Holomorphic symplectic geometry: a problem list

A list of open problems on holomorphic symplectic, contact and Poisson manifolds

A travel guide to the canonical bundle formula

We survey known results on the canonical bundle formula and its applications in algebraic geometry.Comment: 17 pages, to appear in the Proceedings of the conference Birational Geometry and Moduli Space

On the Frobenius integrability of certain holomorphic p-forms

Abstract. The goal of this note is to exhibit the integrability properties (in the sense of the Frobenius theorem) of holomorphic p-forms with values in certain line bundles with nonpositive curvature on a compact Kähler manifold. There are in fact very strong restrictions, both on the holomorphic form and on the curvature of the line bundle. In particular, these observations provide interesting information on the structure of projective manifolds which admit a contact structure: either they are Fano manifolds or they are biholomorphic to the projectivization of the cotangent bundle of another suitable projective manifold. 1. Main results Recall that a holomorphic line bundle L on a compact complex manifold is said to be pseudo-effective if c1(L) contains a closed positive (1, 1)-current T, or equivalently, if L possesses a (possibly singular) hermitian metric h such that the curvature current T = Θh(L) = −i∂ ∂ log h is nonnegative. Our main result is Main Theorem. Let X be a compact Kähler manifold. Assume that there exists a pseudo-effective line bundle L on X and a nonzero holomorphic section θ ∈ H0 (X, Ω p X ⊗ L−1), where 0 � p � n = dimX. Let Sθ be the coherent subsheaf of germs of vector fields ξ in the tangent sheaf TX, such that the contraction iξθ vanishes. Then Sθ is integrable, namely [Sθ, Sθ] ⊂ Sθ, and L has flat curvature along the leaves of the (possibly singular) foliation defined by Sθ. Before entering into the proof, we discuss several consequences. If p = 0 or p = n, the result is trivial (with Sθ = TX and Sθ = 0, respectively). The most interesting case is p = 1. Corollary 1. In the above situation, if the line bundle L → X is pseudoeffective and θ ∈ H 0 (X, Ω 1 X ⊗ L−1) is a nonzero section, the subsheaf Sθ defines a holomorphic foliation of codimension 1 in X, that is, θ ∧ dθ = 0. We now concentrate ourselves on the case when X is a contact manifold, i.e. dim X = n = 2m + 1, m � 1, and there exists a form θ ∈ H 0 (X, Ω 1 X ⊗ L−1), called the contact form, such that θ ∧(dθ) m ∈ H 0 (X, KX ⊗L −m−1) has no zeroes. Then Sθ is a codimension 1 locally free subsheaf of TX and there are dual exact2 Frobenius integrability of certain holomorphic p-forms sequence

Improving the dynamics of Northern Hemisphere high-latitude vegetation in the ORCHIDEE ecosystem model

Processes that describe the distribution of vegetation and ecosystem succession after disturbance are an important component of dynamic global vegetation models (DGVMs). The vegetation dynamics module (ORC-VD) within the process-based ecosystem model ORCHIDEE (Organizing Carbon and Hydrology in Dynamic Ecosystems) has not been updated and evaluated since many years and is known to produce unrealistic results. This study presents a new parameterization of ORC-VD for mid- to high-latitude regions in the Northern Hemisphere, including processes that influence the existence, mortality and competition between tree functional types. A new set of metrics is also proposed to quantify the performance of ORC-VD, using up to five different data sets of satellite land cover, forest biomass from remote sensing and inventories, a data-driven estimate of gross primary productivity (GPP) and two gridded data sets of soil organic carbon content. The scoring of ORC-VD derived from these metrics integrates uncertainties in the observational data sets. This multi-data set evaluation framework is a generic method that could be applied to the evaluation of other DGVM models. The results of the original ORC-VD published in 2005 for mid- to high-latitudes and of the new parameterization are evaluated against the above-described data sets. Significant improvements were found in the modeling of the distribution of tree functional types north of 40° N. Three additional sensitivity runs were carried out to separate the impact of different processes or drivers on simulated vegetation distribution, including soil freezing which limits net primary production through soil moisture availability in the root zone, elevated CO₂ concentration since 1850, and the effects of frequency and severity of extreme cold events during the spin-up phase of the model