Complex and Differential Geometry

Cover
Springer Proceedings in Mathematics
8
Complex and Differential Geometry
ISBN 9783642202995
Foreword
Contents
Participants
Surfaces of general type with geometric genus zero: a survey
1 Introduction
2 Notation
3 The classification problem and “simpler” sub-problems
3.1 Update on surfaces with
4 Other reasons why surfaces with pg = 0 have been of interest in the last 30 years
4.1 Bloch’s conjecture
4.2 Pluricanonical maps
4.3 Differential topology
5 Construction techniques
5.1 Quotients by a finite (resp. : infinite) group
5.1.1 Ball quotients
5.1.2 Product quotient surfaces
5.2 Galois coverings and their deformations
6 Keum-Naie surfaces and primary Burniat surfaces
References
Holomorphic symplectic geometry: a problem list
1 Introduction
2 Compact hyperk¨ahler manifolds
2.1 Basic definitions
2.2 Examples
2.3 The period map
2.4 Cohomology
2.5 Boundedness
2.6 Lagrangian fibrations
2.7 Projective families
3 Compact Poisson manifolds
4 Compact contact manifolds
References
Generalized Lagrangian mean curvature flow in K¨ahler manifolds that are almost Einstein
1 Introduction
2 Lagrangian mean curvature flow in K¨ahler-Einstein manifolds
3 Generalized Lagrangian mean curvature flow in K¨ahler manifolds that are almost Einstein
4 A variational approach to the generalized mean curvature flow
5 The case of almost Calabi-Yau manifolds
References
Einstein metrics and preserved curvature conditions for the Ricci flow
1 Introduction
2 Proof of Theorem 3
References
Differential Harnack Estimates for Parabolic Equations
1 Introduction
2 Proof of Theorem 1 and Application
3 Proof of Theorem 2
4 A Remark on the Conjugate Heat Equation
References
Euler characteristic of a complete intersection
1 Introduction
2 Blow–up of the Fulton–Johnson class
3 Differential forms
4 Euler characteristic computation
4.1 Hypersurface
4.2 Higher codimension complete intersections
5 The xy–genus
References
Cremona special sets of points in products of projective spaces
1 Introduction
2 The Cremona action ofWp
3 Examples of Cremona special sets
4 Association
References
Stable bundles and polyvector fields
1 Introduction
2 Polyvector fields
2.1 The construction
2.2 Injectivity
2.3 The Schouten-Nijenhuis bracket
3 Orthogonal bundles on the moduli space
3.1 Courant algebroids
3.2 A family of Courant algebroids
3.3 The orthogonal structure
4 A vanishing theorem
5 Generators and relations
5.1 Generators
5.2 Some relations
References
Buser-Sarnak invariant and projective normality of abelian varieties
1 Introduction and Statement of Results
2 Volume of subvarieties near a complex subtorus
3 Seshadri number along the diagonal of
4 Projective normality
References
Complete K¨ahler-Einstein Manifolds
1 The Classification Problem
2 Open manifolds
3 Complete Ricci-flat open manifolds
3.1 The assumptions of the classification result
3.2 Parametrizing complete Ricci-flat K¨ahler metrics
3.3 Asymptotic description of the metrics
4 Crepant Resolutions
References
Fixed point subalgebras of Weil algebras: from geometric to algebraic questions
1 Introduction
2 Starting points: product preserving functors
3 To the definition of the Weil algebra
4 Weil contact elements
5 Subalgebra of fixed points
Appendix: The computation method and two examples
References
Self-similar solutions and translating solutions
1 Introduction
2 Self-similar solutions to translating solutions
3 The geometric picture for
References
Aspects of conformal holonomy
1 Introduction
2 Conformal tractor holonomy
2.1 Standard tractors and connection.
2.2 Tractor holonomy.
3 Almost Einstein structures and holonomy
4 Decomposable conformal holonomy
4.1 The special Einstein product.
4.2 The collapsing sphere product.
4.3 The classification in Riemannian signature.
5 The case of unitary conformal holonomy
5.1 Fefferman construction reviewed.
5.2 Holonomy characterisation
5.3 Fefferman-Einstein metrics.
6 The generalised Fefferman construction
7 Overdetermined PDE and BGG-sequences
References
Bifurcation braid monodromy of plane curves
1 Introduction
2 Singularity theory
3 Braid monodromy maps and groups
4 Computation of bifurcation braid monodromy
5 Monodromy for spaces of plane projective curves
6 Braid monodromies versus geometric monodromies
References
A survey of Torelli and monodromy results for holomorphic-symplectic varieties
1 Introduction
1.1 Torelli Theorems
1.2 The fundamental exceptional chamber
1.3 Torelli and monodromy in the polarized case
1.4 The K3[n]-type
2 The Global Torelli Theorem
3 The Hodge theoretic Torelli Theorem
3.1 Parallel transport operators between inseparable marked pairs
3.2 Proof of the Hodge theoretic Torelli Theorem 1.3
4 Orientation
5 A modular description of each fiber of the period map
5.1 Exceptional divisors
5.1.1 The fundamental exceptional chamber versus the birational K¨ahler cone
5.1.2 The divisorial Zariski decomposition
5.2 A K¨ahler-type chamber decomposition of the positive cone
5.3 ML as the moduli space of K¨ahler-type chambers
6 Mon2Hdg (X) is generated by reflections and Mon2Bir (X)
6.1 Reflections
6.2 Stably prime-exceptional line bundles
6.3 Hyperbolic reflection groups
6.4 Mon2Hdg (X) is a semi-direct product ofWExc and Mon2Bir (X)
6.5 Morrison’s movable cone conjecture
7 The monodromy and polarized monodromy groups
7.1 Polarized parallel transport operators
7.2 Deformation types of polarized marked pairs
8 Monodromy quotients of type IV period domains
9 The K3[n] deformation type
9.1 Characterization of parallel-transport operators of K3[n]-type
9.1.1 First two characterizations of Mon2(K3[n])
9.1.2 A third characterization of Mon2(K3[n])
9.1.3 Generators for the cohomology ring H(X;Z)
9.1.4 Parallel transport operators of
9.2 A numerical determination of the fundamental exceptional chamber
9.2.1 Monodromy-reflective classes of K3[n]-type
9.2.2 Stably prime-exceptional classes of K3[n]-type
10 Open problems
References
On singularities of generically immersive holomorphic maps between complex hyperbolic space forms
1 Background
2 Singular loci in the finite-volume case
3 Contracting leafwise totally geodesic isometric embeddings
4 A commutation formula
5 Consequences of the commutation formula
References
Generically nef vector bundles and geometric applications
1 Introduction
2 The movable cone
3 Generically nef vector bundles
4 The cotangent bundle
5 The tangent bundle
References
Dolbeault cohomology of nilmanifolds with left-invariant complex structure
1 Introduction
1.1 Notations
2 Real nilmanifolds and Nomizu’s result on de Rham cohomology
3 Left-invariant complex structures and Dolbeault cohomology
3.1 Reminder on Dolbeault cohomology
3.2 The inductive proof
3.2.1 When is a nilmanifold with left-invariant complex structure an iterated (principal) bundle?
3.3 Console and Fino’s result on openness
3.4 Some new results and open questions
4 Applications
4.1 Prescribing cohomology behaviour and the Fr¨olicher spectral sequence
4.2 Deformations of complex structures
References
Smooth rationally connected threefolds contain all smooth curves
1 Rationally connected varieties
1.1 RC and SRC
1.2 Maps from curves
2 Toric varieties
2.1 Maps to toric varieties
2.2 Embedding a curve
References
Submanifolds in Poisson geometry: a survey
1 Poisson geometry
1.1 Submanifolds and symplectic leaves
1.2 Lie algebroids and Dirac manifolds
2 Coisotropic submanifolds
3 Poisson-Dirac submanifolds
3.1 Lie-Dirac submanifolds
3.2 Cosymplectic submanifolds
4 Pre-Poisson submanifolds
4.1 Embeddings of pre-Poisson submanifolds
4.2 Quotients of pre-Poisson submanifolds
4.3 Relation to subgroupoids of
References