Anti-self-dual bihermitian structures on Inoue surfaces

Fujiki, Akira; Pontecorvo, Massimiliano

doi:10.4310/jdg/1284557925

Cited by 27 publications

(56 citation statements)

References 34 publications

(82 reference statements)

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“…In fact, the arguments above are completely the same even if we start from a properly blown-up half Inoue surface instead of just a half Inoue surface as we did in [1]. , and the (fiberwise) quotient byι of the restricted family toT ι is naturally identified with the Kuranishi family g of the pair (S, C) considered above.…”

Section: Proof Of the Main Resultsmentioning

confidence: 75%

“…Remark 3.1 Lemma 3.4 and Proposition 3.14 in [1] follow clearly from Proposition 1.3 and Lemma 3.1 above. In fact, the arguments above are completely the same even if we start from a properly blown-up half Inoue surface instead of just a half Inoue surface as we did in [1].…”

Section: Proof Of the Main Resultsmentioning

confidence: 77%

“…That T is smooth of dimension m was shown in Proposition 3.12 of [1]. The hypersufaces A i , 1 ≤ i ≤ m, of the theorem are T ( p) in [1,Prop.3.12 ] defined for each node p of C. In fact C is a cycle of rational curves with m irreducible components. So it admits precisely m nodes.…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

“…Then similarlỹ C t := u −1 t (C t ) has two connected components, each mapped isomorphically onto C t . Moreover, by Proposition 3.13 of [1]C t is the anti-canonical curve onS t . Let L t be the unique non-trivial holomorphic line bundle on S t with L ⊗2 t trivial, i.e., L t = L S t , which depends holomorphically on t with L o = L. C t is then an L t -twisted anti-canonical curve on S t .…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

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Non-upper-semicontinuity of algebraic dimension for families of compact complex manifolds

Fujiki

Pontecorvo

2010

Math. Ann.

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In this note we show that in a certain subfamily of the Kuranishi family of any half Inoue surface the algebraic dimensions of the fibers jump downwards at special points of the parameter space showing that the upper semicontinuity of algebraic dimensions in any sense does not hold in general for families of compact non-K hler manifolds. In the K hler case, the upper semicontinuity always holds true in a certain weak sense.

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Section: Proof Of the Main Resultsmentioning

confidence: 75%

Section: Proof Of the Main Resultsmentioning

confidence: 77%

Section: Proof Of the Main Resultsmentioning

confidence: 99%

Section: Proof Of the Main Resultsmentioning

confidence: 99%

See 2 more Smart Citations

Non-upper-semicontinuity of algebraic dimension for families of compact complex manifolds

Fujiki

Pontecorvo

2010

Math. Ann.

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“…The first lcK examples in class VII + 0 are by LeBrun [32] on parabolic Inoue surfaces; later, a twistor construction by Fujiki and Pontecorvo [19] produced these very special lcK metrics on all (minimal) hyperbolic as well as on all half-Inue surfaces.…”

Section: Proposition 13 [4851] a Complex Manifold M Is Lck If And Omentioning

confidence: 99%

On a question of Vaisman concerning complex surfaces

Pontecorvo

2013

Annali di Matematica

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The last years have seen striking improvements on Vaisman's\ud question about existence of locally conformally K\"ahler (lcK)\ud metrics on compact complex surfaces.\ud The aim of this paper is two-fold.\ud We review results of different authors which,\ud for all known examples of compact complex surfaces,\ud give a complete answer to the Vaisman's question.\ud We also point out a relation between lcK surfaces\ud and generalized K\"ahler geometry\ud in four-dimension and prove a new result concerning\ud %generalized K\"ahler structures on\ud Hyperbolic Inoue surfaces.\ud We conclude with a simple observation on a question of Brunella

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Classification of Generalized Kähler‐Ricci Solitons on Complex Surfaces

Streets

Ustinovskiy

2020

Comm Pure Appl Math

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Using toric geometry we give an explicit construction of the compact steady solitons for pluriclosed flow first constructed in by the first author in 2019. This construction also reveals that these solitons are generalized Kähler in two distinct ways, with vanishing and nonvanishing Poisson structure. This gives the first examples of generalized Kähler structures with nonvanishing Poisson structure on nonstandard Hopf surfaces, completing the existence question for such structures. Moreover, this gives a complete answer to the existence question for generalized Kähler-Ricci solitons on compact complex surfaces. In the setting of generalized Kähler geometry with vanishing Poisson structure, we show that these solitons are unique. We show that these solitons are global attractors for the generalized Kähler-Ricci flow among metrics with maximal symmetry.

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Anti-self-dual bihermitian structures on Inoue surfaces

Cited by 27 publications

References 34 publications

Non-upper-semicontinuity of algebraic dimension for families of compact complex manifolds

Non-upper-semicontinuity of algebraic dimension for families of compact complex manifolds

On a question of Vaisman concerning complex surfaces

Classification of Generalized Kähler‐Ricci Solitons on Complex Surfaces

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