Cohomogeneity one Kaehler and Kaehler-Einstein manifolds with one singular orbit, II

Dmitri Alekseevsky; Fabio Zuddas

arxiv: 1906.10633 · v1 · pith:V2LMHEO6new · submitted 2019-06-25 · 🧮 math.DG

Cohomogeneity one Kaehler and Kaehler-Einstein manifolds with one singular orbit, II

Dmitri Alekseevsky , Fabio Zuddas This is my paper

Pith reviewed 2026-05-25 15:49 UTC · model grok-4.3

classification 🧮 math.DG

keywords cohomogeneity one manifoldsKähler-Einstein metricsflag manifoldsKoszul numbersclassical Lie groupsinvariant Kähler structuressingular orbits

0 comments

The pith

For SU_n, Sp_n and Spin_n, existence of invariant Kähler-Einstein metrics on standard cohomogeneity one manifolds reduces to arithmetic properties of the Koszul numbers of the base flag manifold.

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

This paper takes the geometric conditions for invariant Kähler-Einstein metrics on standard G-manifolds with one singular orbit, derived in part I as the existence of a suitable interval in the T-Weyl chamber, and specializes them to the classical groups. For these groups the interval condition is shown to be equivalent to arithmetic properties of the Koszul numbers attached to the flag manifold S_0 = G/H. When the arithmetic test passes, the metric itself is recovered by inverting a single explicit function of one variable. A reader cares because the reformulation turns an analytic search over Weyl chambers into a finite, checkable computation on integers associated with the homogeneous space.

Core claim

The necessary and sufficient conditions for an invariant Kähler-Einstein metric on the standard cohomogeneity one manifold M_φ reduce, for G equal to SU_n, Sp_n or Spin_n, to easily checked arithmetic properties of the Koszul numbers of the flag manifold S_0 = G/H; when these properties hold, the explicit construction of the metric reduces to the calculation of the inverse function to a given function of one variable.

What carries the argument

Reformulation of the linear interval condition in the T-Weyl chamber of the flag manifold F = G ×_H PV as arithmetic properties of the Koszul numbers of S_0 = G/H.

If this is right

When the arithmetic conditions on the Koszul numbers hold, an invariant Kähler-Einstein metric exists on the manifold M_φ.
The metric can be constructed explicitly by inverting a given one-variable function.
The existence question becomes decidable by direct inspection of a finite list of integers attached to the flag manifold.
The same arithmetic test supplies both the existence criterion and the starting data for the explicit construction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The reduction may permit exhaustive enumeration of such metrics over all flag manifolds of the classical groups by checking Koszul-number tables.
If analogous arithmetic simplifications exist for exceptional groups, the same method could decide Kähler-Einstein metrics on a wider class of cohomogeneity-one spaces.
The one-variable inversion step could be implemented numerically to produce sample metrics whose curvature properties can then be verified directly.

Load-bearing premise

The geometric interval conditions from part I are exactly equivalent to the arithmetic properties of the Koszul numbers for these classical groups, with no further hidden constraints from their root systems.

What would settle it

A flag manifold of SU_n (or Sp_n or Spin_n) for which the Koszul numbers satisfy the stated arithmetic relations yet no interval in the T-Weyl chamber meets the linear condition required by part I, or the converse.

read the original abstract

F. Podest\`a and A. Spiro introduced a class of $G$-manifolds $M$ with a cohomogeneity one action of a compact semisimple Lie group $G$ which admit an invariant Kaehler structure $(g,J)$ (``standard $G$-manifolds") and studied invariant Kaehler and Kaehler-Einstein metrics on $M$. In the first part of this paper, we gave a combinatoric description of the standard non compact $G$-manifolds as the total space $M_{\varphi}$ of the homogeneous vector bundle $M = G\times_H V \to S_0 =G/H$ over a flag manifold $S_0$ and we gave necessary and sufficient conditions for the existence of an invariant Kaehler-Einstein metric $g$ on such manifolds $M$ in terms of the existence of an interval in the $T$-Weyl chamber of the flag manifold $F = G \times _H PV$ which satisfies some linear condition. In this paper, we consider standard cohomogeneity one manifolds of a classical simply connected Lie group $G = SU_n, Sp_n. Spin_n$ and reformulate these necessary and sufficient conditions in terms of easily checked arithmetic properties of the Koszul numbers associated with the flag manifold $S_0 = G/H$. If this conditions is fulfilled, the explicit construction of the Kaehler-Einstein metric reduces to the calculation of the inverse function to a given function of one variable.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper turns the geometric interval conditions from part I into explicit arithmetic checks on Koszul numbers for SU_n, Sp_n and Spin_n, which simplifies verification and metric construction to inverting one function.

read the letter

The main thing here is the explicit translation. The authors take the necessary-and-sufficient interval conditions on the T-Weyl chamber that were stated in part I and convert them, via direct root-system calculations, into arithmetic properties of the Koszul numbers attached to the flag manifold S0 = G/H. For the three classical families this produces concrete, checkable criteria; once they hold, the Kähler-Einstein metric is obtained by inverting a single one-variable function. That computational shortcut is the concrete advance over the earlier geometric formulation. It is useful inside the narrow subfield because it replaces repeated work with Weyl chambers by arithmetic tests that can be done by hand or with standard Lie-algebra software. The calculations appear to be carried out case-by-case for the root systems of SU_n, Sp_n and Spin_n, so the equivalence rests on those explicit verifications rather than on a new general argument. The paper does not claim to produce new examples outside the standard cohomogeneity-one setting or to settle existence questions for non-classical groups. Its soundness therefore inherits directly from part I: if the original interval conditions are correct and complete, the arithmetic versions are too; the stress-test note finds no visible internal obstruction in the translation step. The limitation is simply scope and dependence: the work is incremental refinement rather than an independent existence theorem, and it applies only to the listed classical groups. Readers already working on invariant Kähler metrics on cohomogeneity-one manifolds will find the arithmetic test practical for checking specific cases. Others will not need it. I would send the paper to referees who know the part-I framework; the calculations are concrete enough to be checked and the reduction to a one-variable inverse is a clear practical gain.

Referee Report

0 major / 3 minor

Summary. The paper is the second installment in a series on cohomogeneity one Kähler and Kähler-Einstein manifolds with one singular orbit. Building on part I, it considers standard G-manifolds M_φ = G ×_H V for classical simply connected groups G = SU_n, Sp_n, Spin_n and reformulates the necessary and sufficient geometric interval conditions (in the T-Weyl chamber of the associated flag manifold F) for the existence of an invariant Kähler-Einstein metric as arithmetic properties of the Koszul numbers of the flag manifold S_0 = G/H. When these arithmetic conditions hold, explicit metric construction reduces to inverting a single one-variable function.

Significance. If the claimed equivalence between the part-I geometric conditions and the stated arithmetic properties of Koszul numbers holds without hidden representation-theoretic obstructions, the result supplies a readily verifiable arithmetic criterion that simplifies the application of the general theory to the classical groups and reduces metric construction to a one-variable inversion. This is a practical advance for explicit constructions in the setting of invariant Kähler-Einstein metrics on cohomogeneity-one spaces.

minor comments (3)

The abstract and introduction refer to 'the Koszul numbers associated with the flag manifold S_0 = G/H' without an explicit definition or reference to their standard normalization in the literature; a brief reminder of the definition (or citation to the convention used) would aid readability.
Section 3 (or the section containing the explicit root-system computations) should include a short table or list summarizing the arithmetic conditions for each classical group (SU_n, Sp_n, Spin_n) to make the translation from the part-I interval conditions immediately verifiable.
The statement that the construction 'reduces to the calculation of the inverse function to a given function of one variable' would benefit from an explicit formula or reference to the function in question, even if it is inherited from part I.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment and recommendation of minor revision. No specific major comments appear in the report, so we have no individual points requiring response or revision.

Circularity Check

0 steps flagged

No significant circularity; explicit reformulation via root-system computation

full rationale

The paper's central step is an explicit translation, for the classical groups SU_n, Sp_n and Spin_n, of the interval conditions already stated in part I into arithmetic statements on Koszul numbers. This translation is performed by direct computation with the root systems and does not reduce any claimed prediction or existence criterion to a fitted parameter, a self-definition, or a load-bearing self-citation whose validity is presupposed. The derivation chain therefore remains independent of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.0 · 5812 in / 1143 out tokens · 26513 ms · 2026-05-25T15:49:17.525873+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages · 1 internal anchor

[1]

Yugoslav Geometrical seminar, Divcibare, 10-17 October, 3-35 (1993)

Alekseevsky D.: Flag manifolds , 11. Yugoslav Geometrical seminar, Divcibare, 10-17 October, 3-35 (1993)

work page 1993
[2]

global Anal

Alekseevsky D., Zuddas F.: Cohomogeneity one Kahler and Kahler-Einstein manifolds with one singular orbit I , Ann. global Anal. Geom. 52:1, 99-128 (2017)

work page 2017
[3]

Groups (2018), pp

Alekseevsky D., Chrysikos J.: Spin structures on compact homogeneous pseudo-Riemannian manifolds , Transf. Groups (2018), pp. 1-31

work page 2018
[4]

Alekseevsky D., Spiro A.: Invariant CR structures on com- pact homogeneous manifolds, Hokk. Math. J, v. 32, no.2, 209- 276 (2003)

work page 2003
[5]

V., Perelomov, A

Alekseevsky D. V., Perelomov, A. M.: Invariant Kaehler- Einstein metrics on compact homogeneous spaces , Funct. Anal. Appl., 20 (3), 171-182 (1986)

work page 1986
[6]

Alekseevsky D., Cortes V., Hasegawa K., Kamishima Y., Ho- mogeneous locally conformally K¨ ahler and Sasaki manifold s, Int. J. Math., 26, n5, ( 2015)

work page 2015
[7]

5, 6, 957-974 (2006)

Arvanitoyeorgos A.: Geometry of ﬂag manifolds , Interna- tional Journal of Geometric Methods in Modern Physics Vol.3, Nos. 5, 6, 957-974 (2006)

work page 2006
[8]

, Ricci-ﬂat metrics on vector bun- dles over ﬂag manifolds , arXiv:1905.00412 (01/05/2019)

Achmed-Zade I., Bykov D. , Ricci-ﬂat metrics on vector bun- dles over ﬂag manifolds , arXiv:1905.00412 (01/05/2019)

work page arXiv 1905
[9]

2, (2003) 480491

Azad H., Biswas I., Quasi-potentials and Kahler Einstein metrics on ﬂag manifolds II , Journal of Algebra 269 no. 2, (2003) 480491

work page 2003
[10]

Besse A.: Einstein manifolds, Ergeb. Math. Grenzgeb. (3) 10, Springer, Berlin, 1987

work page 1987
[11]

Berard- Bergery L.: Sur des nouvelles varietes Riemanni- ennes d’Einstein, Publ. de Inst. E. Cartan, No. 6, 1-60 (1982)

work page 1982
[12]

Borel A., Hirzebruch F.: Characteristic classes and homoger- neous spaces, Amer. J . Math. 80, 458-538 (1958) 26

work page 1958
[13]

Boyer C.P., Galicki K., Sasakian geometry , Oxford Mathe- matical Monographs, Oxford University Press, Oxford, 2008

work page 2008
[14]

Van Coevering C., Calabi-Yau metrics on canonical bundles of ﬂag varieties , arXiv:1807.07256v1 (19/07/2018)

work page internal anchor Pith review Pith/arXiv arXiv 2018
[15]

Y.: K¨ ahler Einstein metrics of cohomo- geneity one and bundle construction for Einstein Hermitian metrics, Math

Dancer A., Wang M. Y.: K¨ ahler Einstein metrics of cohomo- geneity one and bundle construction for Einstein Hermitian metrics, Math. Ann. 312, 503-526 (1998)

work page 1998
[16]

Y.: The initial value problem for cohomogeneity one Einstein metrics , J

Eschenburg J.-H., Wang M. Y.: The initial value problem for cohomogeneity one Einstein metrics , J. Geom. Anal.,10, No.1, 109-137 (2000)

work page 2000
[17]

Gorbatsevich V.V., Onishchik A.L., Vinberg E.B.: Structure of Lie groups and Lie algebras , Encycl. Math. Sci., Lie groups and Lie algebras, III, Springer Verlag

work page
[18]

19, 763-786 (1982)

Huckleberry A., Snow D.: Almost homogeneous K¨ ahler man- ifolds with hypersurface orbits , Osaka J.math. 19, 763-786 (1982)

work page 1982
[19]

17th Int

Koiso N., Sakane Y.: Non-homogeneous K¨ ahler-Einstein met- rics on compact complex manifolds , Curvature and topology of Riemannian manifolds, Proc. 17th Int. Taniguchi Symp., Katata/Jap. 1985, Lect. Notes Math. 1201, 165-179 (1986)

work page 1985
[20]

Koiso N., Sakane Y.: Non homogeneous K¨ ahler Einstein met- rics on compact complex manifolds II , Osaka J. Math. 25, 933-959 (1988)

work page 1988
[21]

2, 213- 225 (1987)

Page D.N., Pope C.N.: Inhomogeneous Einstein metrics on complex line bundles , Classical Quantum Gravity 4 no. 2, 213- 225 (1987)

work page 1987
[22]

Podest` a F., Spiro A.: Kaehler manifolds with large isometry group, Osaka J. Math. Volume 36, Number 4, 805-833 (1999)

work page 1999
[23]

Sakane Y.: Examples of compact Einstein-K¨ ahler manifolds with positive Ricci tensor , Osaka J. Math. 23, 585-616 (1986)

work page 1986
[24]

M.: Homogeneous vector bundles, Group actions and invariant theory (Montreal, PQ, 1988), CMS Conf

Snow D. M.: Homogeneous vector bundles, Group actions and invariant theory (Montreal, PQ, 1988), CMS Conf. Proc., 10, 193-205, Amer. Math. Soc., Providence, RI (1989)

work page 1988
[25]

16, 265-324 (2011)

Sparks J.: Sasaki-Einstein Manifolds, Surveys Diﬀ.Geom. 16, 265-324 (2011)

work page 2011
[26]

Verdiani L.: Invariant metrics on cohomogeneity one mani- folds, Geometriae Dedicata , 77 (1), 77-110 (1999) 27

work page 1999

[1] [1]

Yugoslav Geometrical seminar, Divcibare, 10-17 October, 3-35 (1993)

Alekseevsky D.: Flag manifolds , 11. Yugoslav Geometrical seminar, Divcibare, 10-17 October, 3-35 (1993)

work page 1993

[2] [2]

global Anal

Alekseevsky D., Zuddas F.: Cohomogeneity one Kahler and Kahler-Einstein manifolds with one singular orbit I , Ann. global Anal. Geom. 52:1, 99-128 (2017)

work page 2017

[3] [3]

Groups (2018), pp

Alekseevsky D., Chrysikos J.: Spin structures on compact homogeneous pseudo-Riemannian manifolds , Transf. Groups (2018), pp. 1-31

work page 2018

[4] [4]

Alekseevsky D., Spiro A.: Invariant CR structures on com- pact homogeneous manifolds, Hokk. Math. J, v. 32, no.2, 209- 276 (2003)

work page 2003

[5] [5]

V., Perelomov, A

Alekseevsky D. V., Perelomov, A. M.: Invariant Kaehler- Einstein metrics on compact homogeneous spaces , Funct. Anal. Appl., 20 (3), 171-182 (1986)

work page 1986

[6] [6]

Alekseevsky D., Cortes V., Hasegawa K., Kamishima Y., Ho- mogeneous locally conformally K¨ ahler and Sasaki manifold s, Int. J. Math., 26, n5, ( 2015)

work page 2015

[7] [7]

5, 6, 957-974 (2006)

Arvanitoyeorgos A.: Geometry of ﬂag manifolds , Interna- tional Journal of Geometric Methods in Modern Physics Vol.3, Nos. 5, 6, 957-974 (2006)

work page 2006

[8] [8]

, Ricci-ﬂat metrics on vector bun- dles over ﬂag manifolds , arXiv:1905.00412 (01/05/2019)

Achmed-Zade I., Bykov D. , Ricci-ﬂat metrics on vector bun- dles over ﬂag manifolds , arXiv:1905.00412 (01/05/2019)

work page arXiv 1905

[9] [9]

2, (2003) 480491

Azad H., Biswas I., Quasi-potentials and Kahler Einstein metrics on ﬂag manifolds II , Journal of Algebra 269 no. 2, (2003) 480491

work page 2003

[10] [10]

Besse A.: Einstein manifolds, Ergeb. Math. Grenzgeb. (3) 10, Springer, Berlin, 1987

work page 1987

[11] [11]

Berard- Bergery L.: Sur des nouvelles varietes Riemanni- ennes d’Einstein, Publ. de Inst. E. Cartan, No. 6, 1-60 (1982)

work page 1982

[12] [12]

Borel A., Hirzebruch F.: Characteristic classes and homoger- neous spaces, Amer. J . Math. 80, 458-538 (1958) 26

work page 1958

[13] [13]

Boyer C.P., Galicki K., Sasakian geometry , Oxford Mathe- matical Monographs, Oxford University Press, Oxford, 2008

work page 2008

[14] [14]

Van Coevering C., Calabi-Yau metrics on canonical bundles of ﬂag varieties , arXiv:1807.07256v1 (19/07/2018)

work page internal anchor Pith review Pith/arXiv arXiv 2018

[15] [15]

Y.: K¨ ahler Einstein metrics of cohomo- geneity one and bundle construction for Einstein Hermitian metrics, Math

Dancer A., Wang M. Y.: K¨ ahler Einstein metrics of cohomo- geneity one and bundle construction for Einstein Hermitian metrics, Math. Ann. 312, 503-526 (1998)

work page 1998

[16] [16]

Y.: The initial value problem for cohomogeneity one Einstein metrics , J

Eschenburg J.-H., Wang M. Y.: The initial value problem for cohomogeneity one Einstein metrics , J. Geom. Anal.,10, No.1, 109-137 (2000)

work page 2000

[17] [17]

Gorbatsevich V.V., Onishchik A.L., Vinberg E.B.: Structure of Lie groups and Lie algebras , Encycl. Math. Sci., Lie groups and Lie algebras, III, Springer Verlag

work page

[18] [18]

19, 763-786 (1982)

Huckleberry A., Snow D.: Almost homogeneous K¨ ahler man- ifolds with hypersurface orbits , Osaka J.math. 19, 763-786 (1982)

work page 1982

[19] [19]

17th Int

Koiso N., Sakane Y.: Non-homogeneous K¨ ahler-Einstein met- rics on compact complex manifolds , Curvature and topology of Riemannian manifolds, Proc. 17th Int. Taniguchi Symp., Katata/Jap. 1985, Lect. Notes Math. 1201, 165-179 (1986)

work page 1985

[20] [20]

Koiso N., Sakane Y.: Non homogeneous K¨ ahler Einstein met- rics on compact complex manifolds II , Osaka J. Math. 25, 933-959 (1988)

work page 1988

[21] [21]

2, 213- 225 (1987)

Page D.N., Pope C.N.: Inhomogeneous Einstein metrics on complex line bundles , Classical Quantum Gravity 4 no. 2, 213- 225 (1987)

work page 1987

[22] [22]

Podest` a F., Spiro A.: Kaehler manifolds with large isometry group, Osaka J. Math. Volume 36, Number 4, 805-833 (1999)

work page 1999

[23] [23]

Sakane Y.: Examples of compact Einstein-K¨ ahler manifolds with positive Ricci tensor , Osaka J. Math. 23, 585-616 (1986)

work page 1986

[24] [24]

M.: Homogeneous vector bundles, Group actions and invariant theory (Montreal, PQ, 1988), CMS Conf

Snow D. M.: Homogeneous vector bundles, Group actions and invariant theory (Montreal, PQ, 1988), CMS Conf. Proc., 10, 193-205, Amer. Math. Soc., Providence, RI (1989)

work page 1988

[25] [25]

16, 265-324 (2011)

Sparks J.: Sasaki-Einstein Manifolds, Surveys Diﬀ.Geom. 16, 265-324 (2011)

work page 2011

[26] [26]

Verdiani L.: Invariant metrics on cohomogeneity one mani- folds, Geometriae Dedicata , 77 (1), 77-110 (1999) 27

work page 1999