# Summer Lectures in Geometry

## Past sessions

### Arrangements, Hypergeometric Functions, and KZ-Type Equations - III

Conformal blocks in conformal field theory and their quantizations are solutions of Knizhnik-Zamolodchikov differential and difference equations. The geometric version of these objects is multidimensional hypergeometric functions and hypergeometric equaitons. The interrelations of these subjects, the theory of solvable models in statistical mechanics, and representation theory will be discussed.

### Non-abelian Cohomology and Gerbes - II

The purpose of these lectures is to introduce the audience to the topological concepts of stack, gerbe and bundle gerbe, and the non-abelian cohomology classes to which they give rise.

#### Lecture 2

Introduction to stacks and gerbes.

#### References

• L. Breen, On the classification of 2-gerbes and 2-stacks, Asterisque 225, (1994).
• J. Giraud, Cohomologie nonabelienne, Springer-Verlag, 1971.
• I. Moerdijk, On the classification of regular Lie groupoids, preprint math.DG/0203099.
• M. Murray, Bundle gerbes, J. London Math. Soc. 54 (1996).

### Arrangements, Hypergeometric Functions, and KZ-Type Equations - II

Conformal blocks in conformal field theory and their quantizations are solutions of Knizhnik-Zamolodchikov differential and difference equations. The geometric version of these objects is multidimensional hypergeometric functions and hypergeometric equaitons. The interrelations of these subjects, the theory of solvable models in statistical mechanics, and representation theory will be discussed.

### Periodic Orbits and Symplectic Topology

Introduction: Hamiltonian flows, the Arnold and Weinstein conjectures, examples: convex Hamiltonians and flows on twisted cotangent bundles, review of results.

Floer homology: review of Morse theory, Floer homology and symplectic homology, application: the Arnold conjecture.

Almost existence theorems for periodic orbits: symplectic capacities almost existence and the Hofer-Zehnder capacity, application: Viterbo's proof of Weinstein conjecture and almost existence in twisted cotangent bundles.

The Hamiltonian Seifert conjecture: the Seifert conjecture and counterexamples, Hamiltonian dynamical systems without periodic orbits.

### Arrangements, Hypergeometric Functions, and KZ-Type Equations - I

Conformal blocks in conformal field theory and their quantizations are solutions of Knizhnik-Zamolodchikov differential and difference equations. The geometric version of these objects is multidimensional hypergeometric functions and hypergeometric equaitons. The interrelations of these subjects, the theory of solvable models in statistical mechanics, and representation theory will be discussed.

### Non-abelian Cohomology and Gerbes - I

The purpose of these lectures is to introduce the audience to the topological concepts of stack, gerbe and bundle gerbe, and the non-abelian cohomology classes to which they give rise.

#### Lecture 1

Sheaves, torsors and non-abelian cohomology in degree 1.

#### References

• L. Breen, On the classification of 2-gerbes and 2-stacks, Asterisque 225, (1994).
• J. Giraud, Cohomologie nonabelienne, Springer-Verlag, 1971.
• I. Moerdijk, On the classification of regular Lie groupoids, preprint math.DG/0203099.
• M. Murray, Bundle gerbes, J. London Math. Soc. 54 (1996).

### Introduction to the application of $$\overline{\partial}$$ estimates to complex geometry - III

The series of three lectures will discuss the recent applications of $$L^2$$ estimates of to geometric problems. The main technique of these applications is the use of multiplier ideal sheaves. The use of $$\overline{\partial}$$ multiplier ideal sheaves is a completely new way of deriving a priori estimates of partial differential equations. So far the technique has been developed only for the equation but should be adaptable to other systems of partial differential equations arising from geometric problems.

When a priori estimates cannot be readily derived by the usual methods of integration by parts, one multiplies the quantity to be estimated by a function to make the a priori estimate hold. The set of all such multipliers form an ideal sheaf. Global geometric conditions are studied which can force the ideal to be the unit ideal, thereby making the desired a priori estimate automatically hold. This method is a powerful tool for many geometric problems.

Without the assumption of any pre-requisites, this series of lectures starts with the derivation of $$L^2$$ estimates of $$\overline{\partial}$$. Then the two kinds of multiplier ideal sheaves, Kohn's and Nadel's, are introduced, along with the problems and motivations from which they originate.

As examples of the geometric application of multiplier ideal sheaves, the following kinds of problems in algebraic geometry are discussed:

1. Fujita's conjecture which says that if L is a positive holomorphic line bundle over a compact complex manifold $$X$$ of complex dimension n with canonical line bundle $$K_X$$, then $$m L+K_X$$ is generated by global holomorphic sections when $$m\geq n+1$$ and is very ample when $$m\geq n+2$$ (in the sense that any basis of the space of global holomorphic sections define a holomorphic embedding of $$X$$ into a complex projective space).
2. Effective Matsusaka's Theorem which says that if $$L$$ is a positive holomorphic line bundle over a compact complex manifold $$X$$ of complex dimension $$n$$, then $$m L$$ is very ample when $$m$$ is greater than some explicit effective function of the Chern numbers $$L^n$$ and $$L^{n-1} K_X$$.
3. Deformational invariance of plurigenera which says that $$m$$-genus of a compact complex projective algebraic manifold $$X$$ ( i.e., the dimension of the space of global holomorphic sections of $$m K_X$$ over $$X$$) is unchanged when $$X$$ is holomorphically deformed.

### Introduction to the Hartshorne Conjecture - III

In the early 60's Hartshorne studied the subvarieties that are the generalization of ample divisors for higher codimensions. Motivated by his study Hartshorne proposed the following conjecture: Let $$X$$ be a smooth projective variety, $$A$$ and $$B$$ be two smooth subvarieties of $$X$$ with ample normal bundle and such that $$\dim A + \dim B \geq \dim X$$. Then $$A$$ intersects $$B$$. We will use this problem to illustrate the interplay of complex, differential and algebraic geometry. We will always target a diverse audience. To that effect we review notions of algebraic geometry : line bundles, vector bundles, $$P^n$$-bundles and the ampleness property. From complex differential geometry: Kahler manifolds, Hermitean metrics, connections, curvature, the positivity of vector bundles and vanishing theorems. From several complex variables: strongly q-convex spaces, the finiteness of cohomology groups and cycle spaces of complex manifolds. The notions and results mentioned above will then be applied to explain the reason of the conjecture, why the conjecture might not be true and to prove special cases. In particular we will do the case where the ambient variety $$X$$ is a hypersurface in $$P^n$$ (done by Barlet), a $$P^ 2$$-bundle over a surface (done by Barlet, Schneider and Peternel) and a $$P^1$$-bundle over a threefold.

### Introduction to the application of $$\overline{\partial}$$ estimates to complex geometry - II

The series of three lectures will discuss the recent applications of $$L^2$$ estimates of to geometric problems. The main technique of these applications is the use of multiplier ideal sheaves. The use of $$\overline{\partial}$$ multiplier ideal sheaves is a completely new way of deriving a priori estimates of partial differential equations. So far the technique has been developed only for the equation but should be adaptable to other systems of partial differential equations arising from geometric problems.

When a priori estimates cannot be readily derived by the usual methods of integration by parts, one multiplies the quantity to be estimated by a function to make the a priori estimate hold. The set of all such multipliers form an ideal sheaf. Global geometric conditions are studied which can force the ideal to be the unit ideal, thereby making the desired a priori estimate automatically hold. This method is a powerful tool for many geometric problems.

Without the assumption of any pre-requisites, this series of lectures starts with the derivation of $$L^2$$ estimates of $$\overline{\partial}$$. Then the two kinds of multiplier ideal sheaves, Kohn's and Nadel's, are introduced, along with the problems and motivations from which they originate.

As examples of the geometric application of multiplier ideal sheaves, the following kinds of problems in algebraic geometry are discussed:

1. Fujita's conjecture which says that if L is a positive holomorphic line bundle over a compact complex manifold $$X$$ of complex dimension n with canonical line bundle $$K_X$$, then $$m L+K_X$$ is generated by global holomorphic sections when $$m\geq n+1$$ and is very ample when $$m\geq n+2$$ (in the sense that any basis of the space of global holomorphic sections define a holomorphic embedding of $$X$$ into a complex projective space).
2. Effective Matsusaka's Theorem which says that if $$L$$ is a positive holomorphic line bundle over a compact complex manifold $$X$$ of complex dimension $$n$$, then $$m L$$ is very ample when $$m$$ is greater than some explicit effective function of the Chern numbers $$L^n$$ and $$L^{n-1} K_X$$.
3. Deformational invariance of plurigenera which says that $$m$$-genus of a compact complex projective algebraic manifold $$X$$ ( i.e., the dimension of the space of global holomorphic sections of $$m K_X$$ over $$X$$) is unchanged when $$X$$ is holomorphically deformed.

### Introduction to the application of $$\overline{\partial}$$ estimates to complex geometry - I

The series of three lectures will discuss the recent applications of $$L^2$$ estimates of to geometric problems. The main technique of these applications is the use of multiplier ideal sheaves. The use of $$\overline{\partial}$$ multiplier ideal sheaves is a completely new way of deriving a priori estimates of partial differential equations. So far the technique has been developed only for the equation but should be adaptable to other systems of partial differential equations arising from geometric problems.

When a priori estimates cannot be readily derived by the usual methods of integration by parts, one multiplies the quantity to be estimated by a function to make the a priori estimate hold. The set of all such multipliers form an ideal sheaf. Global geometric conditions are studied which can force the ideal to be the unit ideal, thereby making the desired a priori estimate automatically hold. This method is a powerful tool for many geometric problems.

Without the assumption of any pre-requisites, this series of lectures starts with the derivation of $$L^2$$ estimates of $$\overline{\partial}$$. Then the two kinds of multiplier ideal sheaves, Kohn's and Nadel's, are introduced, along with the problems and motivations from which they originate.

As examples of the geometric application of multiplier ideal sheaves, the following kinds of problems in algebraic geometry are discussed:

1. Fujita's conjecture which says that if L is a positive holomorphic line bundle over a compact complex manifold $$X$$ of complex dimension n with canonical line bundle $$K_X$$, then $$m L+K_X$$ is generated by global holomorphic sections when $$m\geq n+1$$ and is very ample when $$m\geq n+2$$ (in the sense that any basis of the space of global holomorphic sections define a holomorphic embedding of $$X$$ into a complex projective space).
2. Effective Matsusaka's Theorem which says that if $$L$$ is a positive holomorphic line bundle over a compact complex manifold $$X$$ of complex dimension $$n$$, then $$m L$$ is very ample when $$m$$ is greater than some explicit effective function of the Chern numbers $$L^n$$ and $$L^{n-1} K_X$$.
3. Deformational invariance of plurigenera which says that $$m$$-genus of a compact complex projective algebraic manifold $$X$$ ( i.e., the dimension of the space of global holomorphic sections of $$m K_X$$ over $$X$$) is unchanged when $$X$$ is holomorphically deformed.

### Introduction to the Hartshorne Conjecture - II

In the early 60's Hartshorne studied the subvarieties that are the generalization of ample divisors for higher codimensions. Motivated by his study Hartshorne proposed the following conjecture: Let $$X$$ be a smooth projective variety, $$A$$ and $$B$$ be two smooth subvarieties of $$X$$ with ample normal bundle and such that $$\dim A + \dim B \geq \dim X$$. Then $$A$$ intersects $$B$$. We will use this problem to illustrate the interplay of complex, differential and algebraic geometry. We will always target a diverse audience. To that effect we review notions of algebraic geometry : line bundles, vector bundles, $$P^n$$-bundles and the ampleness property. From complex differential geometry: Kahler manifolds, Hermitean metrics, connections, curvature, the positivity of vector bundles and vanishing theorems. From several complex variables: strongly q-convex spaces, the finiteness of cohomology groups and cycle spaces of complex manifolds. The notions and results mentioned above will then be applied to explain the reason of the conjecture, why the conjecture might not be true and to prove special cases. In particular we will do the case where the ambient variety $$X$$ is a hypersurface in $$P^n$$ (done by Barlet), a $$P^ 2$$-bundle over a surface (done by Barlet, Schneider and Peternel) and a $$P^1$$-bundle over a threefold.

### Survey of the geometric and analytic results in contact structures - III

The relative index. The theory of relative index is presented in detail. Using this concept we give an analytic proof that the set of fillable deformations of the CR-structure on certain three manifolds is a closed set. If time permits we will discuss connections between the relative index and the contact mapping class group.

### Symplectic fibrations and quantum homology - IV

By the very definition of a symplectic form, every symplectic manifold is locally fibered. Donaldson's theorem on Lefschetz pencils shows that a somehow similar statement (with singularities) is actually true at a global level. This means that symplectic fibrations play a fundamental role in symplectic topology and geometry. The goal of these lectures is to introduce to the theory of non- singular symplectic fibrations in arbitrary dimensions. Here, by symplectic fibration, we mean a symplectic manifold which is fibered by symplectic submanifolds. I will show that this is essentially equivalent to a topological fibration $$(M,\omega) \to P \to B$$ with structure group equal to the group of Hamiltonian diffeomorphisms of $$M$$ (at least when $$B$$ is a simply connected symplectic manifold). I will discuss the topology of such fibrations, showing in particular that their rational cohomology splits in a many interesting cases (this is a strong generalisation of works of Kirwan and Atiyah-Bott). This gives hard obstructions to the construction of new symplectic manifolds by twisted products of two symplectic manifolds. I will sketch the proof, based on a geometric interpretation of the Seidel map in quantum homology. If time permits, I will mention the consequences of this on the discovery of new rigidity phenomenon in Hamiltonian dynamics. Most of this work is the result of a collaboration with McDuff and Polterovich.

### Survey of the geometric and analytic results in contact structures - II

Filling three dimensional CR-manifolds. Three dimensional CR-manifolds have, in some sense, too many deformations because most cannot be realized as the boundaries of Stein spaces. This problem is related to that of finding symplectic fillings for contact manifolds. We give examples and consider the general features of this pathology. Lempert's algebraic approximation theorem gives a way to address this problem. We introduce this approach and consider the case of hypersurfaces in lines bundles over $$P^1$$ in detail. We define the relative index which provides a measure of the change in the algebra of CR-functions under a deformation of the CR-structure.

### Symplectic fibrations and quantum homology - III

By the very definition of a symplectic form, every symplectic manifold is locally fibered. Donaldson's theorem on Lefschetz pencils shows that a somehow similar statement (with singularities) is actually true at a global level. This means that symplectic fibrations play a fundamental role in symplectic topology and geometry. The goal of these lectures is to introduce to the theory of non- singular symplectic fibrations in arbitrary dimensions. Here, by symplectic fibration, we mean a symplectic manifold which is fibered by symplectic submanifolds. I will show that this is essentially equivalent to a topological fibration $$(M,\omega) \to P \to B$$ with structure group equal to the group of Hamiltonian diffeomorphisms of $$M$$ (at least when $$B$$ is a simply connected symplectic manifold). I will discuss the topology of such fibrations, showing in particular that their rational cohomology splits in a many interesting cases (this is a strong generalisation of works of Kirwan and Atiyah-Bott). This gives hard obstructions to the construction of new symplectic manifolds by twisted products of two symplectic manifolds. I will sketch the proof, based on a geometric interpretation of the Seidel map in quantum homology. If time permits, I will mention the consequences of this on the discovery of new rigidity phenomenon in Hamiltonian dynamics. Most of this work is the result of a collaboration with McDuff and Polterovich.

### Introduction to the Hartshorne Conjecture - I

In the early 60's Hartshorne studied the subvarieties that are the generalization of ample divisors for higher codimensions. Motivated by his study Hartshorne proposed the following conjecture: Let $$X$$ be a smooth projective variety, $$A$$ and $$B$$ be two smooth subvarieties of $$X$$ with ample normal bundle and such that $$\dim A + \dim B \geq \dim X$$. Then $$A$$ intersects $$B$$. We will use this problem to illustrate the interplay of complex, differential and algebraic geometry. We will always target a diverse audience. To that effect we review notions of algebraic geometry : line bundles, vector bundles, $$P^n$$-bundles and the ampleness property. From complex differential geometry: Kahler manifolds, Hermitean metrics, connections, curvature, the positivity of vector bundles and vanishing theorems. From several complex variables: strongly q-convex spaces, the finiteness of cohomology groups and cycle spaces of complex manifolds. The notions and results mentioned above will then be applied to explain the reason of the conjecture, why the conjecture might not be true and to prove special cases. In particular we will do the case where the ambient variety $$X$$ is a hypersurface in $$P^n$$ (done by Barlet), a $$P^ 2$$-bundle over a surface (done by Barlet, Schneider and Peternel) and a $$P^1$$-bundle over a threefold.

### Survey of the geometric and analytic results in contact structures - I

Several complex variables. After a quick introduction to complex structures and holomorphic functions of several variables we turn to the special features of higher dimensions: the Hartogs phenomenon, CR-structures, pseudoconvexity and the Lewy extension theorem. We define the $$d$$-barb complex and the Kohn-Rossi cohomology, discussing its connection to interior singularities. The lecture concludes with an introduction to 3-dimensional CR-manifolds.

### Symplectic fibrations and quantum homology - II

By the very definition of a symplectic form, every symplectic manifold is locally fibered. Donaldson's theorem on Lefschetz pencils shows that a somehow similar statement (with singularities) is actually true at a global level. This means that symplectic fibrations play a fundamental role in symplectic topology and geometry. The goal of these lectures is to introduce to the theory of non- singular symplectic fibrations in arbitrary dimensions. Here, by symplectic fibration, we mean a symplectic manifold which is fibered by symplectic submanifolds. I will show that this is essentially equivalent to a topological fibration $$(M,\omega) \to P \to B$$ with structure group equal to the group of Hamiltonian diffeomorphisms of $$M$$ (at least when $$B$$ is a simply connected symplectic manifold). I will discuss the topology of such fibrations, showing in particular that their rational cohomology splits in a many interesting cases (this is a strong generalisation of works of Kirwan and Atiyah-Bott). This gives hard obstructions to the construction of new symplectic manifolds by twisted products of two symplectic manifolds. I will sketch the proof, based on a geometric interpretation of the Seidel map in quantum homology. If time permits, I will mention the consequences of this on the discovery of new rigidity phenomenon in Hamiltonian dynamics. Most of this work is the result of a collaboration with McDuff and Polterovich.

### Symplectic fibrations and quantum homology - I

By the very definition of a symplectic form, every symplectic manifold is locally fibered. Donaldson's theorem on Lefschetz pencils shows that a somehow similar statement (with singularities) is actually true at a global level. This means that symplectic fibrations play a fundamental role in symplectic topology and geometry. The goal of these lectures is to introduce to the theory of non- singular symplectic fibrations in arbitrary dimensions. Here, by symplectic fibration, we mean a symplectic manifold which is fibered by symplectic submanifolds. I will show that this is essentially equivalent to a topological fibration $$(M,\omega) \to P \to B$$ with structure group equal to the group of Hamiltonian diffeomorphisms of $$M$$ (at least when $$B$$ is a simply connected symplectic manifold). I will discuss the topology of such fibrations, showing in particular that their rational cohomology splits in a many interesting cases (this is a strong generalisation of works of Kirwan and Atiyah-Bott). This gives hard obstructions to the construction of new symplectic manifolds by twisted products of two symplectic manifolds. I will sketch the proof, based on a geometric interpretation of the Seidel map in quantum homology. If time permits, I will mention the consequences of this on the discovery of new rigidity phenomenon in Hamiltonian dynamics. Most of this work is the result of a collaboration with McDuff and Polterovich.

For detailed overviews of each course see http://camgsd.ist.utl.pt/encontros/slg/.