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Calabi flow on toric varieties with bounded Sobolev constant, I

Hongnian Huang (2016)

Complex Manifolds

Let (X, P) be a toric variety. In this note, we show that the C0-norm of the Calabi flow φ(t) on X is uniformly bounded in [0, T) if the Sobolev constant of φ(t) is uniformly bounded in [0, T). We also show that if (X, P) is uniform K-stable, then the modified Calabi flow converges exponentially fast to an extremal Kähler metric if the Ricci curvature and the Sobolev constant are uniformly bounded. At last, we discuss an extension of our results to a quasi-proper Kähler manifold.

Calculation of Rozansky-Witten invariants on the Hilbert schemes of points on a K3 surface and the generalised kummer varieties.

Nieper-Wißkirchen, Marc A. (2003)

Documenta Mathematica

Canonical metrics on some domains of $ℂ^{n}$

Fabio Zuddas (2008/2009)

Séminaire de théorie spectrale et géométrie

The study of the existence and uniqueness of a preferred Kähler metric on a given complex manifold $M$ is a very important area of research. In this talk we recall the main results and open questions for the most important canonical metrics (Einstein, constant scalar curvature, extremal, Kähler-Ricci solitons) in the compact and the non-compact case, then we consider a particular class of complex domains $D$ in $ℂ^{n}$ , the so-called Hartogs domains, which can be equipped with a natural Kaehler metric $g$ ....

Carleman Approximation on Totally Real Subsets of Class Ck.

Manne Per E. (1994)

Mathematica Scandinavica

Characteristic distributions on 4-dimensional almost complex manifolds

Boris S. Kruglikov (2003)

Banach Center Publications

In this paper the Nijenhuis tensor characteristic distributions on a non-integrable four-dimensional almost complex manifold is investigated for integrability, singularities and equivalence.

Characteristic divisors on complex manifolds.

E. Viehweg, Hélène Esnault, J. Seade (1992)

Journal für die reine und angewandte Mathematik

Characteristic varieties and vanishing cycles.

V. Ginsburg (1986)

Inventiones mathematicae

Classical Poincaré metric pulled back off singularities using a Chow-type theorem and desingularization

Caroline Grant Melles, Pierre Milman (2006)

Annales de la faculté des sciences de Toulouse Mathématiques

We construct complete Kähler metrics on the nonsingular set of a subvariety $X$ of a compact Kähler manifold. To that end, we develop (i) a constructive method for replacing a sequence of blow-ups along smooth centers, with a single blow-up along a product of coherent ideals corresponding to the centers and (ii) an explicit local formula for a Chern form associated to this ‘singular’ blow-up. Our metrics have a particularly simple local formula of a sum of the original metric and of the pull back...

Closed transverse $(p, p)$ -forms on compact complex manifolds

Lucia Alessandrini, Marco Andreatta (1987)

Compositio Mathematica

Codimension one foliations on complex tori

Marco Brunella (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

We prove a structure theorem for codimension one singular foliations on complex tori, from which we deduce some dynamical consequences.

Cohomology and splitting of Hermitian-Einstein vector bundles.

Bernard Shiffman (1991)

Mathematische Annalen

Cohomology of the Lie Algebra of Vector Fields on a Complex Manifold.

Toru Tsujishita (1976)

Inventiones mathematicae

Compact complex manifolds with numerically effective cotangent bundles.

Kratz, Henrik (1997)

Documenta Mathematica

Compact Kähler manifolds with compactifiable universal cover

Benoît Claudon, Andreas Höring (2013)

Bulletin de la Société Mathématique de France

In this appendix, we observe that Iitaka’s conjecture fits in the more general context of special manifolds, in which the relevant statements follow from the particular cases of projective and simple manifolds.

Compact Kähler manifolds with nonnegatively curvature operator.

Huai-Dong Cao, Bennett Chow (1986)

Inventiones mathematicae

Compact lcK manifolds with parallel vector fields

Andrei Moroianu (2015)

Complex Manifolds

We show that for n > 2 a compact locally conformally Kähler manifold (M2n , g, J) carrying a nontrivial parallel vector field is either Vaisman, or globally conformally Kähler, determined in an explicit way by a compact Kähler manifold of dimension 2n − 2 and a real function.

Compactness for embedded pseudoholomorphic curves in 3-manifolds

Chris Wendl (2010)

Journal of the European Mathematical Society

We prove a compactness theorem for holomorphic curves in 4-dimensional symplectizations that have embedded projections to the underlying 3-manifold. It strengthens the cylindrical case of the SFT compactness theorem [BEH+C03] by using intersection theory to show that degenerations of such sequences never give rise to multiple covers or nodes, so transversality is easily achieved. This has application to the theory of stable finite energy foliations introduced in [HWZ03], and also suggests a new...

Complex analytic geometry of complex parallelizable manifolds

Jörg Winkelmann (1998)

Mémoires de la Société Mathématique de France

Complex Hyperbolic Surfaces of Abelian Type

Holzapfel, R. (2004)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 11G15, 11G18, 14H52, 14J25, 32L07.We call a complex (quasiprojective) surface of hyperbolic type, iff – after removing finitely many points and/or curves – the universal cover is the complex two-dimensional unit ball. We characterize abelian surfaces which have a birational transform of hyperbolic type by the existence of a reduced divisor with only elliptic curve components and maximal singularity rate (equal to 4). We discover a Picard modular surface of...

Complex structures on product of circle bundles over complex manifolds

Parameswaran Sankaran, Ajay Singh Thakur (2013)

Annales de l’institut Fourier

Let ${\bar{L}}_{i} \to X_{i}$ be a holomorphic line bundle over a compact complex manifold for $i = 1, 2$ . Let $S_{i}$ denote the associated principal circle-bundle with respect to some hermitian inner product on ${\bar{L}}_{i}$ . We construct complex structures on $S = S_{1} \times S_{2}$ which we refer to as scalar, diagonal, and linear types. While scalar type structures always exist, the more general diagonal but non-scalar type structures are constructed assuming that ${\bar{L}}_{i}$ are equivariant ${(ℂ^{*})}^{n_{i}}$ -bundles satisfying some additional conditions. The linear type complex structures...

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