Projective Manifold Research Articles

Projective Compactification of Dolbeault Moduli Spaces

AbstractWe construct a relative projective compactification of Dolbeault moduli spaces of Higgs bundles for reductive algebraic groups on families of projective manifolds that is compatible with the Hitchin morphism.

Cycle Intersection for SOp,q-Flag Domains

A real form G0 of a complex semisimple Lie group G has only finitely many orbits in any given compact G-homogeneous projective algebraic manifold Z=G/Q. A maximal compact subgroup K0 of G0 has special orbits C which are complex submanifolds in the open orbits of G0. These special orbits C are characterized as the closed orbits in Z of the complexification K of K0. These are referred to as cycles. The cycles intersect Schubert varieties S transversely at finitely many points. Describing these points and their multiplicities was carried out for all real forms of SLn,ℂ by Brecan (Brecan, 2014) and (Brecan, 2017) and for the other real forms by Abu-Shoga (Abu-Shoga, 2017) and Huckleberry (Abu-Shoga and Huckleberry). In the present paper, we deal with the real form SOp,q acting on the SO (2n, C)-manifold of maximal isotropic full flags. We give a precise description of the relevant Schubert varieties in terms of certain subsets of the Weyl group and compute their total number. Furthermore, we give an explicit description of the points of intersection in terms of flags and their number. The results in the case of G/Q for all real forms will be given by Abu-Shoga and Huckleberry.

Birational Calabi-Yau manifolds have the same small quantum products

We show that any two birational projective Calabi-Yau manifolds have isomorphic small quantum cohomology algebras after a certain change of Novikov rings. The key tool used is a version of an algebra called symplectic cohomology, which is constructed using Hamiltonian Floer cohomology. Morally, the idea of the proof is to show that both small quantum products are identical deformations of symplectic cohomology of some common open affine subspace.

Remarks on toric manifolds whose Chern characters are positive

We show that the second Chern character of any projective toric manifold of Picard number three is not positive. In connection with this result, we give various examples of the positivity of higher Chern characters of projective toric manifolds.Communicated by Stephen L. Kleiman

On Ricci pseudo-symmetric super quasi-Einstein Hermitian manifolds

The present paper deals the study of a Bochner Ricci pseudosymmetric super quasi-Einstein Hermitian manifold and a holomorphically projective Ricci pseudo-symmetric super quasi-Einstein Hermitian manifold.

Moduli spaces of sheaves that are semistable with respect to a Kähler polarisation

Using an existence criterion for good moduli spaces of Artin stacks by Alper-Fedorchuk-Smyth we construct a proper moduli space of rank two sheaves with fixed Chern classes on a given complex projective manifold that are Gieseker-Maruyama-semistable with respect to a fixed K\"ahler class.

RC-positive metrics on rationally connected manifolds

Abstract In this paper, we prove that if a compact Kähler manifold X has a smooth Hermitian metric $\omega $ such that $(T_X,\omega )$ is uniformly RC-positive, then X is projective and rationally connected. Conversely, we show that, if a projective manifold X is rationally connected, then there exists a uniformly RC-positive complex Finsler metric on $T_X$ .

The fundamental group of compact Kähler threefolds

Let $X$ be a compact K{\"a}hler manifold of dimension three. We prove that there exists a projective manifold $Y$ such that $\pi\_1(X)\simeq \pi\_1(Y)$. We also prove the bimeromorphic existence of algebraic approximations for compact K{\"a}hler manifolds of algebraic dimension $\dim(X)-1$. Together with the work of Graf and the third author, this settles in particular the bimeromorphic Kodaira problem for compact K{\"a}hler threefolds.

Characterization of generic projective space bundles and algebraicity of foliations

In this paper we consider various notions of positivity for distributions on complex projective manifolds. We start by analyzing distributions having big slope with respect to curve classes, obtaining characterizations of generic projective space bundles in terms of movable curve classes. We then apply this result to investigate algebraicity of leaves of foliations, providing a lower bound for the algebraic rank of a foliation in terms of invariants measuring positivity. We classify foliations attaining this bound, and describe those whose algebraic rank slightly exceeds this bound.

An $$L^2$$ Dolbeault lemma and its applications to vanishing theorems

In this paper, we will first build an $$L^2$$ Dolbeault lemma by analytic methods and Hormander $$L^2$$ estimates. Then as applications, we will prove some log Nadel type vanishing theorems on compact Kahler manifolds and some log Kawamata–Viehweg type vanishing theorems on projective manifolds. Some log Nakano–Demailly type vanishing theorems for vector bundles will be also discussed by the same methods; one of which generalizes the original Nakano vanishing theorem on compact Kahler manifolds.

On holomorphic distributions on Fano threefolds

This paper is devoted to the study of holomorphic distributions of dimension and codimension one on smooth weighted projective complete intersection Fano manifolds X which is threedimensional and with Picard number equal to one. We study the relations between algebro-geometric properties of the singular set of singular holomorphic distributions and their associated sheaves. We characterize either distributions whose tangent sheaf or conormal sheaf are arithmetically Cohen Macaulay (aCM) on X. We also prove that a codimension one locally free distribution with trivial canonical bundle on any Fano threefold, with Picard number equal to one, has a tangent sheaf which either splits or it is stable.

ORBIFOLD ASPECTS OF CERTAIN OCCULT PERIOD MAPS

We first characterize the automorphism groups of Hodge structures of cubic threefolds and cubic fourfolds. Then we determine for some complex projective manifolds of small dimension (cubic surfaces, cubic threefolds, and nonhyperelliptic curves of genus 3 or 4), the action of their automorphism groups on Hodge structures of associated cyclic covers, and thus confirm conjectures made by Kudla and Rapoport in (Pacific J. Math.260(2) (2012), 565–581).

Canonical complex extensions of Kähler manifolds

Given a complex manifold $X$, any K\"ahler class defines an affine bundle over $X$, and any K\"ahler form in the given class defines a totally real embedding of $X$ into this affine bundle. We formulate conditions under which the affine bundles arising this way are Stein and relate this question to other natural positivity conditions on the tangent bundle of $X$. For compact K\"ahler manifolds of non-negative holomorphic bisectional curvature, we establish a close relation of this construction to adapted complex structures in the sense of Lempert--Sz\H{o}ke and to the existence question for good complexifications in the sense of Totaro. Moreover, we study projective manifolds for which the induced affine bundle is not just Stein but affine and prove that these must have big tangent bundle. In the course of our investigation, we also obtain a simpler proof of a result of Yang on manifolds having non-negative holomorphic bisectional curvature and big tangent bundle.

Approximation and interpolation of regular maps from affine varieties to algebraic manifolds

We consider the analogue for regular maps from affine varieties to suitable algebraic manifolds of Oka theory for holomorphic maps from Stein spaces to suitable complex manifolds. The goal is to understand when the obstructions to approximation or interpolation are purely topological. We propose a definition of an algebraic Oka property, which is stronger than the analytic Oka property. We review the known examples of algebraic manifolds satisfying the algebraic Oka property and add a new class of examples: smooth nondegenerate toric varieties. On the other hand, we show that the algebraic analogues of three of the central properties of analytic Oka theory fail for all compact manifolds and manifolds with a rational curve; in particular, for projective manifolds.

Rigidity of certain admissible pairs of rational homogeneous spaces of Picard number 1 which are not of the subdiagram type

Recently, Mok and Zhang (2019) introduced the notion of admissible pairs (X0, X) of rational homogeneous spaces of Picard number 1 and proved rigidity of admissible pairs (X0, X) of the subdiagram type whenever X0 is nonlinear. It remains unsolved whether rigidity holds when (X0, X) is an admissible pair NOT of the subdiagram type of nonlinear irreducible Hermitian symmetric spaces such that (X0, X) is nondegenerate for substructures. In this article we provide sufficient conditions for confirming rigidity of such an admissible pair. In a nutshell our solution consists of an enhancement of the method of propagation of sub-VMRT (varieties of minimal rational tangents) structures along chains of minimal rational curves as is already implemented in the proof of the Thickening Lemma of Mok and Zhang (2019). There it was proven that, for a sub-VMRT structure $$\overline{\omega} : \mathscr{C}(S) \rightarrow S$$ on a uniruled projective manifold $$(X,\,{\cal K})$$ equipped with a minimal rational component and satisfying certain conditions so that in particular S is “uniruled” by open subsets of certain minimal rational curves on X, for a “good” minimal rational curve l emanating from a general point x ∈ S, there exists an immersed neighborhood Nl of l which is in some sense “uniruled” by minimal rational curves. By means of the Algebraicity Theorem of Mok and Zhang (2019), S can be completed to a projective subvariety Z ⊂ X. By the author’s solution of the Recognition Problem for irreducible Hermitian symmetric spaces of rank ⩾ 2 (2008) and under Condition (F), which symbolizes the fitting of sub-VMRTs into VMRTs, we further prove that Z is the image under a holomorphic immersion of X0 into X which induces an isomorphism on second homology groups. By studying ℂ*-actions we prove that Z can be deformed via a one-parameter family of automorphisms to converge to X0 ⊂ X. Under the additional hypothesis that all holomorphic sections in Γ(X0, Tx∣x0) lift to global holomorphic vector fields on X, we prove that the admissible pair (X0, X) is rigid. As examples we check that (X0, X) is rigid when X is the Grassmannian G(n, n) of n-dimensional complex vector subspaces of W ≅ ℂ2n, n ⩾ 3, and when X0 ⊂ X is the La grangian Grassmannian consisting of Lagrangian vector subspaces of (W, σ) where σ is an arbitrary symplectic form on W.

The rigidity on the second fundamental form of projective manifolds

Let $M$ be a complex $n$-dimensional projective manifold in $\mathbb{P}^{n+r}$ endowed with the Fubini-Study metric of constant holomorphic sectional curvature $1$, $\sigma$ its second fundamental form, and $\underline{|\sigma|}^2$ the mean value of the squared length of $\sigma$ on $M$. We derive a formula for $\underline{|\sigma|}^2$ and classify them when $\underline{|\sigma|}^2\leq2n$. We present several applications to these results. The first application is to confirm a conjecture of Loi and Zedda, which characterizes the linear subspace and the quadric in terms of the $L^2$-norm of $\sigma$. The second application is to improve a result of Cheng solving an old conjecture of Oguie from pointwise case to mean case. The third application is to give an optimal second gap value on $\underline{|\sigma|}^2$, which can be viewed as a complex analog to those on minimal submanifolds in the unit spheres.

On Certain Complex Projective Manifolds with Hodge Numbers h10=4 and h20=5

On Certain Complex Projective Manifolds with Hodge Numbers h10=4 and h20=5

Extremal conformal structures on projective surfaces

We introduce a new functional $\mathcal{E}_{\mathfrak{p}}$ on the space of conformal structures on an oriented projective manifold $(M,\mathfrak{p})$. The nonnegative quantity $\mathcal{E}_{\mathfrak{p}}([g])$ measures how much $\mathfrak{p}$ deviates from being defined by a $[g]$-conformal connection. In the case of a projective surface $(\Sigma,\mathfrak{p})$, we canonically construct an indefinite Kahler--Einstein structure $(h_{\mathfrak{p}},\Omega_{\mathfrak{p}})$ on the total space $Y$ of a fibre bundle over $\Sigma$ and show that a conformal structure $[g]$ is a critical point for $\mathcal{E}_{\mathfrak{p}}$ if and only if a certain lift $\widetilde{[g]} : (\Sigma,[g]) \to (Y,h_{\mathfrak{p}})$ is weakly conformal. In fact, in the compact case $\mathcal{E}_{\mathfrak{p}}([g])$ is -- up to a topological constant -- just the Dirichlet energy of $\widetilde{[g]}$. As an application, we prove a novel characterisation of properly convex projective structures among all flat projective structures. As a by-product, we obtain a Gauss--Bonnet type identity for oriented projective surfaces.

Rigidity of pairs of rational homogeneous spaces of Picard number $1$ and analytic continuation of geometric substructures on uniruled projective manifolds

Building on the geometric theory of uniruled projective manifolds by Hwang–Mok, which relies on the study of varieties of minimal rational tangents (VMRTs) from both the algebro-geometric and the differential-geometric perspectives, Mok, Hong–Mok and Hong–Park have studied standard embeddings between rational homogeneous spaces $X = G/P$ of Picard number $1$. Denoting by $S \subset X$ an arbitrary germ of complex submanifold which inherits from $X$ a geometric structure defined by taking intersections of VMRTs with tangent subspaces and modeled on some rational homogeneous space $X_0 = G_0 / P_0$ of Picard number $1$ embedded in $X = G/P$ as a linear section through a standard embedding, we say that $(X_0, X)$ is rigid if there always exists some $\gamma \in \mathrm{Aut}(X)$ such that $S$ is an open subset of $\gamma (X_0)$. We prove that a pair $(X_0, X)$ of sub-diagram type is rigid whenever $X_0$ is nonlinear, which in the Hermitian symmetric case recovers Schubert rigidity for nonlinear smooth Schubert cycles, and which in the general rational homogeneous case goes beyond earlier works dealing with images of holomorphic maps. Our methods apply to uniruled projective manifolds $(X, \mathcal{K})$, for which we introduce a general notion of sub-VMRT structures $\varpi : \mathscr{C} (S) \to S$, proving that they are rationally saturated under an auxiliary condition on the intersection $\mathscr{C} (S) := \mathscr{C} (X) \cap \mathbb{P} T (S)$ and a nondegeneracy condition for substructures expressed in terms of second fundamental forms on VMRTs. Under the additional hypothesis that minimal rational curves are of degree $1$ and that distributions spanned by sub-VMRTs are bracket generating, we prove that $S$ extends to a subvariety $Z \subset X$. For its proof, starting with a “Thickening Lemma” which yields smooth collars around certain standard rational curves, we show that the germ of submanifold $(S; x_0)$ and, hence, the associated germ of sub-VMRT structure on $(S; x_0)$ can be propagated along chains of “thickening” curves issuing from $x_0$, and construct by analytic continuation a projective family of chains of rational curves compactifying the latter family, thereby constructing the projective completion $Z$ of $S$ as its image under the evaluation map.

The complex $$K^{*}$$ ring of the complex projective Stiefel manifold

The topological $$K^{*}$$ ring is one of the important tools used for understanding the topology of a manifold. We settle a problem of computing the $$K^{*}$$ ring of complex projective Stiefel manifold using combinatorics. We calculate $$K^{*}$$ ring of the right generalized projective Stiefel manifold, denoted by $$P_{\ell }W_{n,k}$$ , which gives us description of complex $$K^{*}$$ ring of complex projective Stiefel manifold $$PW_{n,k}$$, which is a particular case of $$P_{\ell }W_{n,k}$$.