Study Kodaira dimensions on compact almost complex manifolds.
problem Understanding invariants on almost complex manifolds.
method Introduce plurigenera, Kodaira dimension, and Iitaka dimension based on Hodge theory. Prove Hartogs extension theorem using foliation-by-disks technique.
result Show that plurigenera and Kodaira dimension are birational invariants in almost complex category, especially in dimension 4.
We derive sufficient conditions for the vanishing of plurigenera, pm(J),m>0, on compact (l|k)-strong, ωl∧∂∂ˉωk=0, Kaehler manifolds with torsion. In particular, we show that the plurigenera of compact (l|k)-strong manifolds, k<n-1, for which the holonomy of the unique Hermitian connectio…
The study bounds growth of Hodge numbers and computes L2-Betti numbers for irregular varieties.
problem Bounding growth of normalized Hodge numbers and computing L2-Betti numbers for irregular varieties. method Analysis of abelian covers, weak generic Nakano vanishing theorem, and convergence of plurigenera.
result Optimal bounds on the growth of normalized Hodge numbers and computation of L2-Betti numbers. We show various vanishing theorems for the cohomology groups of compact hermitian manifolds for which the Bismut connection has (restricted) holonomy contained in SU(n) and classify all such manifolds of dimension four. In this way we provide necessary conditions for the existence of such structures on hermitian manifo…
We give a self contained proof using Seiberg Witten invariants that for Kähler surfaces with non negative Kodaira dimension (including those with pg=0) the canonical class of the minimal model and the (−1)-curves, are oriented diffeomorphism invariants up to sign. This implies that the Kodaira dimension is deter…
Overview of algebraic geometry for almost complex manifolds.
problem Developing algebraic geometry for almost complex manifolds without genericity.
method Reviewing results based on pseudoholomorphic maps and intersection theory.
result Introduction of birational morphism between almost complex manifolds.
Study Kodaira dimensions on almost complex manifolds, proving integrability and structural descriptions.
problem Understanding Kodaira dimensions on almost complex manifolds.
method Using pseudoholomorphic pluricanonical maps, defining new dimensions, and applying probabilistic combinatorics.
result Almost complex structures with top Kodaira dimension are integrable, and for compact 4-manifolds, they have elliptic fibration structures.
Study extends complex sections on non-holomorphic objects on Kähler manifolds.
problem Extension of smooth sections on non-holomorphic objects on Kähler manifolds.
method Use of asymptotically holomorphic line bundles, two twisted Laplace-type operators, and Bochner-Kodaira-Nakano-type inequalities.
result Extensions of smooth sections with control of their L2-norms for non-integrable objects. The paper introduces lattice homology for integrally closed submodules and applies it to geometric invariants.
problem Computing numerical invariants of geometric objects.
method Introduces lattice homology for integrally closed submodules and applies it to geometric invariants.
result Well-defined lattice homology associated to quotient modules of type M/N.