Complex and Differential Geometry: Conference held at Leibniz Universität Hannover, September 14 – 18, 2009

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