute)p-adic Hodge theory in the non-commutative setting

[ Abstract ]

We will explain what is the natural replacement of the notion of Hodge structure in the p-adic setting, and how to construct such a structure for non-commutative manifolds (something which at present cannot be done for the usual Hodge structures, but works perfectly well for the p-adic analog).

We will explain what is the natural replacement of the notion of Hodge structure in the p-adic setting, and how to construct such a structure for non-commutative manifolds (something which at present cannot be done for the usual Hodge structures, but works perfectly well for the p-adic analog).

### 2007/09/26

16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Floor diagrams and enumeration of tropical curves

**Grigory Mikhalkin**(Toronto大学)Floor diagrams and enumeration of tropical curves

[ Abstract ]

The enumerative problems considered in this talk are finding the number of curves in projective spaces (over complex, real and tropical numbers) of given genus and degree constrained by certain incidence conditions (e.g. passing via points or lines). Floor diagrams are a combinatorial tool that reduces an enumerative problem in dimension n to the corresponding problem n dimension n-1. Floor diagrams give a constructive (and rather efficient) way to find all tropical curves for a given enumerative problem. And once we have a tropical solution of the problem we can use it to solve the corresponding problems over the complex and real numbers.

The enumerative problems considered in this talk are finding the number of curves in projective spaces (over complex, real and tropical numbers) of given genus and degree constrained by certain incidence conditions (e.g. passing via points or lines). Floor diagrams are a combinatorial tool that reduces an enumerative problem in dimension n to the corresponding problem n dimension n-1. Floor diagrams give a constructive (and rather efficient) way to find all tropical curves for a given enumerative problem. And once we have a tropical solution of the problem we can use it to solve the corresponding problems over the complex and real numbers.

### 2007/09/12

15:00-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Classification of p-divisible groups by displays and duality

Applications of the theory of displays

Presentation of mapping class groups from algebraic geometry

**E. Lau**(Univ. of Bielefeld) 15:00-15:45Classification of p-divisible groups by displays and duality

**T. Zink**(Univ. of Bielefeld) 16:00-16:45Applications of the theory of displays

**E. Looijenga**(Univ. of Utrecht) 17:00-18:00Presentation of mapping class groups from algebraic geometry

[ Abstract ]

A presentation of the mapping class group of a genus g surface with one hole is due to Wajnryb with later improvements due to M. Matsumoto. The generators are Dehn twists defined by 2g+1 closed curves on the surface. The relations involving only two Dehn twists are the familiar Artin relations, we show that those involving more than two can be derived from algebro-geometry considerations.

A presentation of the mapping class group of a genus g surface with one hole is due to Wajnryb with later improvements due to M. Matsumoto. The generators are Dehn twists defined by 2g+1 closed curves on the surface. The relations involving only two Dehn twists are the familiar Artin relations, we show that those involving more than two can be derived from algebro-geometry considerations.

### 2007/08/29

17:00-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Computations on the moduli spaces of weighted log pairs

**Valery Alexeev**(Georgia大学)Computations on the moduli spaces of weighted log pairs

### 2007/08/02

16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Dynamics of automorphisms on algebraic varieties

**De-Qi Zhang**(Singapore大学)Dynamics of automorphisms on algebraic varieties

[ Abstract ]

The building blocks of automorphisms / endomorphisms of compact varieties are determined --- an algebro geometric approach towards dynamics.

The building blocks of automorphisms / endomorphisms of compact varieties are determined --- an algebro geometric approach towards dynamics.

### 2007/06/22

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Projective varieties with nef anti-canonical divisors

**Qi Zhang**(Missouri大学)Projective varieties with nef anti-canonical divisors

[ Abstract ]

Projective varieties with nef anti-canonical divisors appear naturally in the minimal model program and the theory of classification of higher-dimensional algebraic varieties. In this talk we describe a comprehensive approach to birational geometry of log canonical pair (X, D) with nef anti-canonical class -(K_X+D). In particular, We present two theorems on the birational structure of the varieties. We will also discuss some recent results and new aspects of the subject.

Projective varieties with nef anti-canonical divisors appear naturally in the minimal model program and the theory of classification of higher-dimensional algebraic varieties. In this talk we describe a comprehensive approach to birational geometry of log canonical pair (X, D) with nef anti-canonical class -(K_X+D). In particular, We present two theorems on the birational structure of the varieties. We will also discuss some recent results and new aspects of the subject.

### 2007/05/15

14:30-16:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Riemann-Roch for determinantal gerbes and smooth manifolds

**Mikhail Kapranov**(Yale 大学)Riemann-Roch for determinantal gerbes and smooth manifolds

[ Abstract ]

A version of the Riemann-Roch theorem for curves due to Deligne, describes the determinant of the cohomology of a vector bundle E on a curve.

If one realizes E via the Krichever construction, the determinant of the cohomology becomes a Hom-space in the determinantal gerbe for the vector space over the field of power series. So one has a ``local" Riemann-Roch problem of description of this gerbe itself. The talk will present the results of a joint work with E. Vasserot describing the class of such a gerbe in a family which geometrically can be seen as a circle fibration. This can be further generalized to the case of a fibration with fibers being smooth compact manifolds of any dimension d (joint work with P. Bressler, B. Tsygan and E. Vasserot).

A version of the Riemann-Roch theorem for curves due to Deligne, describes the determinant of the cohomology of a vector bundle E on a curve.

If one realizes E via the Krichever construction, the determinant of the cohomology becomes a Hom-space in the determinantal gerbe for the vector space over the field of power series. So one has a ``local" Riemann-Roch problem of description of this gerbe itself. The talk will present the results of a joint work with E. Vasserot describing the class of such a gerbe in a family which geometrically can be seen as a circle fibration. This can be further generalized to the case of a fibration with fibers being smooth compact manifolds of any dimension d (joint work with P. Bressler, B. Tsygan and E. Vasserot).

### 2007/05/07

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Pathologies on ruled surfaces in positive characteristic

**謝啓鴻(Xie Qihong)**(東大・数理)Pathologies on ruled surfaces in positive characteristic

[ Abstract ]

We discuss some pathologies of log varieties in positive characteristic. Mainly, we show that on ruled surfaces there are counterexamples of several logarithmic type theorems. On the other hand, we also give a characterization of the counterexamples of the Kawamata-Viehweg vanishing theorem on a geometrically ruled surface by means of the Tango invariant of the base curve.

We discuss some pathologies of log varieties in positive characteristic. Mainly, we show that on ruled surfaces there are counterexamples of several logarithmic type theorems. On the other hand, we also give a characterization of the counterexamples of the Kawamata-Viehweg vanishing theorem on a geometrically ruled surface by means of the Tango invariant of the base curve.

### 2007/03/26

15:30-17:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Existence of minimal models and flips (3rd talk of three)

**Professor Caucher Birkar**(University of Cambridge)Existence of minimal models and flips (3rd talk of three)

### 2007/03/22

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

On boundedness of log Fano varieties (2nd talk of three)

**Professor Caucher Birkar**(University of Cambridge)On boundedness of log Fano varieties (2nd talk of three)

### 2007/03/20

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

)

Singularities and termination of flips (1st talk of three)

**Professor Caucher Birkar**(University of Cambridge)

Singularities and termination of flips (1st talk of three)

### 2007/01/26

16:30-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

University of Utrecht

)

Irreducibility of strata and leaves in the moduli space of

abelian varieties I (a survey of results)

**Professor Frans Oort**

(Department of MathematicsUniversity of Utrecht

)

Irreducibility of strata and leaves in the moduli space of

abelian varieties I (a survey of results)

### 2006/12/08

15:00-16:25 Room #126 (Graduate School of Math. Sci. Bldg.)

Universität zu Köln

)

Rationally connected

foliations

**Stefan Kebekus 氏**(Mathematisches InstitutUniversität zu Köln

)

Rationally connected

foliations

### 2006/12/04

16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

When does a curve move on a surface, especially over a finite field?

**Professor Burt Totaro**

(University of Cambridge)When does a curve move on a surface, especially over a finite field?

### 2006/11/13

16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Hom stacks and Picard stacks

**青木昌雄**(京大数理研)Hom stacks and Picard stacks

### 2006/10/18

16:00-18:15 Room #122 (Graduate School of Math. Sci. Bldg.)

Jets of singular foliations and applications

to curves and their moduli spaces

Dual varieties, ramification, and Betti numbers

of projective varieties

**E. Esteves**(IMPA) 16:00-17:00Jets of singular foliations and applications

to curves and their moduli spaces

**F. Zak**(Independent Univ. of Moscow) 17:15-18:15Dual varieties, ramification, and Betti numbers

of projective varieties