Toric Kato manifolds

We introduce and study a special class of Kato manifolds, which we call toric Kato manifolds. Their construction stems from toric geometry, as their universal covers are open subsets of toric algebraic varieties of non-finite type. This generalizes previous constructions of Tsuchihashi and Oda, and in complex dimension 2, retrieves the properly blown-up Inoue surfaces. We study the topological and analytical properties of toric Kato manifolds and link certain invariants to natural combinatorial data coming from the toric construction. Moreover, we produce families of flat degenerations of any toric Kato manifold, which serve as an essential tool in computing their Hodge numbers. In the last part, we study the Hermitian geometry of Kato manifolds. We give a characterization result for the existence of locally conformally K\"ahler metrics on any Kato manifold. Finally, we prove that no Kato manifold carries balanced metrics and that a large class of toric Kato manifolds of complex dimension $\geq 3$ do not support pluriclosed metrics.


Introduction
Kato manifolds are compact complex manifolds of non-Kähler type and were introduced by M. Kato in 1977 [Kat78] as manifolds containing a global spherical shell.More specifically, he showed that any such manifold, which we shall call henceforth a Kato manifold, is constructed in the following way.Let π : B → B be a modification of the standard unit ball in C n at finitely many points and let σ : B → B be a holomorphic embedding.Glue small neighborhoods of the two boundary components of B σ(B) via the local biholomorphism σ • π.The resulting manifold X(π, σ) is a compact complex manifold with infinite cyclic fundamental group.The couple (π, σ) is referred to as a Kato data.
A Kato manifold has an associated germ given by F := π • σ : (C n , 0) → (C n , 0).Many of its properties are encoded in F , however the germ alone does not fully characterize the manifold, unless the complex dimension is 2 and the surface is minimal.This comes from the fact that for n = 2, any modification is a sequence of blow-ups at points, while in higher dimension modifications can be much more complicated.As a result, Kato surfaces are much studied and fairly well understood (see for instance [Nak84], [Dlo84], [DOT03], [Tel05], [FP10] etc.), while in higher dimension very little is known.
One main motivation for studying Kato manifolds comes from the fact that a big part of them carry locally conformally Kähler metrics, but no Vaisman metrics [Bru11], [IOP21].They constitute therefore an immense source of manifolds where one can study the interplay between the existence of special Hermitian metrics and different geometric and cohomological properties.
In [IOP21] we considered the simplest class of Kato manifolds, which corresponds to the case when π : B → B is given by successive blow-ups at special points and σ is a standard chart of a blow-up.We studied several of their analytical invariants, but did not have the tools to compute any Hodge number.
In the present paper, we introduce a much more general class, the toric Kato manifolds.They are determined by a toric Kato data, namely a smooth toric modification π : C n → C n at 0 and a chart σ : C n → C n satisfying a natural (C * ) n -equivariance property.The corresponding germ has the form (1.1) where λ = (λ 1 , . . ., λ n ) ∈ (C * ) n and A = (a jk ) ∈ GL(n, Z).
J.É.P. -M., 2022, tome 9 Starting from the fan of C n , viewed as a toric variety, and the embedding σ, we construct a natural infinite fan Σ A endowed with a Z-action.Then using the germ F , we define a Z-invariant open set in the toric variety of non-finite type X λ,A ⊂ X(Σ A , Z n ).Our starting point is the following description of toric Kato manifolds (see Theorem 4.2 for a more precise statement).
Theorem 1.1.-If X is any toric Kato manifold with germ F λ,A , then we have a biholomorphism: This second point of view of toric Kato manifolds generalizes the toric description of Inoue surfaces [Ino75], [Ino77] given by Oda [Oda78,§14], and a construction given by Tsuchihashi in [Tsu87] of a class of non-Kähler manifolds with infinite cyclic fundamental group.Other generalizations of Tsuchihashi's construction, going in different directions, have been considered in the literature.For instance, Sankaran [San89] generalizes [Tsu87] to higher rank fundamental group, while Battisti and Oeljeklaus [BO15] produce new manifolds by interpolating between the Sankaran construction and the LVMB manifolds of [Bos01].
The above description allows for a better conceptual understanding of toric Kato manifolds, and in particular unveils new ways for computing different invariants using classical techniques from toric geometry.For instance, denoting by a j the number of j-dimensional cones of the fan of C n , we have the following (see 6.1).
Theorem 1.2.-The n-dimensional toric Kato manifold X has the following Betti numbers: (−1) s−j s j a n−s + n s + 1 , 1 j n − 1.
The matrix A in (1.1) has only non-negative coefficients.Consider the maximal sub-matrix P of A which is a permutation matrix.When P = A, the resulting toric Kato manifold is a primary Hopf manifold.When P ∈ GL(n − 1, Z), the universal cover of the corresponding toric Kato manifold X is all of X(Σ A , Z n ), and we call X of parabolic type.In all the other cases, X is a proper subset of X(Σ A , Z n ) and we call X of hyperbolic type.Imitating Nakamura's classification of Inoue surfaces [Nak84] in terms of curves, we have the following characterization (see Theorem 5.2).
Theorem 1.3.-Let X be a toric Kato manifold.
(1) X is a primary Hopf manifold if and only if any of its (C * ) n -invariant curves is elliptic; (2) X is of hyperbolic type if and only if any (C * ) n -invariant curve is rational; (3) X is of parabolic type if and only if X contains a unique (C * ) n -invariant elliptic curve, and at least one rational (C * ) n -invariant curve.
Concerning analytic invariants, we determine explicitly the canonical line bundle in terms of the maximal toric divisor and show that the Kodaira dimension of any toric Kato manifold is negative (Proposition 5.7).Also, we compute several Hodge numbers (see Proposition 8.4, Theorems 8.5 and 8.8).
We note that the primary Hopf manifolds have the same Hodge numbers as above, and that our proof also works for parabolic type manifolds with |λ| small enough (see Remark 8.9).In particular, for all these cases one has b k = p+q=k h p,q for k 2. We suspect that all Kato manifolds should have the Hodge numbers of the above theorem, however showing this is beyond our means since much of our proof is based on the toric description of our manifolds.We should also note that [San89] computes all the Hodge numbers of the toric Kato manifolds with P (A) = ∅, under the additional assumption that A is diagonalizable over C and irreducible over Q.It would be interesting to know if one could generalize his methods to our general setting.
A main tool for computing the Hodge numbers of toric Kato manifolds is developed in Section 7, where we exhibit two different types of flat toric degenerations of toric Kato manifolds.The first one generalizes a previous construction of Nakamura [Nak83] for surfaces.In this case we deform, in a smooth family, any toric Kato manifold to a singular space obtained by identifying two invariant hypersurfaces on a smooth compact toric variety.The second type of degeneration exists only in the hyperbolic case and extends a construction given by Tsuchihashi in [Tsu87].In this case, the singular fiber is again given by the identification of two invariant hypersurfaces in a compact, possibly singular, toric variety.A special feature of this second family is that all the smooth fibers are biholomorphic.
Finally, guided by the general principle of finding special Hermitian metrics on complex manifolds of non-Kähler type, we investigate the case of Kato manifolds.Special Hermitian metrics usually arise by imposing some power of the fundamental form Ω to be in the kernel of some differential operator.The existence of such metrics is far from being guaranteed on a generic compact complex manifold.As a consequence, constant effort has been put to find new specific examples and to unravel the different restrictions that the existence of such metrics might impose.
In this direction, our first result consists in a characterization for the existence of locally conformally Kähler (lcK) metrics (see Theorem 10.3).
We should note here that in the case of a toric Kato manifold X, although the compact torus T n acts on the universal cover X, this action never descends to X if X is not Hopf.Therefore, X is not a toric manifold in any classical sense.On the other hand, if X admits an lcK metric, then X admits a Kähler metric with respect to which it becomes a toric Kähler manifold.In this manner, lcK toric Kato manifolds give interesting generalizations of the class of toric lcK manifolds.
Our last result is concerned with the non-existence of other special Hermitian metrics (see Section 10 for the definitions, Theorems 10.6 and 10.8).
Theorem 1.6 (1) A Kato manifold admits no strongly Gauduchon metric, and in particular no balanced metric and no Hermitian symplectic metric.
(2) If X is a Kato manifold of dimension n 3 satisfying H 1,2 ∂ (X) = 0, then it cannot be endowed with a pluriclosed metric.In particular, a toric Kato manifold of hyperbolic type does not admit pluriclosed metrics, unless it is a surface.
The paper is organized as follows.We begin by some necessary preliminaries on Kato manifolds.The first definition of toric Kato manifolds, together with the elementary but technical properties of toric Kato data, are given in Section 3. In the next section, we describe and prove the second characterization of toric Kato manifolds.Next, in Section 5, we study geometrical properties of toric Kato manifolds, such as the invariant complex submanifolds, or the different natural divisors.Section 6 is dedicated to the topological invariants and Section 8 to the computation of the Hodge numbers.In Section 7, we describe the degenerations of toric Kato manifolds needed for the next section.In Section 9, we tackle the problem of the classification of toric Kato manifolds.Finally, Section 10 deals with the existence of special Hermitian metrics on a general Kato manifold.We conclude the section with an explicit family of examples of toric Kato manifolds in any complex dimension n 4 that do not admit any lcK metrics.
Aknowledgements.-N.Istrati would like to thank Tel Aviv University for the hospitality, where part of this project was carried out.The first three named authors would also like to thank Victor Vuletescu for many stimulating discussions.

Preliminaries on Kato manifolds
In the present paper, a Kato manifold will be a compact complex manifold obtained as follows [Kat78], [Dlo84].We let be the standard open ball in C n .Consider a modification π : B → B at 0 and a holomorphic embedding σ : B → B. Glue the two boundaries of B σ(B) via the real analytic CR-diffeomorphism γ := σ • π.The result is a compact complex manifold X(π, σ), named here a Kato manifold.Any small neighborhood of the image of ∂ B in X(π, σ) is a global spherical shell.We shall refer to the couple (π, σ) as Kato data and to F := π • σ : (B, 0) → (B, 0) as the corresponding germ.
The class of Kato manifolds introduced in [Kat78] is slightly more general, as π is allowed to be a modification at more than one point, so that the resulting manifolds are proper modifications of the ones we described above [Dlo84, Lem.2.7, Part I].However, for the present discussion this generality will not make any difference.
We denote by q : X → X the universal cover of a Kato manifold X = X(π, σ).The deck group Γ of q is canonically isomorphic to Z, and we will denote also by γ the positive generator of Γ, which is indeed induced by the map σ • π.
We start by settling several general facts that will be needed in the paper.
Lemma 2.1.-Let F : (B, 0) → (B, 0) be a holomorphic germ with Proof.-By hypothesis there exists 0 < r < 1 such that F (B) ⊂ B r .By the Schwarz lemma [Sha92, Th. 6], we have F (z) r z for any z ∈ B, hence F m (z) r m for any m > 0 and any z ∈ B. This implies that lim m→∞ sup z∈B F m (z) = 0, and hence diam( m∈N F m (B)) = 0.The conclusion then follows.
Given a map F : (C n , 0) → (C n , 0), define its stable set: Proof.-The previous lemma shows that B ⊂ W s (B).Clearly F m (B) ⊂ W s (F ) for any m > 0 and also for any m < 0. Conversely, if z ∈ W s (F ) then there exists m > 0 such that F m (z) ∈ B, which shows the desired equality.

Toric Kato manifolds
We refer the reader to [Ful93], [Oda78], [Oda88] or any other classical reference for the theory of toric algebraic varieties.Here, we will only fix notation.
We denote by We recall that any toric algebraic variety X is a T N -equivariant partial compactification of T N , and its algebraic structure is encoded by a fan Σ with support in N R .We write X = X(Σ, N ).Traditionally, Σ is a finite collection of rational cones with some compatibility properties, so that X is covered by a finite number of affine open sets (X τ ) τ ∈Σ .However, in the present text, the fan Σ is allowed to be infinite, and then X is of non-finite type.
We denote by |Σ| := τ ∈Σ τ the support of Σ.We will let Σ (k) ⊂ Σ denote the subset of k-dimensional cones of Σ.Also, for a collection of vectors v j ∈ N R , j ∈ J, we will denote by v j | j ∈ J := j∈J R 0 v j the cone generated by it.
Finally, the map restricts to a group homomorphism from C * onto R, and induces a surjective map of fiber T ord : 3.1.Toric Kato data.-We let z = (z 1 , . . ., z n ) denote the standard holomorphic coordinates on C n .In all that follows, C n is viewed as a toric variety with the standard action of T N .We will denote an element of the complex torus by λ = (λ 1 , . . ., λ n ) ∈ T N , and for a toric variety X and x ∈ X, we simply write λx to denote the action of λ on x.
We will call a proper modification π : is a smooth toric modification at 0 and there exists ν ∈ Aut gr.(T N ) such that σ(λz) = ν(λ)σ(z) for any z ∈ B and λ ∈ T N for which this is defined.In this case, we say that σ is ν-equivariant.A Kato manifold will be called a toric Kato manifold if it admits a toric Kato data.
Let π : C n → C n be a toric modification at 0, let Σ be the fan of C n and let Σ be the fan of C n .Note that Σ is uniquely determined by π| π −1 (B) .Then i.e., each new ray of Σ has only positive components.Conversely, any finite refinement with regular cones Σ of Σ satisfying (3.1) defines a toric modification of C n at 0. For every cone τ ∈ Σ (n) , there exists a unique Z-basis e τ := {e τ 1 , . . ., e τ n } of N , such that e τ j ≺ τ , j = 1, . . ., n.If f τ := {f τ 1 , . . ., f τ n } is the corresponding dual basis of M , then the map u j e * j J.É.P. -M., 2022, tome 9 gives rise to a holomorphic chart This chart is uniquely determined up to composing with a permutation of the coordinates.In what follows, we will call such a chart a toric chart.
Express the vectors of e τ in the standard basis e 1 , . . ., e n of N , and put Then A τ completely determines the chart φ τ , so we will also use the notation φ τ = φ Aτ .Note that A τ has only non-negative coefficients, since τ ⊂ C 0 .Moreover, since Using the equivariance of π, we immediately infer the equivariance relation: Any other toric chart φ τ for X τ is given by φ τ = φ τ • s, where s ∈ S n is a permutation and s(z) = (z s(1) , . . ., z s(n) ) = z Ps , with P s = e s(1) • • • e s(n) T .Then the corresponding matrix for φ τ is given by A τ = A τ P s .
Lemma 3.2.-Let (π, σ) be a toric Kato data.Let Σ be the fan determined by π, and let τ ∈ Σ (n) be the cone representing the T N -fixed point σ(0) ∈ B. Then there exists λ 0 ∈ T N and a toric chart φ A : C n → X τ so that σ = λ 0 • φ A .In particular, the germ Proof.-Note that Aut gr.(T N ) ∼ = Aut Z (N ) ∼ = GL(n, Z), and a matrix A determines the group automorphism λ → λ A .Hence, by hypothesis, there exists A ∈ GL(n, Z) so that σ(λz) = λ A σ(z), for all λ ∈ T N and z ∈ B for which this is defined.Let B ∈ GL(n, Z) be a matrix giving a toric chart φ B : a biholomorphism onto its image, and it satisfies, for any λ ∈ T N ∩ B: where λ 1 is some fixed element from T N ∩ B. Thus, putting λ 2 := λ −B −1 A 1 f (λ 1 ) ∈ T N , we find that f (z) = λ 2 z B −1 A on B. Now since f has an holomorphic inverse, it is easy to check that B −1 A is a permutation matrix P s for some s ∈ S n .Therefore A = BP s is a toric matrix and σ = λ 0 • φ A for λ 0 := λ B 2 , as expected.
Lemma 3.3.-If (π, σ) is a toric Kato data with germ F λ 0 ,A and d > 1 is a natural number, then there exist a toric Kato data (π d , σ d ) naturally associated to (π, σ), with corresponding germ F d λ 0 ,A .Moreover, X(π d , σ d ) is a cyclic unramified covering of X(π, σ) of degree d.
Proof.-The Kato data (π d , σ d ) is obtained by composing (π, σ) with itself d times, as it was described in [IOP21, §1], so that the resulting germ is the d-th power of the initial one.From the construction, it is clear that the resulting Kato data is again toric, and that the fan corresponding to π d is given by where Σ 0 := Σ {τ A }, with Σ the fan of π and τ A ∈ Σ (n) the cone representing σ(0).Furthermore, by [IOP21, Lem.1.7], X(π d , σ d ) is a cyclic unramified covering of X(π, σ) of degree d.

Properties of toric Kato matrices
Definition 3.4.-A matrix A ∈ GL(n, Z) will be called a toric Kato matrix if there exists a toric modification π : C n → C n at 0 and a toric chart φ A : Proposition 3.5.-A matrix A ∈ GL(n, Z) is a toric Kato matrix if and only if its columns are either positive or standard vectors.
Proof.-The fact that a toric Kato matrix satisfies the desired properties is a direct consequence of (3.1).For the other implication, we need to prove that for any A ∈ GL(n, Z) with either positive or standard columns, there exists a smooth toric modification π : C n → C n at 0, such that the cone τ A generated by the columns of A belongs to the corresponding fan Σ of C n .This can be achieved by first considering any simplicial fan containing the cone τ A and whose rays are generated either by the standard vectors or positive vectors, and then regularizing it via barycentric subdivision (which adds only positive rays, see, e.g., [Ful93,p. 48]).
3.5 shows that our setting is more general than the one considered in [IOP21].In fact, toric modifications of 0 ∈ C n need not dominate the blow-up of the origin, as the following example shows.
Example 3.6.-Consider the toric Kato matrix Then the cone τ A generated by the three columns of A intersects in its interior the cone generated by e 1 = (1, 0, 0) and e 1 +e 2 +e 3 = (1, 1, 1).Hence no toric modification π : C 3 → C 3 over 0 for which the fan Σ of C 3 contains the cone τ A dominates the blow-up of the origin.
In Figure 1 we give two different explicit realizations of toric modifications π containing τ A .
Let us explain how the fan on the right can be described in terms of blow-ups and contractions.We start from the standard fan of C 3 , where the rays are generated by the canonical basis (1, 0, 0), (0, 1, 0) and (0, 0, 1).When performing a blow-up, we only indicate the generator v of the ray associated to the exceptional divisor obtained via blow-up.This determines uniquely the blow-up, and the new fan Σ is obtained from the previous one Σ by replacing any cone of Σ containing v by all the cones given as the sum of any of its proper faces and v.
Firstly, we perform three point blow-ups corresponding to the rays generated by (1, 1, 1), (2, 2, 1) and (2, 1, 2).Secondly, we perform the blow-up of four equivariant lines, corresponding to the rays generated by (3, 2, 1), (2, 1, 1), (3, 2, 2) and (5, 3, 3) respectively.Next, we notice that the last operation, corresponding to introducing the divisor E given by the ray through (5, 3, 3), is on the one hand the blow-up of the line corresponding to the cone (3, 2, 2), (1, 1, 1) , but also the blow-up of the line corresponding to the cone (2, 1, 2), (3, 2, 1) .Finally, this allows us to contract E by erasing the ray generated by (5, 3, 3) together with the faces connecting this ray to (2, 1, 1) and (3, 2, 2) and the induced changes on the other cones.The composition of the last blow-up and the contraction corresponds to a flip in the language of Minimal Model Program.Let (π, σ = λ 0 • φ A ) be a toric Kato data, and let τ A ∈ Σ (n) be the cone generated by the columns of A, or equivalently, the cone corresponding to the fixed point σ(0).Then A is a positive matrix if and only if τ A ⊂ Int C 0 .In general, we have: be a toric Kato matrix.There exists a unique maximal subset P (A) ⊂ {1, . . ., n} satisfying that there exists a permutation s : P (A) → P (A) so that A j = e s(j) for any j ∈ P (A).Furthermore, there exists m 0 1 so that for any m m 0 and for any j ∈ {1, . . ., n} P (A), the j-th column of A m has strictly positive components.
Proof.-Let us define and let k : S 1 (A) → {1, . . ., n} be the injective map satisfying A j = e k(j) for all j ∈ S 1 (A).Moreover, define inductively, for all m 2, S m (A) : . Then one has: and so there exists m 0 1 with S m (A) = S m0 (A) for all m m 0 .One clearly has then P (A) = S m0 (A) and s = k| P (A) .
On the other hand, it is easy to check that S m (A) = S 1 (A m ) for all m 1.Moreover, by (3.1), a j-th column of A m has only positive components exactly when j / ∈ S 1 (A m ), from which the conclusion follows.
For any matrix A = (a k ) 1 k, n ∈ GL(n, Z) and J ⊂ {1, . . ., n}, we denote by J c = {1, . . ., n} J and we put A J := (a k ) k, ∈J , which is the matrix obtained from A by erasing the lines and columns prescribed by J c .We also use the notation τ J ∈ Σ (|J|)  for the cone generated by e j with j ∈ J and we put N J := span Z {e j |j ∈ J} ⊂ N , respectively T J := N J ⊗ Z C * , C J = N J ⊗ C etc. Finally, we denote by p J the natural projection from N to N J , as well as its R or C-linear extension, and for v ∈ C n , we also write v J := p J (v).
Lemma 3.8.-Let A = (a k ) 1 k, n be a toric Kato matrix and let J ⊆ P (A) be any subset with |J| < n − 1 which is fixed by the permutation s given in Lemma 3.7.Then A J c is again a toric Kato matrix.
Then A has a simple real eigenvalue α > 1 so that for any other α = β ∈ Spec(A), we have |β| < α.Moreover, A admits a Perron eigenvector f A with Af A = αf A so that e * j , f A > 0 for all 1 j n.
Proof.-Let s be the permutation of P (A) given by Lemma 3.7 and let d = ord s.
Then for m = m 0 d big enough, (A m ) P (A) = Id and B := (A m ) P (A) c is a positive matrix.After replacing A by some power, we suppose that this happens for m = 1.For any β ∈ C and k × k matrix M , denote by V M (β) ⊂ C k the generalized eigenspace of M for the eigenvalue β.Put P = P (A) and write p := p P c : C n → C P c for the natural projection.Then p is compatible with the splitting and Spec(A) = Spec(B) ∪ Spec(A P ), where Spec(A P ) = {1}.In particular, we infer that α ∈ Spec(A) is a simple eigenvalue and there exists an eigenvector f A ∈ C n with p(f A ) = f B .For any k ∈ P , we find: from which the conclusion follows.
For a toric Kato matrix A with |P (A)| < n − 1, let f A be a Perron vector for A with positive components.Then A T acting on (C n ) * also has a Perron vector f * A with f * A , f A = 1.We note that since ±1 are the only possible rational eigenvalues of A, f * A cannot be chosen rational.We also observe that since C P (A) is spanned by eigenvectors of A with eigenvalue different from α, for any k ∈ P (A) we have f * A , e k = 0, while for k ∈ P (A) c , we have f * A , e k > 0. Define the half-space: Also, for a toric Kato matrix with P (A) c = {j}, put 3.3.Toric Kato germs.-Note that for any toric Kato germ F = F λ 0 ,A , if m 0 > 0 is such that A m0 has strictly positive components on the columns j ∈ P (A) c , by Lemma 3.7 we have Then we have the following: and let F = F λ ,B be the induced toric Kato germ acting on T P (A) c .Then we have a biholomorphism: In particular, if for any m ∈ N * , we can suppose without loss of generality that A P (A) = id.Let us denote by p := p P (A) c and by p := p P (A) the two projections.Since p Denote by λ := (λ 0 ) P (A) .Then we have where Q m is an integer valued matrix determined by A, whose coefficients are each at least m.From this, it is clear that the fiber of p : we can suppose that w satisfies |λ j | k∈P (A) c |w k | < ε for each j ∈ P (A).Then (3.5) implies that for any z ∈ T N with p(z) = w and each j ∈ P (A), m ∈ N, we have

A different construction of toric Kato manifolds
In this section, we wish to give a different realization of toric Kato manifolds using toric geometry.In particular, we will see that the universal cover of a toric Kato manifold is an open subset of a toric variety of non-finite type.The construction generalizes the one given by Tsuchihashi in [Tsu87], as well as the known toric constructions for Kato surfaces [Oda78].
In all that follows, we fix a smooth toric modification π : C n → C n at 0 with π = id and denote by Σ the fan of C n .We fix a toric chart φ A : Let Σ 0 := Σ {τ A }, and define the infinite fan (4.1) Also note that the map F A extends to an automorphism of the toric variety X(Σ A , N ) as the map induced by N → N , v → Av.Similarly, λ 0 ∈ T N extends to an automorphism of X(Σ A , N ).Therefore, we have a natural action of U := F λ 0 ,A on X λ0,A .Before giving the main theorem of the section, we show the following technical lemma needed in the proof.
Let Σ be a fan in N and let X(Σ, N ) be the associated toric variety.Denote by Ω Σ the closure of Ω in X(Σ, N ), and by V N the closure of V in N R .Then for any cone τ ∈ Σ (k) with 1 k n, one has: Proof.-Let us fix such a cone τ ∈ Σ (k) .In what follows, we let Recall that one has a natural map given by extension by 0: Applying the map ord to the above, j induces a natural injection: given this time by extension by ∞.Moreover, its image is given by On the other hand, the natural inclusion Ω ⊂ T N ⊂ X τ reads, after applying the map ord, as follows: Since orb τ ⊂ X τ , one has orb τ ∩ Ω Σ = orb τ ∩ Ω τ , where the latter closure is taken in X τ .Furthermore, since ord is surjective, this intersection is nonempty precisely when Suppose first that τ ∩ V = ∅ and let u ∈ j R (orb τ R ).In order to show (i), it suffices to prove that u ∈ k(V ).There exists q ∈ N R so that Furthermore, for m 1, since mv ∈ V and using the properties of V , there exists c m > m big enough so that v For the second part, suppose that V is a cone and that there exists with v m ∈ V N for each m 1.Let us fix the standard scalar product on N R and the orthogonal splitting N R = N 0 ⊕ Rτ , so that the map d : N 0 → Hom(τ ⊥ , R), q → •, q becomes a continuous isomorphism.In particular, it follows that u| τ ⊥ ∩Sτ = d(q)| Sτ for a unique q ∈ N 0 .With respect to the splitting, we write On the other hand, we have: Hence there exists m 0 1 so that We are now ready to prove: Theorem 4.2.-Let (π, σ) be a toric Kato data with corresponding germ F λ 0 ,A .Then the group U acts freely and properly discontinuously on X λ 0 ,A and we have a biholomorphism X(π, σ) ∼ = X λ 0 ,A /U.
Furthermore, if D denotes the divisor of X(π, σ) induced by the exceptional divisor of π and D is the preimage of D in the universal cover, then D = D + .Finally, we have the following: Define the fan: The identity map of N induces a (non-proper) map of toric varieties φ : X(Σ + A , N ) → C n = X(Σ, N ) and we have from which it follows that N ) .Since by the proof of Proposition 2.3, Ω 0 is a fundamental domain for the action of U on Ω, it follows that W 0 is a fundamental domain for the action of U on X λ 0 ,A .
Let V := ord(B * ) ⊂ N R and note that it is open and satisfies the condition Let C 0 ⊂ N R be the standard cone, so that V ⊂ C 0 .By Lemma 4.1(ii), we have that Furthermore, since any ray ν ∈ Σ A with ν ⊂ Int C 0 satisfies ν ∩V = ∅, Lemma 4.1(i) implies that orb ν ⊂ B * X(ΣA,N ) .In particular, applying A, we infer that and hence: , which is precisely a fundamental domain for the deck group action on X(π, σ).Clearly this identification of X λ 0 ,A and of X(π, σ) is equivariant, which shows thus that the action of U is free and proper, and that we have the desired isomorphism.Finally, it is clear from the isomorphism that D = D + .
For later use, we also show: where for the case As in the proof of Lemma 3.7, we can suppose without loss of generality that A P (A) c = id and that B := A P (A) c is a positive matrix.We first show that Indeed, the inclusions m∈Z A m C 0 ⊆ H(A) ∪ τ P (A) and τ P (A) ⊆ m∈Z A m C 0 are straightforward.Let now v ∈ H(A).We shall prove there exists > 0 such that Let f * B be a Perron vector for B T with f * B , f B = 1.For the case P (A) c = {j}, we put f B = e j and f * B = e * j .If α > 0 is the dominant eigenvalue of A, or also of B, then we have: which is obvious in the case |P (A)| = n − 1 and implied by the Perron-Frobenius theorem for the case |P (A)| < n − 1.In particular, we have: Since f * B , v > 0, then there is some 0 ∈ N so that for any > 0 , (A v) P (A) c has positive components.For any m ∈ N, put Q m := (a , where a (m) •• denote the components of A m .Then we have: and since Q m is a positive matrix, we obtain that for big enough m, A +m v ∈ C 0 .
In order to compute the intersection, we distinguish between the two cases.Suppose first that |P (A)| < n − 1.Then we wish to show that It is clear that R >0 f A + τ P (A) is invariant by A and is contained in IntC 0 .Moreover, for any vector v ∈ C 0 τ P (A) ⊂ H(A), we have that when m → +∞, and so we have: which proves (4.5).Suppose now that P (A) c = {j}.Then we have: J.É.P. -M., 2022, tome 9 Indeed, clearly τ P (A) ⊂ A m C 0 for any m ∈ Z. Conversely, let v ∈ m∈Z A m C 0 , so that for each m ∈ Z there exist v m ∈ R P (A) and a m ∈ R 0 with In particular, we find that v j = a m e j so that a m = a 0 for any m ∈ Z.If a 0 > 0, then a m Q m e j tends to ∞ with m, so v is unbounded, which is absurd.Thus a 0 = 0 and so v ∈ τ P (A) .Finally, we have m∈Z which concludes the proof.
Example 4.4.-Consider the toric Kato data (π, σ), where π is associated to the fan depicted in the left side of Figure 2, and σ = λ 0 φ A with In this case, P (A) = {1}, and a direct computation shows that the eigenvalues of A are 1, ξ 2 , ξ −2 , where ξ = (1 + √ 5)/2 is the golden ratio.The right side of Figure 2 shows the fan Σ 2 associated to the toric modification π 2 given by Lemma 3.3.

Geometric properties of toric Kato manifolds
In complex dimension n = 2, the germ F λ 0 ,A determines the toric Kato surface (if we ask it to be minimal).Toric Kato surfaces with P (A) = ∅ are precisely the properly blown-up Inoue-Hirzebruch surfaces, in the sense of [FP10].As we will see later, X(F λ 0 ,A ) ∼ = X(F A ) for any compatible λ 0 ∈ T N .In the case |P In analogy to Nakamura's result in dimension n = 2 [Nak84], we can detect the type of a toric Kato manifold by looking at its curves.This shows in particular that the above definition makes sense, and does not depend on the chosen Kato data.In order to make this precise, we first introduce one more definition.
We will call a compact complex subspace Y of a toric Kato manifold X T -invariant if its preimage in the universal cover is invariant under the T n -action, or equivalently, under the (local) T N -action.All such irreducible subspaces correspond to cones in Σ A , and are of two types: either they are immersed toric Kato manifolds, if the corresponding cone is A-invariant, or they are complete toric algebraic varieties, if the corresponding cone is not A-invariant.In particular, we have the following description of T -invariant curves: Theorem 5.2.-Let X be a toric Kato manifold.Then we have the following characterization in terms of T -invariant curves: (1) X is a finite quotient of a diagonal Hopf manifold if and only if all its T -invariant curves are elliptic; (2) X is of hyperbolic type if and only if all T -invariant curves are rational; (3) X is of parabolic type if and only if X contains a unique T -invariant elliptic curve E, and at least one rational T -invariant curve.In this case, E is smooth and all other T -invariant curves are rational.Moreover, the intersection number E • D = 0.
Proof.-Let A ∈ GL(n, Z) denote a Kato matrix of X and suppose, without loss of generality, that A P (A) = id.Any T -invariant curve C of X is given by a cone τ ∈ Σ (n−1) A , either as orb τ /Γ, in case τ is Γ-invariant, or as the image of orb τ ∩ X.Moreover, C is an elliptic curve in the first case, and toric, thus rational, in the second case.
, so all T -invariant curves of X are rational.Suppose now that P (A) c = {j}.Then the only A-invariant cone τ ∈ Σ (n−1) A is τ P (A) , so E := orb τ P (A) /Γ ∼ = C * /λ j is the only elliptic curve in this last case, and is clearly smooth.Moreover, it is clear that there exits at least one τ P (A) = τ ∈ Σ (n−1) A , giving rise to a T -invariant rational curve.
Recall that by Theorem 4.2, D is the image in X of Thus, in order to see that E • D = 0, it is enough to see that orb τ P (A) does not intersect orb ν for any ν ∈ Σ (1) A with ν ⊂ τ P (A) .But this is equivalent to say that any such ν is not a face of τ P (A) , which is obvious.
Next we want to give an explicit description of the divisors defined by It is clear that D X T = D + + k∈P (A) X k , where X k := orb e k X(Σ A ,N ) ∩ X λ 0 ,A .
We will call D T the toric divisor of X.It satisfies q * D T = D X T , where q : X → X is the universal cover map.
Suppose first that the permutation s given by 3.7 corresponding to A is the identity.
Then we find that and it is easy to check that F λ 0 ,A | orb e k = F (λ 0 ) {k} c ,A {k} c .In particular, we have: .
We have thus found: Lemma 5.3.-If A is a toric Kato matrix with s = id, and k ∈ P (A), then X k is Γ-equivariantly biholomorphic to X (λ 0 ) {k} c ,A {k} c , where Γ acts by F λ 0 ,A on the first manifold and by F (λ 0 ) {k} c ,A {k} c on the second one.In particular, In the general case, i.e., when s = id, X k need not be invariant by F := F λ 0 ,A , however F still acts on k∈P (A) X k .In this case, there exists d ∈ N with s d = id.Then it is easy to see that W T (F ) = W T (F d ) and that F d = F λ ,A d , where . Consider the toric Kato data (π d , σ d ) given by Lemma 3.3, with corresponding germ F d .Then clearly Σ A d = Σ A .The above argument thus implies that X k ∼ = X (λ ) {k} c ,A d {k} c .To each cycle J ⊂ P (A) of s then corresponds a compact hypersurface which autointersects transversely such that the divisor D J := Y J on X(π, σ) satisfies q * D J = k∈J X k .Finally, we have found: Proposition 5.4.-Let (π, σ) be a toric Kato data with germ F = F λ 0 ,A and let s be the permutation corresponding to A. Let C := {J ⊂ P (A) | J is a cycle of s}.Then we have: Moreover, for each J ∈ C, Y J is a compact (possibly singular) hypersurface whose normalization is a toric Kato manifold.
We also want to establish when the divisors D, D, D T , D X T are connected.In complex dimension 2, the situation is well known by the works of Nakamura [Nak84] and Dloussky [Dlo84], and can be easily read off from the fan Σ 0 .Namely, in the case of parabolic Inoue surfaces, i.e., when |P (A)| = 1, if P (A) = {j}, then orb e j is fixed by A and disconnected from D, which itself is connected.It follows that both D X T and D T have two connected components, respectively D and orb e j for D T , and D and the elliptic curve orb e j /Γ for D T .In the case of Inoue-Hirzebruch surfaces, D X T = D has two connected components, given by the two connected components of |Σ 0 | {0}.When det A = 1, A fixes the components, so that D has also two connected components.When det A = −1, A interchanges the components, and in this case D is connected.In higher dimension, things are much simpler: Lemma 5.5.-Let X = X(π, σ) be a toric Kato manifold of complex dimension n 3. Then the divisors D X T , D, D T and D are connected. Proof.
-Given a divisor F on a toric variety X(Σ, N ), F = j∈I orb ν j , let G F,Σ = (V, E) be the graph whose vertexes are given by V = {ν j | j ∈ I} and edges given by Let A be a Kato matrix of X.Consider first C n = X( Σ, N ), the exceptional divisor E of π, and the divisors D 1 = E+ n j=1 orb e j and D 2 = E+ j / ∈P (A) orb e j .Since E is connected and intersects each component orb e j , 1 j n, also D 1 and D 2 are connected, so their graphs G D1, Σ , G D2, Σ are connected.Next, consider J.É.P. -M., 2022, tome 9 C n {σ(0)} = X( Σ 0 , N ) ⊂ C n and D j = D j ∩ X( Σ 0 , N ), j = 1, 2. As n 3, we have G Dj , Σ = G D j , Σ0 , so D j are also connected, j = 1, 2. From this we infer that D T = m∈Z γ m D 1 and D = m∈Z γ m D 2 are connected on X( Σ, N ), hence also their respective restrictions to X are connected and their images in X are connected.This concludes the proof.
We also have the following geometric description of Kato manifolds with P (A) = ∅: Proposition 5.6.-Let X A := X(π, σ) be a toric Kato manifold with σ = λ 0 σ A .Let B = A P (A) c and let (π , σ = λ σ B ) be the naturally induced Kato data from (π, σ) and B. Let j : X B := X(π , σ ) → X A be the natural embedding.We denote by D A the divisor of X A induced by the exceptional divisor of π, and by D B = j * D A the corresponding divisor of X B .Then X A D A has a natural structure of a holomorphic vector bundle V of rank |P (A)| over X B D B .Moreover, up to passing to a finite unramified cover of X A , we have we have that Moreover, X({0}, N P (A) c ) = X(Σ B , N P (A) c ) D B,+ , and the natural projection u Z : N → N P (A) c induces a toric morphism u : X(Σ , N ) → X({0}, N P (A) c ) satisfying u(γz) = γ (u(z)) for any z ∈ X(Σ , N ), where γ is the positive generator of Γ acting on X(Σ A , N ) and γ is the positive generator of Γ acting on X(Σ B , N P (A) c ).Since moreover W T (F A ) = T P (A) × W T (F B ) by 3.10, it follows that we have an induced vector bundle: With respect to the standard basis e 1 , . . .e n of N = Z n , we have a natural splitting Up to taking a finite cover of X A , which corresponds to taking a positive power of A, suppose that A P (A) = id.Then the action of Γ preserves each Ce k × W T (F B ) and γ acts on it by where k ∈ N |P (A) c | is the line (A ks ) s∈P (A) c .It follows that Ce k descends to a line bundle L k over X B D B and that V = k∈P (A) L k .
In particular, when P (A) c = {j}, each line bundle L k is given by the multiplier Finally, we also have: Proposition 5.7.-Let X be a toric Kato manifold with toric divisor D T and matrix A, and let L det A be the flat line bundle on X determined by ρ : Γ → C * , ρ(γ) = (−1) det A .Then we have In particular, X has negative Kodaira dimension.

Betti numbers
In this section, we fix a toric Kato manifold X = X(π, σ) of matrix A, with π : C n → C n .Let D be the divisor of X induced by the exceptional divisor of π.We also let Σ denote the fan of C n , we put CP n = C n ∪ H, where H is the divisor at infinity, and we let Σ P be the corresponding fan of CP n .Let us put a j := | Σ (j) |, for each 0 j n.Then we can express the Betti numbers of X purely in terms of the combinatorial data a 0 , . . ., a n .Theorem 6.1.-The n-dimensional toric Kato manifold X has the following Betti numbers: In particular, we have where # denotes the number of irreducible components, and χ(X) denotes the Euler characteristic of X.
Proof.-Let us suppose that n > 2, since the statement for n = 2 is well known.
In this proof, we use the notation H • (M ) to denote the cohomology of M with coefficients in C.
We will first show that b j (X) = b j ( C n ) for 2 j 2n−2.By [Kat78], X = X(π, σ) is diffeomorphic to a Kato manifold X = X(π, σ ), so that σ (0) no longer meets the exceptional divisor of π.In particular, there exists a Hopf manifold Y ∼ = S 2n−1 × S 1 , p ∈ Y and a modification µ : X → Y at p induced by π.Let B ⊂ Y be a ball centered at p, so that V := µ −1 (B) ⊂ X is a neighborhood of the exceptional divisor F of µ which is homotopy equivalent to C n , and let Next, we wish to compare the cohomology of C n with that of CP n .Let U be a neighborhood of H in CP n biholomorphic to a D-bundle over H inside N CP n H, let V := C n ⊂ CP n and let U * := U ∩ V , which is a D * -bundle over H.We have Using this, the Mayer-Vietoris sequence in cohomology then gives: where r U , r V are the corresponding restriction maps, so that r * V = 0 and r * U is injective.From this we infer (6.1) Finally, since CP n = X(Σ P , N ) is a complete toric algebraic variety, it satisfies [Dan78, Th. 10.8] where a j := |Σ (j) P | = a j + n j−1 , j ∈ {1, . . ., n}, a 0 = a 0 .Together with (6.1), this gives the desired formula for b • (X).
In particular, since Furthermore, we have: which concludes the proof.

Toric degenerations
7.1.Nakamura degenerations.-In this section, we describe how the Nakamura degenerations [Nak83] of toric Kato surfaces, generalizing constructions of Miyake-Oda [Oda78], can be generalized to any dimension.Let us fix a toric Kato manifold X = X(π, σ) with germ F = λF A .Recall that X can be seen as a partial compactification of Ω F = W T (F ) inside the toric manifold X(Σ A , N ).Also recall that we denoted by Σ 0 the fan of the toric manifold C n {σ(0)}, which is then a fundamental domain of the action of Γ on Σ A .Here Γ is the deck group of X → X, generated by γ.Let CP n be C n with the hyperplane at infinity H, and let CP n be the blowup of CP n along σ(0), with exceptional divisor E. Then CP n = X(Σ c , N ) is a toric manifold, with fan Σ c given by the completion of Σ 0 with the n-dimensional cones: We now wish to construct a manifold X N as a subset in a toric variety, together with an action of Γ, giving rise to the Nakamura degeneration.Consider then N = N ⊕ Ze, and define an action A on N by putting A| N = A and Ae = e + Ac.Put ν k := A k e, k ∈ Z, and define regular fans Σ 0 , Σ in N as follows: Let us also put λ = (λ, 1) ∈ T N and let γ be the automorphism of X( Σ, N ) given by λ A, where we also denote by A the natural action induced by A on the toric variety.Note that for z ∈ T N , γ(z) = F (z) := λz A .
Let p Z : N → Ze be the natural projection, and let Σ 1 = {0, e } be the fan of C in Ze.Then p Z is a map of fans, and so defines naturally a map of toric varieties p : X( Σ, N ) → X(Σ 1 , Ze).We have and define the smooth manifold together with a free and proper action of Γ = γ and a Γ-invariant map p : X N → ∆ so that p −1 (t) = X(π, σ(t•)) for t = 0 and p −1 (0) is given by Z copies of CP n glued along H and E via γ.Thus we obtain a flat proper holomorphic map . This is precisely the Nakamura degeneration.
Define the toric divisor on X N : We have thus obtained: Theorem 7.1.-Let X(π, σ) be an n-dimensional toric Kato manifold, let CP n = X(Σ c , N ) be defined as above and let S = CP n / E∼ γ H , where γ was defined above.There exists a flat holomorphic proper family p N : (X N , D) → ∆ over some disk ∆ ⊂ C of radius R > 1, so that: Note that by the discussion of Section 9, the smooth fibers of the family X N are all isomorphic if c ∈ im(id −A).7.2.Isotrivial degenerations.-Nakamura degenerations are defined for all Kato manifolds, however generally the smooth fibers are not all isomorphic.On the other hand, for most toric Kato manifolds we can define other degenerations, whose smooth fibers will all be isomorphic, but whose central fiber is less concrete and generally more singular.Moreover, the total space of these degenerations need not be smooth.These follow the same ideas as [Tsu87,§2].
In order to do so, we first need the following: Note that the central fiber of the family X u depends on u, and that any primitive element u ∈ im(id −A) ∩ Int C 0 ∩ N gives rise to such a family.However, we cannot in general ensure that X u is a smooth manifold, or equivalently that the normalization of the central fiber is smooth, as the following result shows: In this case, the family (X u , D) is equivalent to the Nakamura family (X N , D N ), in the sense that there exists a biholomorphism φ X : Proof.-It is clear that if u = c, then X(Σ u , N ) ∼ = CP n , so it is smooth.Conversely, suppose that X(Σ u , N ) is smooth.Since for any j ∈ {1, . . ., n}, −u, e 1 , . . ., e j , . . ., e n is a basis of N , and since u ∈ Int C 0 , this implies that u = c.
where v is chosen to satisfy Ac = Av − v.We have then φ Z ( Σ) = Σ c and φ Z Ax = A φ Z x for any x ∈ N .This implies that φ Z induces a Γ-equivariant biholomorphism φ : X( Σ, N ) → X( Σ c , N ) commuting with the projections to C and satisfying φ • F = F • φ.In particular, it is clear that φ(Ω) = Ω F ∩ p −1 (∆) and that φ * D = D N , so φ also induces a biholomorphism between X N and X c | ∆ , and so a biholomorphism φ X : The conclusion of the discussion is the following: Theorem 7.4.-Let X be a toric Kato manifold of hyperbolic type of matrix A, and let u ∈ im(id −A) ∩ Int C 0 ∩ N be a primitive element.Then there exists a flat holomorphic proper family p u : (X u , D) → C so that: Moreover, X u is smooth if and only if u = c, in which case the family (X u , D) is equivalent to the Nakamura family (X N , D) of Theorem 7.1.

Analytic invariants
In this section, we compute different analytical invariants of toric Kato manifolds, such as the Hodge numbers h 0,q , h 1,q and h p,0 , as well as the cohomology of the logarithmic sheaves Θ X (− log D T ) and Ω 1 (log D T ).To this aim, we will make use of the degenerations of Section 7 and we will extend the method of Tsuchihashi [Tsu87], who computed these invariants when P (A) = ∅ and λ = (1, . . ., 1).Using a different method, Sankaran computed in [San89] all the Hodge numbers of toric Kato manifolds with λ = (1, . . ., 1), under the additional assumption that A is diagonalizable over C and irreducible over Q, so in particular P (A) = ∅, obtaining h 0,1 = h n,n−1 = 1, h p,p = b 2p , h p,q = 0 otherwise.We should note that the number |P (A)| is an invariant of toric Kato manifolds -it can be retrieved, for instance, as the number of non-compact irreducible toric divisors of the universal cover, cf.Section 5.Moreover, even when P (A) = ∅ but 1 ∈ Spec(A) (see Example 9.11), one cannot simply reduce to the case λ = (1, . . ., 1) (see Corollary 9.8).This shows that our context is much more general than both Sankaran's rank one actions or Tsuchihashi's.Sankaran's proof strongly depends on the above mentioned hypotheses, which are quite restrictive from the point of view of toric Kato manifolds.Therefore it would be interesting to know if one can adapt his strategy to our more general situation.
Since Ω is a T-invariant domain of T N , we can express f as a series on Ω: Since the right hand side extends naturally to a meromorphic function on X, f writes on X as where for each m ∈ M , m is the natural meromorphic function induced by m on X(Σ A , N ).Now since for any τ ∈ Σ A , f has no poles on orb τ , it follows that for any m ∈ M with c m = 0 we have m, v 0, ∀v ∈ τ.
Since C := |Σ A | = H(A) is a convex cone by Lemma 4.3, we readily infer: From this it follows that in the parabolic case, f is an entire function in the z j -variable.
Remark 8.2.-For a parabolic toric Kato manifold X with P (A) c = {j}, define where λ j ∈ D is the j-th component of λ 0 .Then it follows that the holomorphic function on the universal cover defined by f (z) = z j on T N gives rise to a holomorphic section of L .
Corollary 8.3.-If X is a toric Kato manifold which is not Hopf and L ∈ Pic 0 (X) is non-trivial, then H 0 (X, L) = 0 if and only if X is of parabolic type and L is a positive power of the line bundle defined in Remark 8.2.In this case, H 0 (X, L) ∼ = C.
Next, let Θ X (− log D T ) denote the locally free subsheaf of the tangent sheaf given by germs of holomorphic vector fields tangent to D T , let Ω 1 X (log D T ) be the dual sheaf and let Ω p (log D T ) := ∧ p (Ω 1 (log D T )) for p 2.
Proposition 8.4.-If X is any toric Kato manifold, then H 0 (X, Ω p X ) = 0 for any p 1.
Proof.-For Hopf manifolds the result is well known, so we only treat the other cases.We start by noting that the map where m is the natural meromorphic function on X associated to m, is an isomorphism of O X -modules [Ful93, Prop.at p. 87].In particular, for any p 1, Ω p X (log A for each 1 j n and let m 1 , . . ., m n be the dual basis of M .For each J = (1 Using Proposition 8.1, we find that where O C (C) denotes the ring of entire functions in z j = e * j .Since Ω p X is a subsheaf of Ω p X (log D T ) and since each ω J has poles along j∈J orb ν j , it is clear that in the case |P (A)| < n − 1 we have H 0 (X, Ω p X ) = H 0 ( X, Ω p X ) = 0 for any p 1.In the case P (A) c = {j}, the same argument shows that H 0 (X, Ω p X ) = H 0 ( X, Ω p X ) = 0 for p 2. For p = 1, we find that and as |λ j | < 1, we find Theorem 8.5.-Let X be a toric Kato manifold of hyperbolic type.Then we have: Proof.-Let A be a toric Kato matrix for X.Since |P (A)| < n − 1, there exists u ∈ im(id −A) ∩ Int C 0 ∩ N a primitive element.Consider the family p u : X u → C of Theorem 7.4, of central fiber S and generic fiber X.Then by upper-semicontinuity [BS76, Th. 4.12] we have: Let us first compute H • (S, O S ).Let n : S = X(Σ u , N ) → S be the normalization map, and let Y = n(E ) = n(H ) be the double locus of S, with inclusion map j : Y → S. Then we have an exact sequence of sheaves on S: and the corresponding long exact sequence in cohomology.Since Y and S are compact toric varieties, we have . This implies that H p (S, O S ) = 0 for p 2 and gives rise to the exact sequence: We infer that H 0 (S, The cohomology of O S together with (8.3) immediately gives the desired result for H p (X, O X ) for p = 1.On the other hand, since h 1,0 (X) = 0 by Proposition 8.4, using again (8.3) we find: hence we also have H 1 (X, O X ) = C.This concludes the proof.
Corollary 8.6.-Let X be a toric Kato manifold of hyperbolic type.Then we have an isomorphism Pic 0 (X Moreover, we have a short exact sequence: Proof.-Since by Theorem 8.5 we have the exponential sequence gives rise to the exact sequence: from which we conclude.
Theorem 8.7.-Let X be a toric Kato manifold with toric divisor D T and matrix A.
Then we have: If moreover X is of hyperbolic type, then we also have: -We only show the statements for the sheaf Θ X (− log D T ), since for Ω 1 X (log D T ) the proof is dual.We have Since X is an open set of the toric manifold X(Σ A , N ), we find Θ X (− log D T ) ∼ = O X ⊗ N and so, by Proposition 8.1, the conclusion follows for the case The Γ-invariance of s reads: Using that |λ j | < 1 and that for any µ ∈ Spec(A), |µ| = 1, we find: Suppose next that |P (A)| < n − 1, so that there exists a primitive element u ∈ im(id −A) ∩ Int C 0 ∩ N , and consider the family p u : (X u , D) → C of Theorem 7.4, of central fiber S and generic fiber X.Consider the sheaf F := Θ X (− log D), and denote by F t its restriction to any fiber p −1 u (t), t ∈ C. Then for any t ∈ C * , we have F t ∼ = Θ X (− log D T ) ⊕ O X , so by upper-semicontinuity [BS76, Th. 4.12] we find: We must thus compute H • (S, F 0 ).Consider again the exact sequence (8.4) and tensorize it with the flat sheaf F 0 , obtaining: As before, passing to the long exact sequence in cohomology we find that H p (S, F 0 ) = 0 for p 2 and so H p (X, Θ X (− log D T )) = 0 for p 2. Moreover, we have the short exact sequence: which implies H 1 (S, F 0 ) = coker( A − id) = coker(A − id) ⊕ C. Thus, from (8.5) and Theorem 8.5 we infer the inequality: At the same time, H 1 (X, Θ X (− log D T )) can be computed, as Γ-equivariant cohomology, by a spectral sequence [Gro57, §5.2].Namely, if q : X → X denotes the covering map, then Θ X (− log D T ) is a Γ-sheaf on X and Θ X (− log D T ) = q Γ * Θ X (− log D T ).Therefore, we have a spectral sequence . Moreover, since Γ ∼ = Z, we have E p,• 2 = 0 for any p 2. Hence this spectral sequence degenerates at the second page, implying that H 1 (X, Θ X (− log D T )) = E 1,0 2 ⊕ E 0,1 2 .In particular, we have But using Proposition 8.1, the fact that S 1 is the Eilenberg-MacLane space of Γ and Poincaré duality for local systems, we find: This, together with (8.7) and (8.6) gives us the desired conclusion.
Theorem 8.8.-Let X be a toric Kato manifold of hyperbolic type, with divisor D induced by the exceptional divisor of π.Then we have: Proof.-Let A be a Kato matrix for X, let s be its corresponding permutation and let C denote the set of cycles of s.Let us write: where each D j is an immersed irreducible hypersurface.Let us put t := #C, so that k = #D + t.Let D j be the normalization of D j for each j = 1, . . ., k and let α : D T = k j=1 D j → X be the natural map given by inclusion and normalization.Then we have an exact sequence: where P is the Poincaré residue map.Each component of D T is either a compact toric variety, or a toric Kato manifold of hyperbolic type, and we have precisely t components of the last type.Using Theorems 8.5 and 8.7, we infer that H p (X, Ω 1 X ) = 0 for p 3 and we find the following exact sequence: J.É.P. -M., 2022, tome 9 Now the point is to show that the map P 1 induced by P is surjective.This will readily imply that h 1,2 (X) = 0. Furthermore, as the Euler characteristic of the exact sequence is 0, and as dim H 0 (X, Ω 1 X (log D T )) = dim H 1 (X, Ω 1 X (log D T )) by Theorem 8.7, we also infer that h 1,1 = k − t = #D.
Let us thus show that P 1 is surjective.We recall that where q is the covering map, and for each j ∈ P (A), X j = orb e j ∩ X.The map P 1 is induced by the Γ-equivariant map of Γ-sheaves on X: so that On the other hand, using again the spectral sequence of [Gro57] which degenerates at the second page, together with Theorems 8.5 and 8.7, we find where Γ acts on C |P (A)| via the permutation s.Hence P 1 is the map induced in group cohomology by Q : M ⊗ C → C |P (A)| , m → ( m, e j ) j∈P (A) .Clearly Q is surjective, and let K = ker Q, so that we have an exact sequence This further induces an exact sequence But H 2 (Γ, K) = 0 since Γ = Z, so indeed P 1 is a surjective map.This concludes the proof of the theorem.
Remark 8.9.-We note here that for toric Kato manifolds X(π, σ A (t•)) of parabolic type, the conclusions of Theorems 8.5, 8.7 and 8.8 also hold if |t| is small enough.The arguments are exactly the same, except that one has to replace the families p u by the Nakamura family p N in Theorems 8.5 and 8.7.Since in the Nakamura family the smooth fibers need not be isomorphic, the semi-continuity argument then works only for parabolic Kato manifolds close to the singular fiber.
Notice that p m m leaves W T (F ) invariant, and hence defines a regular map p m m : .This allows to construct projective limits of the families X Σ m A , N m∈Z and X m F m∈Z .The spaces X Σ A , N and X F sit naturally inside such projective limits as open dense subsets.Their complement can be interpreted in terms of valuations (see, e.g., [Tei18]), and the whole projective limits in terms of hybrid spaces (see, e.g., [Ber09,BJ17,Fav20]). 9.2.Isomorphism classes.-We will say that two toric Kato manifolds X and X are equivariantly isomorphic if there exists a biholomorphism Φ : X → X and a group isomorphism ν ∈ Aut gr.(T N ) so that the lift Φ : X → X satisfies Φ(λx) = ν(λ) Φ(x) for any x ∈ X and λ ∈ T N for which this is defined.
Also, recall that a holomorphic germ Φ satisfying Φ • F = G • Φ is called a semiconjugacy between F and G.When Φ is moreover an invertible germ, we call Φ a conjugacy.
We start by describing the compact hypersurfaces in the universal covering of a toric Kato variety.
Lemma 9.4.-Let X be a toric Kato manifold, and X its universal covering.Then the compact hypersurfaces of X are exactly the toric hypersurfaces whose sum gives D + : they correspond to the rays of the fan Σ A associated to X not contained in τ P (A) .
Proof.-Let H be any compact hypersurface of X. Being compact, it must be contained in the open set is a toric modification above the origin of C n , and we infer that H must correspond to one of the exceptional primes of the toric modification.This means that H corresponds to one of the rays of Σ m A , and so to one of the rays of Σ A .Finally, it is clear that a ray ν of Σ A gives rise to a compact hypersurface in X if and only if ν is not contained in τ P (A) .
Theorem 9.5.-Let X and X be two isomorphic toric Kato manifolds, with associated Kato germs Proof.-Let Ψ : X → X be an isomorphism, and let Ψ : X → X be a lift of Ψ to the universal coverings.
To distinguish collapsing maps for X and X , we denote collapsing maps for X with the letter p, and collapsing maps for X with the letter q.To simplify notations, By checking the lowest degrees of (9.1), we get QA = BQ and χλ Q = µχ B .After simplifying the common factor in (9.1), we get (9.2) Let u i z i , with u i ∈ C n and i ∈ N n {0}, be any (n-uple of) monomial(s) appearing in the formal power series expansion of u.
This monomial contributes with λ i u i z iA in the left-hand-side, and with Bu i z i + h.o.t. in the right-hand-side of (9.2), where h.o.t.denotes a suitable formal power series with monomials z j with |j| > |i|.Since k a hk 1 for all h = 1, . . ., n, we have |iA| |i| for any i ∈ N n .
Assume that this inequality is strict for any i ∈ N n {0}.This implies that u ≡ 0. In fact, if this is not the case, there exists i ∈ N n {0} so that u i = 0. We may assume that |i| is minimal among the multi-indices satisfying the above condition.In this case we get a contradiction, since the left-hand-side of (9.2) contains a non-trivial monomial of order |i|, while the right-hand side does not.
Assume now that |iA| = |i| for some i ∈ N n {0}.Notice that k a jk = 1 for some j ∈ {1, . . ., n} if and only if P (A) c = {j}.We infer that in this case P (A) c = {j} and i = me j for some m > 0. Then the coefficient of the monomial z i in (9.2) gives λ j u i = Bu i , where λ = (λ 1 , . . ., λ n ).Since |λ j | < 1 and B has only eigenvalues of modulus 1, we deduce that u i = 0, and hence u ≡ 0.
Suppose moreover that Φ lifts to an automorphism Φ : C n → C n on the total space C n of the toric modification π.Then Φ induces a (non-equivariant) automorphism of X.
This example also shows that the vector space H 0 (X, Θ X ) is generally larger than the vector space H 0 (X, Θ X (− log D T )) ∼ = ker(A − id) computed in Theorem 8.7.
Since the semi-conjugacy Φ induced by an equivariant isomorphism Ψ as in Theorem 9.5 automatically satisfies the hypothesis of Proposition 9.6, we deduce the following classification of toric Kato manifolds, associated to the same toric modification, up to equivariant isomorphisms.
Corollary 9.8.-Let X = X(π, σ) be a toric Kato manifold with Kato germ F = F λ,A .Let X = X(π, σ ) be another toric Kato manifold associated to the same toric modification π, and denote by G = F µ,B its associated Kato germ.Set λ = e 2πi and µ = e 2πim , where and m belong to C n .
Then X is equivariantly isomorphic to X if and only if there exists Q ∈ GL(n, Z) such that QA = BQ, Q(Σ A ) = Σ B , and so that Proof.-By Proposition 9.6, if X and X are equivariantly isomorphic, then there exists a map Φ(z) = χz Q conjugating F and G and so that QΣ A = Σ B .Conversely, any such map Φ induces a natural equivariant biholomorphism Φ : X → X commuting with the deck group action, and so an equivariant biholomorphism between X and X .By Φ • F = G • Φ, we infer QA = BQ, and χλ Q = µχ B .By applying 1 2πi log to both sides, we get where v is such that χ = e 2πiv .The statement follows.
Notice that, as in Theorem 9.5, we may assume that the matrix Q in the statement of Corollary 9.8 has non-negative entries.As an immediate corollary, we get: Corollary 9.9.-Let X and X be two toric Kato manifolds as in Corollary 9.8, both with the same associated Kato matrix A. If 1 ∈ Spec(A), then X and X are equivariantly isomorphic.
Proof.-It suffices to apply Corollary 9.8 with A = B and Q = id.
The non-triviality of the moduli space of toric Kato varieties with prescribed toric modification is related to the existence of (possibly non-equivariant) invariant families of subvarieties (or more generally foliations) for the associated Kato germs, which in turn corresponds to the existence of families of compact subvarieties (or foliations) on the toric Kato variety.
A direct computation shows that the matrices Q satisfying the conditions of Corollary 9.8 for A = B = A a are of the form Q = A k for some k ∈ N. Then condition (9.3) is satisfied if and only if there exists t ∈ R so that Let X = X(π, σ) be a toric Kato data whose associated Kato germ takes the form F = F λ,A .In this example, 1 ∈ Spec(A), and Im(A − id) is generated by e 1 + e 2 + e 3 and e 1 , hence it coincides with {v ∈ R3 | v 2 − v 3 = 0}.In fact, an eigenvector for 1 is given by (0, 1, −1), and can be also interpreted by the fact that z 2 /z 3 = const defines an invariant family of surfaces for F .These families induce a regular foliation on X, whose geometrical properties depend on the value of λ 2 /λ 3 =: λ.
Let now X = X(π, σ ) be another toric Kato data with associated germ G = F µ,A .The condition (9.3) to have X and isomorphic, when Q = id, gives Hence X and X are isomorphic if λ = µ := µ 2 /µ 3 .
Notice that the matrix A commutes with the permutation matrix If the fan associated to π is invariant by the action of Q (see for example Figure 3), then Φ = χz Q induces an isomorphism between X and X , as long as 9.3.Deformations of Kato manifolds of hyperbolic type Theorem 9.12.-Let X = X(π, σ) be a toric Kato manifold of non-parabolic type, with toric divisor D T .There exists a smooth versal family of deformations of the pair p : (X , D) → B, where B is a centered ball in Moreover, for each u ∈ B, (X u , D ∩ X u ) is the toric Kato manifold X(π, (exp u) • σ) together with its toric divisor.In particular, the family p is versal for each of its fibers. Proof.
-The existence of a smooth versal family of deformations of the pair follows simply from the fact that H 2 (X, Θ X (− log D T )) = 0 and [KNS58], but we can in fact construct such a family explicitly.Let A be a Kato matrix of X.Note first that, since the family (X , D) must come from a trivial family p and a deformation of the action of Γ.Indeed, let γ denote the positive generator of the deck group of X, and consider an action of Γ := γ on X by where exp : Since γ commutes with exp v for any v ∈ ker(A − id), X ⊂ X(Σ A , N ) is invariant under the action of exp(ker(A − id)), so γ is well-defined.Moreover, we take the ball B ⊂ ker(A − id) small enough so that σ(exp u • B) ⊂ B for each u ∈ B. It follows that for each u ∈ B, γ| X×{u} = γ(exp u•) = exp uγ ∈ Aut( X).

Hermitian geometry of Kato manifolds
Kato manifolds cannot admit Kähler metrics since their fundamental group is isomorphic to Z.However, a large class of them carry locally conformally Kähler metrics (lcK), as shown in [IOP21, Th. 2.2].In this section, we give a characterization for the existence of lcK metrics on Kato manifolds.Furthermore, we investigate the possibility of endowing a Kato manifold with other special Hermitian metrics, such as balanced, pluriclosed, Hermitian symplectic or strongly Gauduchon.
Recall the following definition: Definition 10.1.-A locally conformally Kähler metric (lcK) on a complex manifold X is a Hermitian metric Ω satisfying dΩ = θ ∧ Ω for a closed one-form θ.If θ is not exact, Ω is called strictly lcK.
This definition is equivalent to any of the following formulations: J.É.P. -M., 2022, tome 9 (1) There exists a covering with open sets (U i ) i of X and smooth functions f i on every U i such that e −fi Ω is Kähler.
For our characterization purpose, we start by giving a general principle used to modify Kähler metrics on modifications of balls, already used in [Bru11] and [IOP21], which will be needed.
If X is Kähler, then Proposition 2.4 implies that B {σ(0)} is also Kähler, so by Miyaoka's extension theorem [Miy74, Prop.A], B admits a Kähler metric ω.Now putting ρ := log(1 + z 2 ) in Lemma 10.2, we find that C n admits a Kähler metric ω which glues to λ•π * ω F S on CP n B for some λ > 0, where ω F S is the standard Fubini-Study metric on CP n .Thus we have shown that CP n is Kähler.Since on the other hand CP n is a modification of CP n , it is Moishezon, and therefore CP n is projective by Moishezon's theorem [Moȋ67,Chap. I,Th. 11].This concludes the proof.
Remark 10.4.-We note here that a non-Hopf toric Kato manifold X admitting an lcK metric can in no way be a toric lcK manifold in the sense of [Ist19,Def. 4.7], because in this case it would admit a Vaisman metric by [Ist19, Th.A], which is impossible by [IOP21, Prop.2.6].At the same time, the compact torus T acts effectively and holomorphically on the universal cover X, and any lcK metric on X gives rise to a Kähler metric ω on X, which can be made T-invariant by averaging.Since X is moreover simply connected, it follows that ( X, ω, T) is a toric Kähler manifold in the classical sense.In this manner, lcK toric Kato manifolds give interesting generalizations of the class of toric lcK manifolds.
Next, we recall the definitions of the other classes of Hermitian metrics whose existence on Kato manifolds we discuss.
There are certain inclusion relations between the classes of manifolds which admit one of the above metrics.For instance, it is easy to see that any balanced metric is strongly Gauduchon.Similarly, it is easy to see that any Hermitian symplectic metric is pluriclosed.Finally, [YZZ19, Lem.1] shows that any manifold admitting a Hermitian symplectic metric also admits a strongly Gauduchon one.
We have the following non-existence results: Theorem 10.6.-A Kato manifold X admits no strongly Gauduchon metric, and in particular no balanced or Hermitian symplectic metric.
Proof.-Let X π −→ D be the complex analytic deformation of X given by [Kat78, Th. 1], so that π −1 (0) = X and X t := π −1 (t) is a proper modification at a finite number of points of a Hopf manifold for t = 0. Assume that X admits a strongly Gauduchon metric.Then by [Pop14,Th. 3.1], stating that the strongly Gauduchon property is open with respect to holomorphic deformations, X t also admits a strongly Gauduchon metric if t is close to 0. However, according to [Pop13b, Th. 2.2], if µ : Y → Y is a proper modification of complex manifolds and Y carries a strongly Gauduchon metric, then so does Y .Consequently, this would imply that Hopf manifolds admit strongly Gauduchon metrics, which is false by [Pop14,§2].Therefore, X cannot be endowed with strongly Gauduchon metrics, nor with balanced or Hermitian symplectic metrics [YZZ19, Lem.1].
Remark 10.7.-The above result is related to the open question [ST10, Quest.1.7], asking whether there exist any non-Kähler Hermitian symplectic manifolds in dimension 3. We answer it negatively for the class of Kato manifolds.
Theorem 10.8.-If X is a Kato manifold of dimension n 3 satisfying H 1,2 ∂ (X) = 0, then it cannot be endowed with a pluriclosed metric.In particular, if X is a toric Kato manifold of hyperbolic type, then it does not admit a pluriclosed metric unless it is a surface.
Proof.-Suppose that X carries a pluriclosed metric.Let X π −→ D ⊂ C be the same deformation as above.Then by [Cav20, Th. 8.5 (arXiv version)], for any t ∈ D close to 0, π −1 (t) also admits a pluriclosed metric.We recall that for each t = 0, π −1 (t) is a modification at a finite number of points of a Hopf manifold M t .According to the analogue of Miyaoka's theorem for pluriclosed metrics [FT09, Th. 4.3], a complex manifold M without a point admits a pluriclosed metric if and only if M admits a pluriclosed metric.Therefore, the Hopf manifolds M t also admit pluriclosed metrics.However, by [Cav20, Ex. 5.17, Th. 5.16 (arXiv version)], this can only happen if n = 2. Finally, in the toric case the conclusion follows by Theorem 8.8.
Equivalently, define C n = X( Σ, N ), where the fan Σ has the rays of Σ to which we add ν j = f 0 + f j + n k=4 f k , j = 1, 2, 3, and is generated by the n-dimensional cones: It is clear that the strict transform of CP 3 under π is isomorphic to CP , for instance one corresponding to the cone τ 10 , in order to obtain a toric Kato manifold X = X(π, σ).Clearly CP 3 ⊂ X, therefore X is not Kähler and so X admits no lcK metric.

Figure 1 .
Figure 1.Toric Kato data with π not dominating the blow-up of the origin.
on which the group Γ A := {A m | m ∈ Z} acts naturally.Let X(Σ A , N ) be the toric variety associated to the fan Σ A , let D T := ν∈Σ (1) A orb ν be the associated toric divisor and let D + := ν∈Σ (1) A ν ≺τ P (A) orb(ν), where we recall that τ P (A) = e k | k ∈ P (A) .Define the open set
(A)| = 1, toric Kato surfaces are precisely the properly blown-up parabolic Inoue surfaces, and for fixed A, they are parametrized by D * .Finally, the case |P (A)| = 2 gives finite quotients of diagonal Hopf surfaces.In higher dimension n, it is still true that the case |P (A)| = n gives finite quotients of diagonal Hopf manifolds, while the dichotomy persists between the cases |P (A)| = n − 1 and |P (A)| < n − 1.For this reason, we propose the following definition: Definition 5.1.-We call a toric Kato manifold Fix k ∈ P (A), let C {k} c = orb e k ⊂ C n and let π k := π| C {k} c : C {k} c → C {k} c .Then the proof of 3.8 shows that the toric chart φ A of C n induces a toric chart φ A {k} c of C {k} c .Let Σ k be the fan of C {k} c and set and for each k ∈ P (A), L k has negative Chern class determined by A.

F:=
, which is a proper modification of 0 m .Notice also that for any m ∈ Z 1 , the collapsing maps p 0 m 0 W s (F ) correspond to the toric modifications π m given by Lemma 3.3.Remark 9.3.-The natural diagrams involving collapsing maps commute, and in particular, for any m m m , one has p m m = p m m • p m m

Figure 3 .
Figure 3. Fan associated to the toric modification π, invariant by the permutation Q of 9.11.
∂Bt ρ, v(t) = max ∂Bt ψ are strictly increasing.It follows that for any 0 < s < 1, r := u(s) ≡ ρ| ∂Bs < R := u(1) ≡ ρ| ∂B , m := min ∂Bs ψ v(s) < M := v(1).Thus, for any α > 1, we can take λ := α M − m R − r > 0, c ∈ (M − λR, m − λr) = ∅, so that λρ + c > ψ on ∂B and λρ + c < ψ on ∂B s .Therefore, if ψ is the function on C n defined as the regularized maximum between ψ and λρ + c on B B s , equal to ψ on B s and equal to λρ + c on C n B, then the metric π * (dd c ψ) on C n π −1 (0) glues to ω to define the desired metric ω on C n .Now we are ready to characterize the existence of lcK metrics.For this, let us note that any proper modification π : B → B induces naturally proper modifications π : C n → C n and π : CP n → CP n , where CP n is the compactification of C n with a hyperplane at infinity.Theorem 10.3.-Let (π : B → B, σ : B → B) be a Kato data and let X = X(π, σ) be the corresponding Kato manifold.The following are equivalent: J.É.P. -M., 2022, tome 9 3, so π = π • π : C n → C n has the desired properties.Now it suffices to take any toric chart σ with σ(B) ⊂ ( B) and σ(0) / ∈ CP 3 This implies in particular that the group F acts freely and properly on W * (F ) and that π induces the desired isomorphism X(π, σ) D ∼ = W * (F )/ F .Proposition 2.4.-Let (π : B → B, σ : B → B) be a Kato data with germ F = π • σ and let X = X(π, σ) be the corresponding Kato manifold.Then there exists a holomorphic open embedding φ which holds if and only if 2 − m 2 ∈ Z, i.e., if and only if λ 2 = µ 2 .