Given a degeneration of compact projective complex manifolds $X$ over the punctured disc, with meromorphic singularities, and a relatively ample line bundle $L$ on $X$, we study spaces of plurisubharmonic metrics on $L$, with particular focus on (relative) finite-energy conditions. We endow the space $\hat \cE^1(L)$ of relatively maximal, relative finite-energy metrics with a $d_1$-type distance given by the Lelong number at zero of the collection of fibrewise Darvas $d_1$-distances. We show that this metric structure is complete and geodesic. Seeing $X$ and $L$ as schemes $X_\K$, $L_\K$ over the discretely-valued field $\K=\mathbb{C}((t))$ of complex Laurent series, we show that the space $\cE^1(L_\K\an)$ of non-Archimedean finite-energy metrics over $L_\K\an$ embeds isometrically and geodesically into $\hat \cE^1(L)$, and characterize its image. This generalizes previous work of Berman-Boucksom-Jonsson, treating the trivially-valued case. We investigate consequences regarding convexity of non-Archimedean functionals.
We identify the perverse filtration of a Lagrangian fibration with the monodromy weight filtration of a maximally unipotent degeneration of compact hyper-Kähler manifolds.
This note studies the equivalencies among convergences of Ricci-flat Kähler-Einstein metrics on Calabi-Yau manifolds, cohomology classes and potential functions.
Calabi--Yau manifolds have risen to prominence in algebraic geometry, in part because of mirror symmetry and enumerative geometry. After Bershadsky--Cecotti--Ooguri--Vafa (BCOV), it is expected that genus 1 curve counting on a Calabi--Yau manifold is related to a conjectured invariant, only depending on the complex structure of the mirror, and built from Ray--Singer holomorphic analytic torsions. To this end, extending work of Fang--Lu--Yoshikawa in dimension 3, we introduce and study the BCOV invariant of Calabi--Yau manifolds of arbitrary dimension. To determine it, knowledge of its behaviou…
In this paper, we extend Tosatti's method to study the lower boundedness of modified $K$-energy on a Fano manifold and apply this result to study the relative $K$-stability of the deformation space of a Kähler Ricci soliton.
For a one parameter family of Calabi-Yau threefolds, Green, Griffiths and Kerr have expressed the total singularities in terms of the degrees of Hodge bundles and Euler number of the general fiber. In this paper, we show that the total singularities can be expressed by the sum of asymptotic values of BCOV invariants, studied by Fang, Lu and Yoshikawa. On the other hand, by using Yau's Schwarz lemma, we prove Arakelov type inequalities and Euler number bound for Calabi-Yau family over a compact Riemann surface.
We prove that on Fano manifolds, the Kähler-Ricci flow produces a "most destabilising" degeneration, with respect to a new stability notion related to the H-functional. This answers questions of Chen-Sun-Wang and He. We give two applications of this result. Firstly, we give a purely algebro-geometric formula for the supremum of Perelman's μ-functional on Fano manifolds, resolving a conjecture of Tian-Zhang-Zhang-Zhu as a special case. Secondly, we use this to prove that if a Fano manifold admits a Kähler-Ricci soliton, then the Kähler-Ricci flow converges to it modulo the action of automorphis…
Let $π: \mathcal{X}^* \rightarrow B^*$ be an algebraic family of compact Kähler manifolds of complex dimension $n$ with negative first Chern class over a punctured disc $B^*\in \mathbb{C}$. Let $g_t$ be the unique Kähler-Einstein metric on $\mathcal{X}_t= π^{-1}(t)$. We show that as $t\rightarrow 0$, $(\mathcal{X}_t, g_t)$ converges in pointed Gromov-Hausdorff topology to a unique finite disjoint union of complete metric length spaces $\coprod_{α=1}^\mathcal{A} (Y_α, d_α)$ without loss of volume. Each $(Y_α, d_α)$ is a smooth open Kähler-Einstein manifold of complex dimension n outside its clo…
Manifolds with fibered cusps are a class of complete noncompact Riemannian manifolds including all locally symmetric spaces of rank one. We study the spectrum of the Hodge Laplacian with coefficients in a flat bundle on a closed manifold undergoing degeneration to a manifold with fibered cusps. We obtain precise asymptotics for the resolvent, the heat kernel, and the determinant of the Laplacian. Using these asymptotics we obtain a topological description of the analytic torsion on a manifold with fibered cusps in terms of the R-torsion of the underlying manifold with boundary.
We propose a Hodge theory for the spaces $E_2^{p,\,q}$ featuring at the second step either in the Frölicher spectral sequence of an arbitrary compact complex manifold $X$ or in the spectral sequence associated with a pair $(N,\,F)$ of complementary regular holomorphic foliations on such a manifold. The main idea is to introduce a Laplace-type operator associated with a given Hermitian metric on $X$ whose kernel in every bidegree $(p,\,q)$ is isomorphic to $E_2^{p,\,q}$ in either of the two situations discussed. The surprising aspect is that this operator is not a differential operator since it…
We study the degenerations of asymptotically conical Ricci-flat Kähler metrics as the Kähler class degenerates to a semi-positive class. We show that under appropriate assumptions, the Ricci-flat Kähler metrics converge to a incomplete smooth Ricci-flat Kähler metric away from a compact subvariety. As a consequence, we construct singular Calabi-Yau metrics with asymptotically conical behaviour at infinity on certain quasi-projective varieties and we show that the metric geometry of these singular metrics are homeomorphic to the topology of the singular variety. Finally, we will apply our resul…
We construct infinitely many complete Calabi-Yau metrics on $\mathbf{C}^n$ for $n \geq 3$, with maximal volume growth, and singular tangent cones at infinity. In addition we construct Calabi-Yau metrics in neighborhoods of certain isolated singularities whose tangent cones have singular cross section, generalizing work of Hein-Naber.
We establish a lower bound for the Donaldson-Futaki invariant of optimal degenerations produced by the Kähler-Ricci flow in terms of the greatest Ricci lower bound on arbitrary Fano manifolds. As an application, we can generalize the finiteness of the Futaki invariants on Kähler-Ricci solitons obtained by Guo-Phong-Song-Sturm to the space of all Fano manifolds. Also, we discuss the relation to Hisamoto's inequality for the infimum of the $H$-functional.
We consider a condition on the Ricci curvature involving vector fields, which is broader than the Bakry-Émery Ricci condition. Under this condition volume comparison, Laplacian comparison, isoperimetric inequality and gradient bounds are proven on the manifold. Specializing to the Bakry-Émery Ricci curvature condition, we initiate an approach to work on the original manifold, which yields, under a weaker than usual assumption, the results mentioned above for the {\it original manifold}. These results are different from most well known ones in the literature where the conclusions are made on th…
A theorem of Anderson and Bando-Kasue-Nakajima from 1989 states that to compactify the set of normalized Einstein metrics with a lower bound on the volume and an upper bound on the diameter in the Gromov-Hausdorff sense, one has to add singular spaces called Einstein orbifolds, and the singularities form as blow-downs of Ricci-flat ALE spaces. This raises some natural issues, in particular: can all Einstein orbifolds be Gromov-Hausdorff limits of smooth Einstein manifolds? Can we describe more precisely the smooth Einstein metrics close to a given singular one? In this first paper, we prove th…
We survey some recent developments on the problem of understanding degenerations of Calabi-Yau manifolds equipped with their Ricci-flat Kahler metrics, with an emphasis on the case when the metrics are volume collapsing.
In this second article, we prove that any desingularization in the Gromov-Hausdorff sense of an Einstein orbifold is the result of a gluing-perturbation procedure that we develop. This builds on our first paper where we proved that a Gromov-Hausdorff convergence implied a much stronger convergence in suitable weighted Hölder spaces, in which the analysis of the present paper takes place. The description of Einstein metrics as the result of a gluing-perturbation procedure also sheds light on the local structure of the moduli space of Einstein metrics near its boundary. More importantly here, we…
We investigate connections between the sGG property of compact complex manifolds, defined in earlier work by the second author and L. Ugarte by the requirement that every Gauduchon metric be strongly Gauduchon, and a possible degeneration of the Frölicher spectral sequence. In the first approach that we propose, we prove a partial degeneration at $E_2$ and we introduce a positivity cone in the $E_2$-cohomology of bidegree $(n-2,\,n)$ of the manifold that we then prove to behave lower semicontinuously under deformations of the complex structure. In the second approach that we propose, we introd…
Let A be an integral Banach ring, and X/A be a projective scheme of finite type, endowed with a semi-ample line bundle L. We define a class PSH(X,L) of plurisubharmonic metrics on L on the Berkovich analytification X^an and prove various basic properties thereof. We focus in particular on the case where A is a hybrid ring of complex power series and X/A is a smooth variety, so that X^an is the hybrid space associated to a degeneration X of complex varieties over the punctured disk. We then prove that when L is ample, any plurisubharmonic metric on L with logarithmic growth at zero admits a can…
We study the behavior of the degeneration at the second step of the Frölicher spectral sequence of a $\mathcal{C}^{\infty}$ family of compact complex manifolds. Using techniques from deformation theory and adapting them to pseudo-differential operators we prove a result \textit{à la Kodaira-Spencer} for the dimension of the second step of the Frölicher spectral sequence and we prove that, under a certain hypothesis, the degeneration at the second step is an open property under small deformations of the complex structure. We also provide an example in which we show that, if we omit that hypothe…
As an application of the theory of linear parabolic differential equations on noncompact Riemannian manifolds, developed in earlier papers, we prove a maximal regularity theorem for nonuniformly parabolic boundary value problems in Euclidean spaces. The new feature of our result is the fact that, besides of obtaining an optimal solution theory, we consider the `natural' case where the degeneration occurs only in the normal direction.
