Dolbeault cohomology of complex manifolds with torus action

Roman Krutowski; Taras Panov

REVIEW 2 major objections 5 minor 26 references

Dolbeault cohomology of complex manifolds with torus action

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Pith's one-line read Complex moment-angle manifolds with maximal torus action have diagonal basic Dolbeault cohomology.

desk verdict A genuinely useful paper that likely proves the right theorem, but the central reduction to the polytopal case has a real proof gap that needs fixing before the main theorem is fully established. read the letter →

arxiv 1908.06356 v4 pith:ZHFA77IX submitted 2019-08-18 math.DG math.AGmath.ATmath.CV

Roman Krutowski , Taras Panov This is my paper · ORCID

classification math.DGmath.AGmath.ATmath.CV MSC 32J1832L0532M0532Q5537F7557R1914M25

keywords basicDolbeaultcohomologycanonicalholomorphicfoliationmoment-anglemanifoldsmaximaltorusactionFujikiHodgedecompositionstellarsubdivisionstransverselyKählerfoliations

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The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Complex moment-angle manifolds are non-Kähler, but their canonical holomorphic foliation still has well-behaved cohomology. The paper proves that the basic Dolbeault cohomology of this foliation has a Hodge decomposition, $H^r = \bigoplus_{p+q=r} H^{p,q}$, and that all non-zero classes sit on the diagonal $p=q$. The proof introduces the notion of a Fujiki foliation and reduces the general case to the polytopal case by a foliated analogue of toric blow-up, where the foliation is transversely Kähler and the result was known. The same argument carries over to every compact complex manifold with a maximal holomorphic torus action, including LVM- and LVMB-manifolds. The payoff is a complete algebraic description of the basic Dolbeault cohomology ring and a DGA model for the ordinary Dolbeault cohomology of these spaces.

What carries the argument

The machinery has three parts. First, a Fujiki foliation is defined as a holomorphic foliation that is the image, under a surjective holomorphic foliated map, of a transversely Kähler foliation; an injectivity lemma shows that such maps induce injections of basic Dolbeault cohomology, and hence a Hodge decomposition. Second, generalized toric blow-up is defined concretely: a stellar subdivision of the fan at a cone $\tau$ gives a map $f_\tau:(U(K_\tau),\mathcal F_{\Sigma_\tau})\to(U(K),\mathcal F_\Sigma)$ that sends leaves to leaves. Third, Theorem 4.9 supplies the combinatorial bridge: any complete simplicial fan, even with non-rational generators, can be made polytopal by stellar subdivisions whose new rays are integer combinations of the old generators within each cone. The composition is the foliated version of the toric Chow lemma, a surjection from a transversely Kähler moment-angle manifold to the original one.

What would settle it

Construct a complete simplicial fan with non-rational rays, form the polytopal fan $\Sigma_h$ from the hyperplanes through its codimension-one cones, and check whether every perturbation of the new rays to nearby rays rational in each cone's lattice still gives a polytopal fan; the first fan for which no such perturbation and no sequence of stellar subdivisions produces a polytopal fan is a direct counterexample to Theorem 4.9 and breaks Theorem 4.10.

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Extended reading notes

Core claim

The central result is Theorem 4.11: for any complex moment-angle manifold $Z_K$ with its canonical holomorphic foliation $\mathcal F_h$, the pair $(Z_K,\mathcal F_h)$ is a Fujiki foliation, so there is a Hodge decomposition $$H^r_{\mathcal F_h}(Z_K;\mathbb C)=\bigoplus_{p+q=r}$H^{{p,q}}$_{\mathcal F_h}(Z_K)$$ and $H^{p,q}_{\mathcal F_h}(Z_K)=0$ whenever $p\neq q$. Consequently the basic Dolbeault cohomology algebra is $$$H^{{*,*}}$_{\mathcal F_h}(Z_K)\cong \mathbb C[v_1,\dots,v_m]/(I_K+J)$$ with every generator $v_i$ of type $(1,1)$; here $I_K$ is the Stanley–Reisner ideal of the simplicial complex $K$ and $J$ is the ideal of linear relations among the fan generators. Theorem 5.1 extends this description to all compact complex manifolds with maximal holomorphic torus action, a class that includes LVM- and LVMB-manifolds, by a transverse equivalence with moment-angle manifolds. Theorem 6.1 then feeds the basic ring into a DGA model for ordinary Dolbeault cohomology.

Load-bearing premise

Everything rests on Theorem 4.9: every complete simplicial fan, even one with non-rational generators, can be made polytopal by stellar subdivisions whose new rays lie in the integer span of each cone's old generators; the proof is sketched cone-wise, citing the rational cases, and the foliated surjection needed for the Fujiki argument does not exist if this fails.

Editorial extensions

If this is right

The basic Hodge numbers of the canonical foliation vanish off the diagonal, so the basic Dolbeault cohomology of $Z_K$ is completely determined by its diagonal entries.
For every compact complex manifold with maximal holomorphic torus action, the basic Dolbeault ring is $\mathbb C[v_1,\dots,v_m]/(I_K+J)$ with all generators of type $(1,1)$.
The ordinary Dolbeault cohomology of a complex moment-angle manifold is quasi-isomorphic to a DGA built from the basic Dolbeault ring and an $R$-invariant subspace $W\subset\Omega^1(Z_K)$ of dimension $(m-n)/2$.
A foliated analogue of the toric Chow lemma holds: every complex moment-angle manifold is the holomorphic foliated image of a transversely Kähler one.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The foliated blow-up construction is not tied to moment-angle geometry: it suggests a recipe for reducing any holomorphic foliation with a torus-invariant transversal structure to a transversely Kähler one, and would then yield Hodge decompositions in other families of non-Kähler manifolds.
If the polytopal subdivision theorem extends to fans with additional symmetry, the same Fujiki-foliation argument would likely give equivariant Hodge decompositions for the basic cohomology, not just the plain one.
A numerically testable consequence is that ordinary Dolbeault cohomology depends only on the simplicial complex and the fan relations, so changing the complex structure on the same smooth moment-angle manifold cannot change the Dolbeault groups as long as the marked fan is unchanged.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper's main claim is a Hodge decomposition for the basic Dolbeault cohomology of the canonical foliation on complex moment-angle manifolds, for arbitrary (possibly non-rational) fans, together with an explicit ring presentation and a DGA model. That answers a published conjecture by Battaglia and Zaffran, and the new notion of a Fujiki foliation is a reasonable and potentially reusable tool. The polytopal case was already known, so the whole weight falls on the reduction step: Theorem 4.10 constructs a transversely Kähler cover from a polytopal subdivision of the fan. This is a good idea and the surrounding setup is careful.

The soft spot is real and it is in Lemma 4.6. The proof says we can argue cone-wise: for each maximal cone, apply the rational de Concini–Procesi result to that cone and its rational subdivision. But a stellar subdivision of a 2-dimensional cone in the boundary of one maximal cone also subdivides the adjacent maximal cone sharing that face. Applying the rational result independently to two neighbouring cones can produce incompatible subdivisions of the common facet, and the union may not be a fan. No gluing argument is provided. So Theorem 4.9 is not established as written. This is load-bearing: Theorem 4.11 for non-polytopal fans depends on it. The statement may well be true and patchable, but the proof needs to be supplied or replaced with a citation to a result that covers the non-rational case.

A smaller point: Lemma 3.2 uses the fact that a surjective holomorphic map is a locally trivial bundle on a dense open set; that is standard but should be stated. The positivity argument itself looks fine once that's in place. The paper leans on earlier work by the same authors, but those are independent published results, not circular.

If I were refereeing this, I would not desk-reject it. The main idea is sound and the answer to the conjecture is significant. But I would send it back with a request for a complete proof of Lemma 4.6 or a precise reference. As it stands, the reduction step is a sketch, and a sketch at the hinge of the paper is not enough.

Referee Report

2 major / 5 minor

Summary. The paper studies the basic Dolbeault cohomology of the canonical holomorphic foliation on complex moment-angle manifolds and, more generally, on complex manifolds with a maximal torus action. The main results are: (1) a Hodge decomposition for the basic Dolbeault cohomology of the canonical foliation, with all nonzero groups of type (p,p); (2) an explicit description of the basic Dolbeault algebra as C[v_1,...,v_m]/(I_K+J) with all generators of type (1,1); (3) an extension of this description to all complex manifolds with a maximal torus action; and (4) a DGA model for the ordinary Dolbeault cohomology of moment-angle manifolds. The central mechanism is the introduction of "Fujiki foliations" and a reduction, via generalized toric blow-ups and stellar subdivisions of fans, to the transversely Kähler (polytopal) case, where the decomposition is already known by work of El Kacimi-Alaoui and Ishida.

Significance. If the main results hold, the paper resolves a question of Battaglia and Zaffran about the Hodge numbers of the canonical foliation and gives a complete and elegant algebraic description of its basic Dolbeault cohomology. The notion of Fujiki foliation, together with the foliated toric blow-up construction, is a natural and potentially reusable tool for studying foliated cohomology outside the transversely Kähler setting. The extension to all maximal torus actions and the DGA model for ordinary Dolbeault cohomology are substantial additional contributions. The paper is carefully written and builds appropriately on prior work, including the authors' earlier computation of basic de Rham cohomology and Ishida's classification of maximal torus actions.

major comments (2)

[Section 4.3, Lemma 4.6 and Theorem 4.9] The proof of Lemma 4.6 is incomplete: it reduces to the rational case cone-wise, asserting that for each maximal cone one can apply the de Concini–Procesi result to the rational fan formed by the cone and its rational subdivision. However, stellar subdivisions performed independently in adjacent maximal cones need not agree on a shared facet; a 2-dimensional cone lying in the common facet would be subdivided from the two sides in potentially incompatible ways unless the sequences of stellar subdivisions are chosen globally and compatibly. The cited rational result applies to a globally rational fan, not to an arbitrary collection of independent maximal cones. Because Theorem 4.10 constructs the required foliated surjection onto a transversely Kähler moment-angle manifold only from the polytopal fan supplied by Theorem 4.9, this gap is load-bearing for the Hodge decomposition and the diagonal vanishing in Theorem 4.11. The authors should supply a global proof of the non-rational stellar-subdivision statement, or cite a theorem that covers it.
[Section 3, Lemma 3.2] The proof of injectivity of f* uses a local trivialization U' ≅ U × W and then asserts that the restriction of the transverse Kähler form ω to each slice {x}×W makes the restricted foliation transversely Kähler. Since the foliation F' is not assumed to be tangent to the slices, the restriction of a foliation to a submanifold requires justification; one must argue that the intersection of the leaves of F' with the fiber is a foliation on the fiber to which El Kacimi–Alaoui's theory applies. A similar issue arises in the claim that f^*(σ) is positive on U×{y}. This is likely standard in the Riemannian foliation setting, but it should be stated explicitly and proved or referenced, as the argument underpins the Hodge decomposition for Fujiki foliations.

minor comments (5)

[Abstract and Introduction] There are several typographical errors: "Dolbealut" in the abstract, "to rus" in the abstract, and inconsistent spacing in "L VM-" in the introduction. These should be corrected.
[Section 4.3, proof of Theorem 4.9] The statement that "polytopality is an open condition for simplicial fans" after replacing a ray by a rational perturbation would benefit from a reference or a short justification, because moving a ray changes the fan combinatorially in a neighborhood of that ray.
[Section 4.2, Construction 4.2] The notation α_i ∈ N should specify that N denotes the positive integers, and the fact that the resulting stellar subdivision depends on the chosen α_i (suppressed in the notation) should be stated explicitly for clarity.
[Sections 4.4 and 5] The ideal J in Theorem 4.12 is defined using u ∈ (t/r)^*, while in Theorem 5.1 it is u ∈ (g/r)^*; a brief reminder of the identification of these dual spaces would help the reader compare the two statements.
[Section 6, Theorem 6.1] The differential d_{Z_K} on the model is described only through its action on W^{1,0} and W^{0,1}; a more explicit description of the quasi-isomorphism, or at least a sentence explaining how the differential is determined, would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the new theorem reduces to independently established transverse Kähler and de Rham results, not to its own conclusion.

full rationale

The derivation chain for the main Hodge-decomposition theorem is not circular. Theorem 4.11 first treats the polytopal case, citing external or prior results for transverse Kählerity ([24, Prop. 4.4]), the Hodge decomposition for transversely Kähler foliations ([10, Thm. 3.4.6]), and the diagonal vanishing in that case ([15, Thm. 8.1]). For the general case it constructs, via stellar subdivisions and the generalized toric blow-down (Constructions 4.2 and 4.3), a foliated surjection from a transversely Kähler moment-angle manifold (Theorem 4.10), then proves injectivity of the induced basic Dolbeault map in Lemma 3.2 using the standard positivity/Serre-duality argument adapted from [11]. None of these steps assumes the Hodge decomposition or the diagonal vanishing it derives. The same-author citations [17] and [24] are prior independent results: [17] computes the basic de Rham cohomology ring, not the Dolbeault bigrading, and [24, Prop. 4.4] is a parameter-free transverse-Kählerity statement. Theorem 4.12 combines the de Rham description from [17] with the newly proved Hodge decomposition, so the algebra presentation is not being used to force the Hodge statement. Possible concerns about the cone-wise proof of Lemma 4.6 are correctness or completeness concerns about the subdivision construction, not circularity: the proof does not reduce the theorem to its own conclusion. Therefore no step identified meets the standard of exhibiting a claim that is equivalent by construction to its input.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on classification theorems and prior cohomology computations, primarily from Ishida and from the authors' own earlier work. The only newly introduced mathematical object, Fujiki foliations, is a definition rather than an unproved postulate. The reduction to the polytopal case is the main load-bearing step and depends on the non-rational polytopal subdivision theorem, whose proof is partially cited rather than fully derived.

assumptions (6)

domain assumption Every complete simplicial fan can be subdivided to a polytopal fan by a sequence of stellar subdivisions with new rays chosen rationally relative to the cones they subdivide.
Invoked in Theorem 4.10; the proof in Section 4.3 adapts rational results to non-rational fans and is the key reduction step, so if it fails the main result fails.
domain assumption Classification of complex manifolds with maximal torus action as quotients U(K)/H (Ishida [14]).
Used throughout Section 5 to pass from moment-angle manifolds to the full class of manifolds with maximal torus action.
domain assumption Hodge decomposition for basic Dolbeault cohomology of transversely Kähler foliations (El Kacimi-Alaoui [10]).
The base case for the Fujiki foliation reduction; invoked in Theorem 4.11.
domain assumption For polytopal fans, the basic Dolbeault cohomology groups vanish off the diagonal (Ishida [15, Theorem 8.1]).
Used in Theorem 4.11 to conclude H^{p,q}=0 for p different from q once the Hodge decomposition is known.
domain assumption The basic de Rham cohomology ring of the canonical foliation has the presentation C[v_1,...,v_m]/(I_K+J) (Ishida, Krutowski, Panov [17, Theorem 3.4]).
Prior work by the same authors; combined with the diagonal vanishing to obtain the Dolbeault ring in Theorem 4.12.
standard math A surjective holomorphic map between compact complex manifolds is a proper submersion over a dense open subset, hence a locally trivial fibre bundle over that subset.
Used in the proof of Lemma 3.2 to justify the generic-fibre bundle description.

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Pith. "Pith review of Dolbeault cohomology of complex manifolds with torus action." pith.science (2026). https://pith.science/paper/ZHFA77IX

@misc{pith2026190806356,
  author       = {Pith},
  title        = {Pith review of: Dolbeault cohomology of complex manifolds with torus action},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZHFA77IX}},
  note         = {Machine review of arXiv:1908.06356}
}

read the original abstract

We describe the basic Dolbealut cohomology algebra of the canonical foliation on a class of complex manifolds with a torus symmetry group. This class includes complex moment-angle manifolds, LVM- and LVMB-manifolds and, in most generality, complex manifolds with a maximal holomorphic torus action. We also provide a dga model for the ordinary Dolbeault cohomology algebra. The Hodge decomposition for the basic Dolbeault cohomology is proved by reducing to the transversely Kaehler (equivalently, polytopal) case using a foliated analogue of toric blow-up.

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Works this paper leans on

26 extracted references · 26 canonical work pages

[1]

A note on the factorization theorem of toric birational maps after Morelli and its toroi dal extension

Abramovich, Dan; Matsuki, Kenji; Rashid, Suliman. A note on the factorization theorem of toric birational maps after Morelli and its toroi dal extension. Tohoku Math. J. (2) 51 (1999), no. 4, 489–537

work page 1999
[2]

Derived subdivisions make every PL sphere polytopal

Adiprasito, Karim A.; Izmestiev, Ivan. Derived subdivisions make every PL sphere polytopal. Israel J. Math. 208 (2015), no. 1, 443–450

work page 2015
[3]

Generalized toric varieties for simple nonrational convex polytopes

Battaglia, Fiammetta; Prato, Elisa. Generalized toric varieties for simple nonrational convex polytopes. Internat. Math. Res. Notices 2001 (2001), no. 24, 1315–1337

work page 2001
[4]

Nonrational symplectic toric reduction

Battaglia, Fiammetta; Prato, Elisa. Nonrational symplectic toric reduction. J. Geom. Phys. 135 (2019), 98–105

work page 2019
[5]

Foliations modeling nonrational simplicial toric varieties

Battaglia, Fiammetta; Zaﬀran, Dan. Foliations modeling nonrational simplicial toric varieties. Int. Math. Res. Not. 2015 (2015), no. 22, 11785–11815

work page 2015
[6]

Bosio, Fr´ ed´ eric.Vari´ et´ es complexes compactes: une g´ en´ eralisation de la construction de Meersseman et L´ opez de Medrano–Verjovsky. Ann. Inst. Fourier 51 (2001), 1259– 1297

work page 2001
[7]

Toric topology

Buchstaber, Victor; Panov, Taras. Toric topology. Mathematical Surveys and Mono- graphs, 204. American Mathematical Society, Providence, R I, 2015. xiv+518 pp. DOLBEAULT COHOMOLOGY OF MANIFOLDS WITH TORUS ACTION 15

work page 2015
[8]

The geometry of toric varieties

Danilov, Vladimir. The geometry of toric varieties . Uspekhi Mat. Nauk 33 (1978), no. 2, 85–134 (Russian); Russian Math. Surveys 33 (1978), no . 2, 97–154 (English translation)

work page 1978

Show all 26 references

[9]

Complete symmetric varieties

De Concini, Corrado; Procesi Claudio. Complete symmetric varieties. II. Intersection theory. In: Algebraic Groups and Related Topics (Kyoto/Nagoya, 1983). Adv. Stud. Pure Math., 6, North-Holland, Amsterdam, 1985, pp. 481–513

1983
[10]

Op´ erateurs transversalement elliptiques sur un feuillet age riemannien et applications (French)

El Kacimi-Alaoui, Aziz. Op´ erateurs transversalement elliptiques sur un feuillet age riemannien et applications (French). Compositio Math. 73 (1990), no. 1, 57–106

1990
[11]

D´ ecomposit ion de Hodge basique pour un feuilletage riemannien (French)

El Kacimi-Alaoui, Aziz; Hector, Gilbert. D´ ecomposit ion de Hodge basique pour un feuilletage riemannien (French). Ann. Inst. Fourier (Gren oble) 36 (1986), no. 3, 207– 227

1986
[12]

Closedness of the Douady spaces of compact K¨ ahler spaces

Fujiki, Akira. Closedness of the Douady spaces of compact K¨ ahler spaces. Publ. Res. Inst. Math. Sci. 14 (1978), no. 1, 1–52

1978
[13]

Torus invariant transversely K¨ ahler foliations

Ishida, Hiroaki. Torus invariant transversely K¨ ahler foliations. Trans. Amer. Math. Soc. 369 (2017), no. 7, 5137–5155

2017
[14]

Complex manifolds with maximal torus actions

Ishida, Hiroaki. Complex manifolds with maximal torus actions. J. Reine Angew. Math. 751 (2019), 121–184

2019
[15]

Towards transverse toric geometry

Ishida, Hiroaki. Towards transverse toric geometry . Preprint (2018); arXiv:1807.10449

2018 arXiv
[16]

Transversely K¨ ahler structures on central foliations of complex manifolds

Ishida, Hiroaki; Kasuya, Hisashi. Transversely K¨ ahler structures on central foliations of complex manifolds. Ann. Mat. Pura Appl. (4) 198 (2019), no. 1, 61–81

2019
[17]

Basic cohomology of canon- ical holomorphic foliations on complex moment-angle manif olds

Ishida, Hiroaki; Krutowski, Roman; Panov, Taras. Basic cohomology of canon- ical holomorphic foliations on complex moment-angle manif olds. Preprint (2018); arXiv:1811.12038

2018 arXiv
[18]

The deﬁnition of a non-commutative toric variety

Katzarkov, Ludmil; Lupercio, Ernesto; Meersseman, La urent; Verjovsky, Alberto. The deﬁnition of a non-commutative toric variety. In: Algebraic Topology: Applica- tions and New Directions . Contemp. Math., 620, Amer. Math. Soc., Providence, RI, 2014, pp. 223–250

2014
[19]

Basic Kirwan injectivity and its applications

Lin, Yi; Yang, Xiangdong. Basic Kirwan injectivity and its applications . Preprint (2019); arXiv:1902.06187

2019 arXiv
[20]

A new geometric construction of compact complex manifolds in any dimension

Meersseman, Laurent. A new geometric construction of compact complex manifolds in any dimension. Math. Ann. 317 (2000), 79–115

2000
[21]

Riemannian foliations

Molino, Pierre. Riemannian foliations. Progress in Mathematics, 73. Birkh¨ auser Boston, Inc., Boston, MA, 1988. xii+339 pp

1988
[22]

Convex Bodies and Algebraic Geometry

Oda, Tadao. Convex Bodies and Algebraic Geometry. An Introduction to th e Theory of Toric Varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Resu lts in Mathematics and Related Areas (3)], 15. Springer, New Yor k, 1988

1988
[23]

Complex-analytic structures on moment-angle man- ifolds

Panov, Taras; Ustinovsky, Yuri. Complex-analytic structures on moment-angle man- ifolds. Mosc. Math. J. 12 (2012), no. 1, 149–172

2012
[24]

Complex geometry of moment- angle manifolds

Panov, Taras; Ustinovskiy, Yury; Verbitsky, Misha. Complex geometry of moment- angle manifolds. Math. Z. 284 (2016), no. 1–2, 309–333

2016
[25]

Presymplectic convexity and (ir)rational polytopes

Ratiu, Tudor; Zung, Nguyen Tien. Presymplectic convexity and (ir)rational polytopes. J. Symplectic Geom. 17 (2019), no. 5, 1479–1511

2019
[26]

Hodge theory and complex algebraic geometry

Voisin, Claire. Hodge theory and complex algebraic geometry. I. Cambridge Studies in Advanced Mathematics, 76. Cambridge Univ. Press, Cambri dge, 2007. x+322 pp. F aculty of Mathematics, National Research University High er School of Economics, Moscow, Russia E-mail address : ...

2007

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

A note on the factorization theorem of toric birational maps after Morelli and its toroi dal extension

Abramovich, Dan; Matsuki, Kenji; Rashid, Suliman. A note on the factorization theorem of toric birational maps after Morelli and its toroi dal extension. Tohoku Math. J. (2) 51 (1999), no. 4, 489–537

work page 1999

[2] [2]

Derived subdivisions make every PL sphere polytopal

Adiprasito, Karim A.; Izmestiev, Ivan. Derived subdivisions make every PL sphere polytopal. Israel J. Math. 208 (2015), no. 1, 443–450

work page 2015

[3] [3]

Generalized toric varieties for simple nonrational convex polytopes

Battaglia, Fiammetta; Prato, Elisa. Generalized toric varieties for simple nonrational convex polytopes. Internat. Math. Res. Notices 2001 (2001), no. 24, 1315–1337

work page 2001

[4] [4]

Nonrational symplectic toric reduction

Battaglia, Fiammetta; Prato, Elisa. Nonrational symplectic toric reduction. J. Geom. Phys. 135 (2019), 98–105

work page 2019

[5] [5]

Foliations modeling nonrational simplicial toric varieties

Battaglia, Fiammetta; Zaﬀran, Dan. Foliations modeling nonrational simplicial toric varieties. Int. Math. Res. Not. 2015 (2015), no. 22, 11785–11815

work page 2015

[6] [6]

Bosio, Fr´ ed´ eric.Vari´ et´ es complexes compactes: une g´ en´ eralisation de la construction de Meersseman et L´ opez de Medrano–Verjovsky. Ann. Inst. Fourier 51 (2001), 1259– 1297

work page 2001

[7] [7]

Toric topology

Buchstaber, Victor; Panov, Taras. Toric topology. Mathematical Surveys and Mono- graphs, 204. American Mathematical Society, Providence, R I, 2015. xiv+518 pp. DOLBEAULT COHOMOLOGY OF MANIFOLDS WITH TORUS ACTION 15

work page 2015

[8] [8]

The geometry of toric varieties

Danilov, Vladimir. The geometry of toric varieties . Uspekhi Mat. Nauk 33 (1978), no. 2, 85–134 (Russian); Russian Math. Surveys 33 (1978), no . 2, 97–154 (English translation)

work page 1978