Projective Manifold Research Articles

Equivariant asymptotics of Szegö kernels under Hamiltonian $${{\varvec{U}}}(\mathbf{2})$$ U ( 2 ) -actions

Let M be complex projective manifold and A a positive line bundle on it. Assume that a compact and connected Lie group G acts on M in a Hamiltonian manner and that this action linearizes to A. Then, there is an associated unitary representation of G on the associated algebro-geometric Hardy space. If the moment map is nowhere vanishing, the isotypical components are all finite dimensional; they are generally not spaces of sections of some power of A. One is then led to study the local and global asymptotic properties the isotypical component associated with a weight $$k \, \varvec{ \nu }$$ , when $$k\rightarrow +\infty $$ . In this paper, part of a series dedicated to this general theme, we consider the case $$G=U(2)$$ .

Variance of the volume of random real algebraic submanifolds

Let $\mathcal{X}$ be a complex projective manifold of dimension $n$ defined over the reals and let $M$ denote its real locus. We study the vanishing locus $Z_{s_d}$ in $M$ of a random real holomorphic section $s_d$ of $\mathcal{E} \otimes \mathcal{L}^d$, where $ \mathcal{L} \to \mathcal{X}$ is an ample line bundle and $ \mathcal{E}\to \mathcal{X}$ is a rank $r$ Hermitian bundle. When $r \in \{1,\dots , n − 1\}$, we obtain an asymptotic of order $d^{r− \frac{n}{2}}$, as $d$ goes to infinity, for the variance of the linear statistics associated to $Z_{s_d}$, including its volume. Given an open set $U \subset M$, we show that the probability that $Z_{s_d}$ does not intersect $U$ is a $O$ of $d^{-\frac{n}{2}}$ when $d$ goes to infinity. When $n\geq 3$, we also prove almost sure convergence for the linear statistics associated to a random sequence of sections of increasing degree. Our framework contains the case of random real algebraic submanifolds of $\mathbb{RP}^n$ obtained as the common zero set of $r$ independent Kostlan–Shub–Smale polynomials.

Global Koppelman Formulas on (Singular) Projective Varieties

Let i:Xrightarrow {mathbb {P}}^N be a projective manifold of dimension n embedded in projective space {mathbb {P}}^N, and let L be the pullback to X of the line bundle {mathcal {O}}_{{mathbb {P}}^N}(1). We construct global explicit Koppelman formulas on X for smooth (0,*)-forms with values in L^s for any s. The same construction works for singular, even non-reduced, X of pure dimension, if the sheaves of smooth forms are replaced by suitable sheaves {mathcal {A}}_X^* of (0,*)-currents with mild singularities at X_{sing}. In particular, if sge mathrm{reg,}X -1, where mathrm{reg,}X is the Castelnuovo–Mumford regularity, we get an explicit representation of the well-known vanishing of H^{0,q}(X, L^{s-q}), qge 1. Also some other applications are indicated.

Projective embedding of pairs and logarithmic K-stability

Let $\hat{L}$ be the projective completion of an ample line bundle $L$ over $D$, a smooth projective manifold. Hwang-Singer \cite{HwangS} have constructed complete CSCK metric on $\hat{L}\backslash D$. When the corresponding \kahler form is in the cohomology class of a rational divisor $A$ and when $L$ has negative CSCK metric on $D$, we show that the Kodaira embedding induced by orthonormal basis of the Bergman space of $kA$ is almost balanced. As a corollary, $(\hat{L},D,cA,0)$ is K-semistable.

Mellin transformation, propagation, and abelian duality spaces

For arbitrary field coefficients K, we show that K-perverse sheaves on a complex affine torus satisfy the so-called propagation package, i.e., the generic vanishing property and the signed Euler characteristic property hold, and the corresponding cohomology jump loci satisfy the propagation property and codimension lower bound. The main ingredient used in the proof is Gabber–Loeser's Mellin transformation functor for K-constructible complexes on a complex affine torus, and the fact that it behaves well with respect to perverse sheaves.As a concrete topological application of our sheaf-theoretic results, we study homological duality properties of complex algebraic varieties, via abelian duality spaces. We provide new obstructions on abelian duality spaces by showing that their cohomology jump loci satisfy a propagation package. This is then used to prove that complex abelian varieties are the only complex projective manifolds which are abelian duality spaces. We also construct new examples of abelian duality spaces. For example, we show that if a smooth quasi-projective variety X satisfies a certain Hodge-theoretic condition and it admits a proper semi-small map (e.g., a closed embedding or a finite map) to a complex affine torus, then X is an abelian duality space.

Deformation of a generically finite map to a hypersurface embedding

We give a structure theorem for projective manifolds W0 with the property of admitting a 1-parameter deformation where Wt is a hypersurface in a projective smooth manifold Zt.Their structure is the one of special iterated univariate coverings which we call of normal type, which essentially means that the line bundles where the univariate coverings live are tensor powers of the normal bundle to the image X of W0.We give applications to the case where Zt is projective space, respectively an Abelian variety.

A remark on boundedness of manifolds embedded with small codimension

For embedded projective manifolds of small codimension we give an explicit bound for their degree, depending on the Castelnuovo–Mumford regularity of their structure sheaf. As an application, we obtain bounds for the degree of such manifolds whose structure sheaf is arithmetically Cohen–Macaulay (in a weak sense) and whose canonical map is not birational.

The canonical spectrum of projective toric manifolds

Let [Formula: see text] be a complex projective toric manifold. We associated to [Formula: see text], a positive and closed [Formula: see text]-current called the canonical toric Kähler current of [Formula: see text] denoted by [Formula: see text], and a new invariant called the canonical spectrum of [Formula: see text]. This spectrum is obtained as the set of the eigenvalues of a singular Laplacian defined by [Formula: see text] and which is described uniquely by the combinatorial structure of [Formula: see text]. The construction of this Laplacian and the study of its spectral properties are the consequence of a generalized spectral theory of Laplacians on compact Kähler manifolds that we develop in this paper.

Une propriété de continuité associée aux classes de cohomologie de Hodge

The obstructions, for a closed smooth differential form of bidimension (p,p) on a projective manifold, to be cohomologous to an algebraic cycle with complex coefficients, are calculated in terms of the Chow transformation. They can be expressed as an orthogonality condition, on the manifold itself, with families parametrized by the Grassmannian of currents that are completely determined. Each of these currents is ddc-closed and with support in the intersection of the manifold and of the projective subspace associated with the parameter. By the theory of harmonic forms, a period is thus associated with that differential form for each parameter. We study the set of periods, obtained when the parameter varies, and we arrive at a continuity on the Grassmannian, when the cohomology class is rational. The same property can be obtained by going to the space of divisors of the Grassmannian and by using a characterization of Chow forms. We proceed here directly, by calculating the periods by means of the Atiyah–Hirzebruch theorem. This global continuity implies the orthogonality for all parameter.

Ampleness equivalence and dominance for vector bundles

Hartshorne in “Ample vector bundles” proved that E is ample if and only if $${\mathcal O}_{P(E)}(1)$$ is ample. Here we generalize this result to flag manifolds associated to a vector bundle E on a complex projective manifold X: For a partition a we show that the line bundle $$ Q_a^s$$ on the corresponding flag manifold $$\mathcal {F}l_s(E)$$ is ample if and only if $$ {\mathcal S}_aE $$ is ample. In particular $$\det Q$$ on $$ {G}_r(E)$$ is ample if and only if $$\wedge ^rE$$ is ample. We give also a proof of the Ampleness Dominance theorem that does not depend on the saturation property of the Littlewood–Richardson semigroup.

Collapsing hyperkähler manifolds

Given a projective hyperkahler manifold with a holomorphic Lagrangian fibration, we prove that hyperkahler metrics with volume of the torus fibers shrinking to zero collapse in the Gromov-Hausdorff sense (and smoothly away from the singular fibers) to a compact metric space which is a half-dimensional special Kahler manifold outside a singular set of real Hausdorff codimension 2 and is homeomorphic to the base projective space.

Remarks on Ramanujam-Kawamata-Viehweg Vanishing Theorem

In this article we prove a general result on a nef vector bundle E on a projective manifold X of dimension n depending on the vector space Hn,n(X,E): It is also shown that Hn,n(X,E) = 0 for an indecomposable nef rank 2 vector bundles E on some specific type of n dimensional projective manifold X. The same vanishing shown to hold for indecomposable nef and big rank 2 vector bundles on any variety with trivial canonical bundle.

Singular foliations with trivial canonical class

This paper describes the structure of singular codimension one foliations with numerically trivial canonical bundle on complex projective manifolds.

Maximal time existence of unnormalized conical Kähler–Ricci flow

Abstract We generalize the maximal time existence of Kähler–Ricci flow in [G. Tian and Z. Zhang, On the Kähler–Ricci flow on projective manifolds of general type, Chin. Ann. Math. Ser. B 27 (2006), no. 2, 179–192] and [J. Song and G. Tian, The Kähler–Ricci flow through singularities, Invent. Math. 207 (2017), no. 2, 519–595] to the conical case. Furthermore, if the log canonical bundle K M + ( 1 - β ) ⁢ [ D ] {K_{M}+(1-\beta)[D]} is big or big and nef, we can examine the limit behaviors of such conical Kähler–Ricci flow. Moreover, these results still hold when D is a simple normal crossing divisor.

On flops and canonical metrics

This article is concerned with an observation for proving non-existence of canonical Kahler metrics. The idea is to use a rather explicit type of degeneration that applies in many situations. Namely, in a variation on a theme introduced by Ross-Thomas, we consider flops of the deformation to the normal cone. This yields a rather widely applicable notion of stability that is still completely explicit and readily computable, but with wider scope. We describe some applications, among them, a proof of one direction of the Calabi conjecture for asymptotically logarithmic del Pezzo surfaces. 1. Motivation and results A variety is slope stable in the sense of Ross-Thomas if, roughly, it is K-stable with respect to degenerations to the normal cone of its subvarieties. This notion has been studied extensively by a number of authors and has yielded many non-existence results for canonical metrics on projective Kahler manifolds. Our main purpose in this article is to introduce a slight variation on this theme by considering a somewhat more involved notion of stability that involves additional flops on the degeneration to the normal cone but that is still geometric and computable, and is partly inspired by the work of Li-Xu. This gives many new non-existence results, and most notably allows us to resolve one direction of the Calabi conjecture for asymptotically logarithmic del Pezzo surfaces. 1.1. Existence theorem for KEE metrics. Kahler-Einstein edge (KEE) metrics are a nat- ural generalization of Kahler-Einstein metrics: they are smooth metrics on the complement of a divisor, and have a conical singularity of angle 2πβ transverse to that 'complex edge'. When β = 1, of course, this is just an ordinary Kahler-Einstein metric, that extends smoothly across the divisor. One can think of the metric as being 'bent' at an angle 2πβ along the divisor. In the case of Riemann surfaces, KEE metrics are just the familiar constant curvature metrics with isolated cone singularities, that have been studied since the late 19th century, e.g., by Picard (25). A basic question, whose origins trace back to Tian's 1994 lectures in the setting of nonpositive curvature (32), extended in Donaldson's 2009 lectures to the setting of anticanonical divisors on Fano manifolds (10), and further extended in our previous work (5) (see also the survey (27, §8)), is the following: Problem 1.1. Under what analytic conditions on the triple (X,D,β) does a KEE metric exist on the Kahler manifold X bent at an angle 2πβ along the divisor D ⊂ X? This is partly motivated by Troyanov's solution in the Riemann surface case (36), and was settled by Jeffres-Mazzeo-Rubinstein (sufficient condition)and Darvas-Rubinstein (sufficient and necessary conditions) in higher dimensions for smooth D (17, 7). These results give an analytic criterion characterizing existence, once the cohomological condition

Deforming convex projective manifolds

We study a properly convex real projective manifold with (possibly empty) compact, strictly convex boundary, and which consists of a compact part plus finitely many convex ends. We extend a theorem of Koszul which asserts that for a compact manifold without boundary the holonomies of properly convex structures form an open subset of the representation variety. We also give a relative version for non-compact (G,X)-manifolds of the openess of their holonomies.

Semi-stable vector bundles on fibred varieties

We introduce and investigate a semi-stability concept relative to morphisms $$Y\xrightarrow {\scriptscriptstyle \pi } X$$ between projective manifolds, different of Maruyama–Simpson. For torsion-free sheaves on Y and for appropriate polarizations, we relate the (semi-)stability on Y to the slope (semi-)stability on the generic fibre of $$\pi $$ . We apply the result to describe in great detail the moduli spaces of semi-stable vector bundles on Hirzebruch surfaces and on threefolds which are $$\mathbb P^2$$ -bundles over . We establish their irreducibility and rationality, an intensely investigated issue.

Branched Holomorphic Cartan Geometries and Calabi–Yau Manifolds

Abstract We introduce the concept of a branched holomorphic Cartan geometry. It generalizes to higher dimension the definition of branched (flat) complex projective structure on a Riemann surface introduced by Mandelbaum [25]. This new framework is much more flexible than that of the usual holomorphic Cartan geometries. We show that all compact complex projective manifolds admit a branched flat holomorphic projective structure. We also give an example of a non-flat branched holomorphic normal projective structure on a compact complex surface. It is known that no compact complex surface admits such a structure with empty branching locus. We prove that non-projective compact simply connected Kähler Calabi–Yau manifolds do not admit any branched holomorphic projective structure. The key ingredient of its proof is the following result of independent interest: if E is a holomorphic vector bundle over a compact simply connected Kähler Calabi–Yau manifold and E admits a holomorphic connection, then E is a trivial holomorphic vector bundle and any holomorphic connection on E is trivial.

Inertia groups of high-dimensional complex projective spaces

For a complex projective space the inertia group, the homotopy inertia group and the concordance inertia group are isomorphic. In complex dimension 4n+1, these groups are related to computations in stable cohomotopy. Using stable homotopy theory, we make explicit computations to show that the inertia group is non-trivial in many cases. In complex dimension 9, we deduce some results on geometric structures on homotopy complex projective spaces and complex hyperbolic manifolds.

Compact leaves of codimension one holomorphic foliations on projective manifolds

This article studies codimension one foliations on projective man-ifolds having a compact leaf (free of singularities). It explores the interplay between Ueda theory (order of flatness of the normal bundle) and the holo-nomy representation (dynamics of the foliation in the transverse direction). We address in particular the following problems: existence of foliation having as a leaf a given hypersurface with topologically torsion normal bundle, global structure of foliations having a compact leaf whose holonomy is abelian (resp. solvable), and factorization results.