## Algebraic Geometry Seminar

Seminar information archive ～09/27｜Next seminar｜Future seminars 09/28～

Date, time & place | Tuesday 10:30 - 11:30 or 12:00 ハイブリッド開催/002Room #ハイブリッド開催/002 (Graduate School of Math. Sci. Bldg.) |
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**Seminar information archive**

### 2008/11/07

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

Hyperkaehler SYZ conjecture and stability

**Misha Verbitsky**(ITEP and IPMU)Hyperkaehler SYZ conjecture and stability

[ Abstract ]

Let L be a nef bundle on a hyperkaehler manifold. A Hyperkaehler SYZ conjecture postulates that L is semi-ample. As shown by Matsushita, this implies existence of holomorphic Lagrangian fibrations on hyperkaehler manifolds. It was conjectured by many

people, most recently by Tschinkel, Hassett, Huybrechts and Sawon. We prove that a sufficiently big power of L is effective, assuming that L admits a semi-positive metric. A multiplier ideal version of this argument would give effectivity of L^N for any nef L. The proof uses stability and Boucksom's divisorial

Zariski decomposition.

Let L be a nef bundle on a hyperkaehler manifold. A Hyperkaehler SYZ conjecture postulates that L is semi-ample. As shown by Matsushita, this implies existence of holomorphic Lagrangian fibrations on hyperkaehler manifolds. It was conjectured by many

people, most recently by Tschinkel, Hassett, Huybrechts and Sawon. We prove that a sufficiently big power of L is effective, assuming that L admits a semi-positive metric. A multiplier ideal version of this argument would give effectivity of L^N for any nef L. The proof uses stability and Boucksom's divisorial

Zariski decomposition.

### 2008/10/17

13:00-14:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Construction of surfaces of general type with pg=0 via

Q-Gorenstein smoothing

**Yongnam Lee**(Sogang U.)Construction of surfaces of general type with pg=0 via

Q-Gorenstein smoothing

### 2008/04/21

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

Scorza quartics of trigonal spin curves and their varieties of power sums

**高木寛通**(東大数理)Scorza quartics of trigonal spin curves and their varieties of power sums

[ Abstract ]

Our fundamental result is the construction of new subvarieties in the varieties of power sums for the Scorza quartic of any general pairs of trigonal curves and non-effective theta characteristics. This is a generalization of Mukai's description of smooth prime Fano threefolds of genus twelve as the varieties of power sums for plane quartics. Among other applications, we give an affirmative answer to the conjecture of Dolgachev and Kanev on the existence of the Scorza quartic for any general pairs of curves and non-effective theta characteristics.

Our fundamental result is the construction of new subvarieties in the varieties of power sums for the Scorza quartic of any general pairs of trigonal curves and non-effective theta characteristics. This is a generalization of Mukai's description of smooth prime Fano threefolds of genus twelve as the varieties of power sums for plane quartics. Among other applications, we give an affirmative answer to the conjecture of Dolgachev and Kanev on the existence of the Scorza quartic for any general pairs of curves and non-effective theta characteristics.

### 2008/03/14

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

Understanding singular algebraic varieties via string theory

**David Morrison**(UC Santa Barbara)Understanding singular algebraic varieties via string theory

[ Abstract ]

String theory has helped to formulate two major new insights in the study of singular algebraic varieties. The first -- which also arose from symplectic geometry -- is that families of Kaehler metrics are an important tool in uncovering the structure of singular algebraic varieties. The second, more recent insight -- related to independent work in the representation theory of associative algebras -- is that one's understanding of a singular (affine) algebraic variety is enhanced if one can find a non-commutative ring whose center is the coordinate ring of the variety. We will describe both of these insights, and explain how they are related to string theory.

String theory has helped to formulate two major new insights in the study of singular algebraic varieties. The first -- which also arose from symplectic geometry -- is that families of Kaehler metrics are an important tool in uncovering the structure of singular algebraic varieties. The second, more recent insight -- related to independent work in the representation theory of associative algebras -- is that one's understanding of a singular (affine) algebraic variety is enhanced if one can find a non-commutative ring whose center is the coordinate ring of the variety. We will describe both of these insights, and explain how they are related to string theory.

### 2008/01/29

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

Homological methods in non-commutative geometry, part 11 (last lecture)

[ Reference URL ]

http://imperium.lenin.ru/~kaledin/math/tokyo/

**Dmitry KALEDIN**(Steklov研究所, 東大数理)Homological methods in non-commutative geometry, part 11 (last lecture)

[ Reference URL ]

http://imperium.lenin.ru/~kaledin/math/tokyo/

### 2008/01/22

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

Homological methods in non-commutative geometry, part 10

[ Reference URL ]

http://imperium.lenin.ru/~kaledin/math/tokyo/

**Dmitry KALEDIN**(Steklov研究所, 東大数理)Homological methods in non-commutative geometry, part 10

[ Reference URL ]

http://imperium.lenin.ru/~kaledin/math/tokyo/

### 2008/01/15

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

Homological methods in non-commutative geometry, part 9

[ Reference URL ]

http://imperium.lenin.ru/~kaledin/math/tokyo/

**Dmitry KALEDIN**(Steklov研究所, 東大数理)Homological methods in non-commutative geometry, part 9

[ Reference URL ]

http://imperium.lenin.ru/~kaledin/math/tokyo/

### 2008/01/08

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

Homological methods in non-commutative geometry, part 8

**Dmitry KALEDIN**(Steklov研究所, 東大数理)Homological methods in non-commutative geometry, part 8

### 2007/12/11

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

Homological methods in non-commutative geometry, part 7

**Dmitry KALEDIN**(Steklov研究所, 東大数理)Homological methods in non-commutative geometry, part 7

### 2007/11/27

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

Categorical resolutions of singularities

**Alexander Kuznetsov**(Steklov Inst)Categorical resolutions of singularities

[ Abstract ]

I will give a definition of a categorical resolution of singularities and explain how such resolutions can be constructed.

I will give a definition of a categorical resolution of singularities and explain how such resolutions can be constructed.

### 2007/11/08

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

New restrictions on the fundamental groups of complex algebraic varieties

**Alexandru DIMCA**(Univ Nice )New restrictions on the fundamental groups of complex algebraic varieties

[ Abstract ]

My talk will be based on joint work with S. Papadima (Bucarest, Romania) and A. Suciu (Boston, USA). First I will recall the basic facts on characteristic varieties $V_k(M)$ associated to rank one local systems on a complex algebraic variety $M$ which are due to Beauville, Simpson and Arapura. Then I will introduce the resonance varities $R_k(M)$, which may be related to the Isotropic Subspace Theorems by Catanese and Bauer. One of the main new results is that for a class of algebraic varieties (the 1-formal ones), the two types of varieties $V_k(M)$ and $R_k(M)$ are strongly related. Applications to right angle Artin groups, Bestvina-Brady groups and to a conjecture by Kollar will be discussed in the end.

My talk will be based on joint work with S. Papadima (Bucarest, Romania) and A. Suciu (Boston, USA). First I will recall the basic facts on characteristic varieties $V_k(M)$ associated to rank one local systems on a complex algebraic variety $M$ which are due to Beauville, Simpson and Arapura. Then I will introduce the resonance varities $R_k(M)$, which may be related to the Isotropic Subspace Theorems by Catanese and Bauer. One of the main new results is that for a class of algebraic varieties (the 1-formal ones), the two types of varieties $V_k(M)$ and $R_k(M)$ are strongly related. Applications to right angle Artin groups, Bestvina-Brady groups and to a conjecture by Kollar will be discussed in the end.

### 2007/10/30

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

Homological methods in Non-commutative Geometry

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

### 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)