The spaces of K\"ahler and holomorphically tamed symplectic forms on closed 4-manifolds

Shengzhen Ning; Tian-Jun Li

REVIEW 1 major objections 6 minor 81 references

The spaces of K\"ahler and holomorphically tamed symplectic forms on closed 4-manifolds

T0 review · 1 major / 6 minor · reviewed 2026-08-01 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves that holomorphically tamed symplectic forms on closed 4-manifolds are exactly the Kähler forms, class by class, in three regimes: b_2^+=1, Kodaira dimension zero, and negative curvature.

desk verdict A serious, mostly-solid paper with real new theorems; the negative-curvature branch depends on a rigidity citation that a referee should verify. read the letter →

arxiv 2607.18778 v1 pith:NQVQIQLQ submitted 2026-07-21 math.SG math.AGmath.DG

Tian-Jun Li , Shengzhen Ning This is my paper · ORCID

classification math.SGmath.AGmath.DG MSC 53D3532Q1553C5557R17

keywords Kählerformsholomorphicallytamedsymplecticclosed4-manifoldsTorellitheoremperioddomainKodairadimensiondeformation-to-isotopyStreets–Tianconjecture

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks whether a symplectic form that is only tamed by some complex structure must actually be a Kähler form, compatible with a (possibly different) complex structure. It proves that in three broad settings—closed 4-manifolds with positive part of the second Betti number b_2^+=1, manifolds admitting a symplectic form of Kodaira dimension zero, and manifolds admitting a Kähler metric of negative sectional curvature—the two classes coincide for every cohomology class. In the same settings it proves uniqueness up to a homologically trivial diffeomorphism (one acting trivially on homology), connectedness or finiteness of the moduli space of Kähler-type forms, and openness of the space of Kähler-type forms in the space of all symplectic forms in the first two settings. This matters because results about Kähler-type symplectic forms—packing, rigidity, mapping-class-group effects—then transfer to the larger class of merely tamed forms on these manifolds.

What carries the argument

The load-bearing identity is the cone comparison for an integrable complex structure J: the J-tamed cone is the J-compatible cone plus the anti-invariant part of cohomology, so a class is tamed exactly when its (1,1)-projection is compatible. Three mechanisms then carry the proof: the deformation-to-isotopy theorem when b_2^+=1; the Torelli theorem and period-domain topology for K3 surfaces, complex tori, and their blowups, which control the moduli of complex structures; and marked strong rigidity for negatively curved Kähler surfaces, which forces any homotopy equivalence to be homotopic to a holomorphic or anti-holomorphic map and identifies the (1,1)-cohomology of all complex structures o

What would settle it

Take a compact Kähler surface X with negative sectional curvature and choose two complex structures J and J' on the same smooth manifold. Compute the real (1,1)-cohomology subspaces H^{1,1}_J(X;R) and H^{1,1}_{J'}(X;R). The paper predicts they are equal. Any pair with different subspaces would falsify Corollary 2.8 and hence Theorem 1.8(3).

Watch

Extended reading notes

Core claim

The central statement is Theorem 1.8: for every cohomology class a, SK_a(X) is either empty or equal to ST_a(X) when X has b_2^+=1, admits a symplectic form of Kodaira dimension zero, or admits a Kähler metric of negative sectional curvature. Here SK_a consists of symplectic forms compatible with some complex structure (Kähler-type), and ST_a consists of forms tamed by some integrable complex structure. Thus in these regimes a symplectic form tamed by a complex structure and carrying a Kähler-type class is itself Kähler-type. The paper derives from this the uniqueness bound #MK_a(X)≤1, connectedness/finiteness of the moduli space MK(X), and openness of SK(X) in S(X) in the first two cases.

Load-bearing premise

The negative-curvature half rests on the deep rigidity theorem that every compact Kähler surface of negative sectional curvature is marked strongly rigid—that any homotopy equivalence involving it is homotopic to a holomorphic or anti-holomorphic map—and if that theorem failed for even one such surface, the equality, uniqueness, and non-openness conclusions in that regime would collapse.

Editorial extensions

If this is right

On rational and ruled 4-manifolds, K3 surfaces, Enriques and hyperelliptic surfaces, complex tori, and their blowups, every holomorphically tamed symplectic form whose cohomology class admits a Kähler form is itself Kähler-type.
On those manifolds, any two cohomologous Kähler-type symplectic forms are related by a homologically trivial diffeomorphism, so symplectic invariants of a class are canonical up to the Torelli group.
The moduli space MK(X) of Kähler-type forms has finitely many connected components for every closed 4-manifold underlying a Kähler surface; it is connected in the non-positive Kodaira dimension regime and has at most two components in the negative-curvature regime.
The space SK(X) is open in the space of all symplectic forms for b_2^+=1 and for Kodaira dimension zero, but fails to be open for negatively curved Kähler surfaces with b_2^+>1.
Symplectic properties proved for a single holomorphically tamed form on these manifolds transfer automatically to every cohomologous holomorphically tamed form.

Reading between the lines

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

Editorial inference: the class-by-class equality suggests a possible global dichotomy among Kähler surfaces—tame and compatible forms coincide in non-positive Kodaira dimension, while Example 1.3 shows failures can occur in positive Kodaira dimension; the paper leaves open whether the tamed and compatible spaces differ at the level of forms, not just cohomology classes, in general type.
Editorial inference: the period-domain argument for K3 and tori is tailored to the Torelli theorem; a natural test is whether the same equality and uniqueness statements survive for other manifolds with a global Torelli theorem, such as hyperkähler 4-manifolds with different intersection lattices.
Editorial inference: the negative-curvature non-openness result suggests that openness of SK(X) in S(X) may be a Kodaira-dimension phenomenon; one could conjecture that among Kähler surfaces, SK(X) is open exactly when b_2^+=1 or the Kodaira dimension is non-positive, and the paper's results are consistent with that.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Punchline: this is a real contribution. For a broad set of closed 4-manifolds—b+2=1, K3/T4 and their blowups, Enriques and hyperelliptic surfaces and their blowups—the paper proves that holomorphically tamed symplectic forms in a Kähler class are actually Kähler, and that the space of Kähler-type forms modulo homologically trivial diffeomorphisms has at most one point. These are genuine extensions of earlier work, not repackaged results. The proofs are serious and mostly convincing, and the paper is well organized; Section 4, with the period-domain and blowup-family arguments, is the technical heart and is handled carefully.

What is genuinely new: the blowup cases, the Enriques/hyperelliptic uniqueness and connectedness, and the negative-curvature theorems. The minimal K3/T4 uniqueness was known from Tolman and Entov–Verbitsky, but extending it to blowups requires real work with Torelli and the iterative blowup construction, and the connectedness of MK for Kodaira-zero manifolds is a clean new statement. The citation pattern is fine: the paper leans on earlier work of the first author, but those theorems have independent published proofs and are not being re-derived here.

Where the soft spots are: the negative-curvature branch. The stress-test note is right: Theorem 2.7 states that any compact Kähler surface with a Kähler metric of negative sectional curvature is marked strongly rigid, citing Siu and Zheng. But Siu's strong rigidity needs strong negativity of the curvature tensor, which is stronger than negative sectional curvature; Zheng's theorem, as the paper itself describes it, is for nonpositively curved surfaces of general type with c1^2 > 2c2. The paper never shows that negative sectional curvature implies c1^2 > 2c2, nor does it quote Zheng's precise statement. And Corollary 2.8 needs the 'marked' version—applied to the identity map, the resulting biholomorphism must be homotopic to the identity to land in Diff_h. If either piece fails, Theorems 1.8(3), 1.9(2), 1.10(3), and the non-openness part of 1.11 lose their support. This is not an internal contradiction, but it is load-bearing, and a referee should make the authors close that gap with a precise citation or a proof.

The other terse passages—Lemma 2.4's path lifting, Lemma 4.4's continuity argument, sign conventions in Theorem 4.15—look like exposition issues, not real flaws. I would want the negative-curvature gap addressed before publication, but not the whole paper rewritten.

Who this is for: anyone working on symplectic 4-manifolds, Kähler cones, or complex surface moduli. It deserves a serious referee and a revision, not a desk reject. My verdict tracks the reader's ACCEPT with moderate confidence, with the negative-curvature concern given a bit more weight.

Referee Report

1 major / 6 minor

Summary. The paper studies, on a closed oriented 4-manifold X, the spaces ST_a(X) of holomorphically tamed symplectic forms in a fixed cohomology class a and SK_a(X) of Kähler-type symplectic forms in that class. Theorem 1.8 claims ST_a(X)=SK_a(X) whenever SK_a(X) is nonempty, under any of: b^+_2(X)=1; X admits a symplectic form of Kodaira dimension zero; X admits a Kähler metric of negative sectional curvature. Theorems 1.9–1.11 give uniqueness, connectedness, and openness results for these spaces. Section 3 handles the b^+=1 case using the tamed/compatible cone comparison of Li–Zhang, Moser stability, and deformation-to-isotopy. Section 4 is the technical core: for K3 surfaces, T^4, and their blowups, the proof uses Torelli theorems, the topology of the period domain, and a delicate blowup-deformation construction. Section 5 assembles the proofs, with the negative-curvature case depending on a strong-rigidity theorem stated as Theorem 2.7.

Significance. If correct, the results are strong: they establish that holomorphically tamed symplectic forms with Kähler-type cohomology class are actually Kähler in three substantial regimes, and they give uniqueness (at most one Diff_h-orbit in each class), connectedness, and openness results for spaces of Kähler-type forms. The paper is not circular: it builds on previously published theorems (Torelli, deformation-to-isotopy, gauge theory, rigidity) and the new arguments in Section 4 are substantial. The main caveat is that the negative-curvature branch rests entirely on a deep external rigidity theorem whose precise hypotheses are not verified in the text.

major comments (1)

[§2.3, Theorem 2.7] Theorem 2.7 is the sole support for Corollary 2.8, which in turn is the load-bearing step for Theorem 1.8(3), Theorem 1.9(2), Theorem 1.10(3), and the non-openness part of Theorem 1.11. As written, the cited results do not transparently imply the stated theorem. The text itself notes that [Siu80] requires strong negativity of the curvature tensor, which is stronger than negative sectional curvature, and two paragraphs later quotes [Zhe95] as applying to nonpositively curved Kähler surfaces of general type with c_1^2>2c_2. The paper does not prove that negative sectional curvature implies c_1^2>2c_2 for a Kähler surface, nor does it quote a theorem from Zheng that directly gives the 'marked' refinement needed for the identity homotopy equivalence. Because Corollary 2.8 needs a biholomorphism homotopic to the identity map, this missing verification is essential. Please supply the exact the

minor comments (6)

[§2.2, Lemma 2.4] The proof invokes Kodaira–Spencer stability along an arbitrary path of complex structures. A sentence explaining how to subdivide the interval and use local versal deformations would make the argument fully precise.
[§1.4, Theorem 1.10] Formatting: 'The spaceMK(X)' should read 'The space MK(X)'. Also, the remark about fake projective planes says 'This would follow'; if it is meant as a claim, the argument should be supplied or the sentence made explicitly conditional.
[§4, Figure 1] The caption uses 'Φ_k' in places where the text notation is 'Φ_κ'; please correct the mismatch.
[§4.3, Corollary 4.14] In the proof of the converse inclusion, the use of the Seiberg–Witten blowup formula and the SW=Gr theorem is cited rather briefly; a precise reference for the blowup formula used would help.
[§4.2, Theorem 4.6] The sentence 'When X admits no Kähler structure' is confusing in a statement about K3 or T^4, which always admit Kähler structures; the intended degenerate case should be clarified.
[§3, Corollary 3.6] For the hyperelliptic case, the proof explicitly treats cyclic G of order 2,3,4,6 and then appeals to [CC17, Theorem 1] for the parameterization. Since the non-cyclic G' cases are part of the statement, the connection between the cyclic-group discussion and the cited Teichmüller-space description should be spelled out.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained modulo external published theorems and prior independent results.

full rationale

The paper's main claims (Theorems 1.8–1.11) are derived from four types of inputs: (i) the tamed/compatible cone comparison of [LZ09] stated as Theorem 2.1; (ii) Torelli and period-domain analysis for K3 and T4; (iii) external rigidity theorems of Siu and Zheng (Theorem 2.7); and (iv) gauge-theoretic finiteness results from Friedman–Morgan. None of these inputs restate the target equalities, and the spaces SK_a(X) and ST_a(X) are defined independently of the conclusions about them. The only references to the authors' own prior work are [Li08], [LZ09], and [Nin25]; these are published results with independent proofs and are used as lemmas or contextual examples, not as an unverified chain that forces the conclusions. The negative-curvature branch depends on the external strong-rigidity theorem, which is a genuine outside input; any concern about hypotheses matching would be a correctness risk, not circularity. No fitted parameter is renamed as a prediction, and no claimed derivation reduces by construction to its own inputs. The paper's central novelty—the extension to blowups and the negative-curvature regime—is not obtained by re-deriving a prior conclusion from the same conclusion.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

No new free parameters or postulated entities are introduced. The central claims rest on a suite of deep cited theorems in complex surface classification, Torelli theory, gauge theory, and rigidity; these are standard in the field but assumed rather than proved.

assumptions (9)

domain assumption Li-Zhang cone comparison theorem: for integrable J, K^t_J = K^c_J + H^-_J whenever K^c_J is nonempty.
Used throughout to pass from tamed cones to compatible cones; see Section 2.1, Theorem 2.1(2).
standard math Surjectivity of the period map and global Torelli theorems for K3 and T4 surfaces.
Section 4.1 relies on these to identify complex structures with period data and to realize Hodge isometries by diffeomorphisms.
domain assumption Marked strong rigidity for compact Kähler surfaces admitting a metric of negative sectional curvature (Siu, Zheng).
Theorem 2.7 is the load-bearing input for all negative-curvature results.
domain assumption Deformation-to-isotopy theorem for symplectic 4-manifolds with b+2=1 (McDuff, Li-Liu).
Theorem 3.1 is used in Theorem 3.2(3) to turn deformations into isotopies.
standard math Buchdahl-Lamari criterion: a compact complex surface is Kähler iff b1 is even.
Used in Lemma 2.4 and Theorem 2.9 to conclude every complex structure on a Kähler 4-manifold is Kähler.
standard math Kodaira-Spencer stability theorem for complex structures.
Used in Lemma 2.4 and Theorem 4.18 to produce smooth families of Kähler forms.
domain assumption Friedman-Morgan finiteness of deformation-equivalence classes of complex surfaces.
Theorem 2.10 gives finiteness of connected components of I(X), used in Corollary 2.11.
standard math Classification of Kodaira-dimension-zero complex surfaces and Castelnuovo contraction theorem.
Section 4.3.1 reduces every complex structure on a blowup of K3/T4 to blowups of a K3 or T4 minimal model.
domain assumption Seiberg-Witten/Gromov-Taubes blowup formulas for exceptional classes.
Corollary 4.14 uses these to identify the integrable tamed cone with the symplectic cone and to assert λ_i≠0.

how reviews work

0 comments

Cite this review

Pith. "Pith review of The spaces of K\"ahler and holomorphically tamed symplectic forms on closed 4-manifolds." pith.science (2026). https://pith.science/paper/NQVQIQLQ

@misc{pith2026260718778,
  author       = {Pith},
  title        = {Pith review of: The spaces of K\"ahler and holomorphically tamed symplectic forms on closed 4-manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NQVQIQLQ}},
  note         = {Machine review of arXiv:2607.18778}
}

abstract

This paper investigates the uniqueness, connectedness and openness properties of spaces of K\"ahler forms on closed $4$-manifolds, extending discussions in \cite{Li08,Sal13}. Motivated by the Streets--Tian conjecture concerning the existence of K\"ahler metrics on Hermitian-symplectic complex manifolds, we also study holomorphically tamed symplectic forms, which are symplectic forms tamed by some integrable complex structure. We formulate a parallel question and relate it to the corresponding questions for K\"ahler-type symplectic forms.

Figures

Figures reproduced from arXiv: 2607.18778 by the authors.

**Figure 1.** A schematic picture of the decomposition of the period domain Φ with respect to κ. The wall Ψ=0 κ contains the locus Ψ∗ κ , shown as the central triangular region, and separates Φ into the regions Ψ<0 κ and Ψ>0 κ . The central sphere joining the periods φ and φ represents a hyperK¨ahler twistor family, which realizes Ψ>0 κ as a disk bundle over Mκ. When X = K3, Mκ and Φκ are obtained by deleting the dashed curves co… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

81 extracted references · 40 canonical work pages

[1]

Li, Tian-Jun , TITLE =. Third. 2008 , ISBN =

2008
[2]

Wilson, P. M. H. , TITLE =. Invent. Math. , FJOURNAL =. 1992 , NUMBER =. doi:10.1007/BF01231902 , URL =

work page doi:10.1007/bf01231902 1992
[3]

Entov, Michael and Verbitsky, Misha , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2023 , NUMBER =. doi:10.1093/imrn/rnac107 , URL =

work page doi:10.1093/imrn/rnac107 2023
[4]

Shevchishin, Vsevolod and Smirnov, Gleb , TITLE =. J. Symplectic Geom. , FJOURNAL =. 2020 , NUMBER =. doi:10.4310/JSG.2020.v18.n5.a3 , URL =

work page doi:10.4310/jsg.2020.v18.n5.a3 2020
[5]

2025 , journal=

Infinite connected components of the space of symplectic forms on ruled surfaces , author=. 2025 , journal=

2025
[6]

Donaldson, S. K. , TITLE =. Inspired by. 2006 , ISBN =. doi:10.1142/9789812772688\_0007 , URL =

work page doi:10.1142/9789812772688 2006
[7]

Smirnov, Gleb , TITLE =. Geom. Funct. Anal. , FJOURNAL =. 2022 , NUMBER =. doi:10.1007/s00039-022-00600-z , URL =

work page doi:10.1007/s00039-022-00600-z 2022
[8]

Smirnov, Gleb , TITLE =. J. Topol. , FJOURNAL =. 2023 , NUMBER =. doi:10.1112/topo.12304 , URL =

work page doi:10.1112/topo.12304 2023

Show all 81 references

[9]

Compositio Math

Looijenga, Eduard and Peters, Chris , TITLE =. Compositio Math. , FJOURNAL =. 1980/81 , NUMBER =

1980
[10]

Streets, Jeffrey and Tian, Gang , TITLE =. J. Geom. Anal. , FJOURNAL =. 2022 , NUMBER =. doi:10.1007/s12220-022-01036-5 , URL =

2022 doi
[11]

2016 , PAGES =

Huybrechts, Daniel , TITLE =. 2016 , PAGES =. doi:10.1017/CBO9781316594193 , URL =

2016 doi
[12]

Draghici, Tedi , TITLE =. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) , FJOURNAL =. 1999 , NUMBER =

1999
[13]

Streets, Jeffrey and Tian, Gang , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2010 , NUMBER =. doi:10.1093/imrn/rnp237 , URL =

2010 doi
[14]

Wilson, P. M. H. , TITLE =. J. Differential Geom. , FJOURNAL =. 1997 , NUMBER =

1997
[15]

Blaine , TITLE =

Harvey, Reese and Lawson, Jr., H. Blaine , TITLE =. Invent. Math. , FJOURNAL =. 1983 , NUMBER =. doi:10.1007/BF01394312 , URL =

1983 doi
[16]

Sullivan, Dennis , TITLE =. Invent. Math. , FJOURNAL =. 1976 , PAGES =. doi:10.1007/BF01390011 , URL =

1976 doi
[17]

Entov, Michael and Verbitsky, Misha , TITLE =. J. Topol. Anal. , FJOURNAL =. 2016 , NUMBER =. doi:10.1142/S1793525316500229 , URL =

2016 doi
[18]

Selecta Math

Entov, Michael and Verbitsky, Misha , TITLE =. Selecta Math. (N.S.) , FJOURNAL =. 2018 , NUMBER =. doi:10.1007/s00029-017-0353-3 , URL =

2018 doi
[19]

Entov, Michael and Verbitsky, Misha , TITLE =. J. Assoc. Math. Res. , FJOURNAL =. 2023 , NUMBER =

2023
[20]

Amerik, Ekaterina and Verbitsky, Misha , TITLE =. J. Geom. Phys. , FJOURNAL =. 2015 , PAGES =. doi:10.1016/j.geomphys.2015.07.006 , URL =

2015 doi
[21]

Enrietti, Nicola and Fino, Anna and Vezzoni, Luigi , TITLE =. J. Symplectic Geom. , FJOURNAL =. 2012 , NUMBER =. doi:10.4310/jsg.2012.v10.n2.a3 , URL =

2012 doi
[22]

Li, Tian-Jun and Zhang, Weiyi , TITLE =. Comm. Anal. Geom. , FJOURNAL =. 2009 , NUMBER =. doi:10.4310/CAG.2009.v17.n4.a4 , URL =

2009 doi
[23]

and Picard, Sebastien and Zhang, Xiangwen , TITLE =

Fei, Teng and Phong, Duong H. and Picard, Sebastien and Zhang, Xiangwen , TITLE =. Comm. Anal. Geom. , FJOURNAL =. 2024 , NUMBER =. doi:10.4310/cag.241212005221 , URL =

2024 doi
[24]

Tolman, Susan , TITLE =. Invent. Math. , FJOURNAL =. 2017 , NUMBER =. doi:10.1007/s00222-017-0741-5 , URL =

2017 doi
[25]

Sheridan, Nick and Smith, Ivan , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2020 , NUMBER =. doi:10.1090/jams/946 , URL =

2020 doi
[26]

Almost toric fibrations on

Pranav Chakravarthy and Yoel Groman , year=. Almost toric fibrations on. arXiv:2502.04304 , archivePrefix=

arXiv
[27]

Smirnov, Gleb , TITLE =. J. Symplectic Geom. , FJOURNAL =. 2025 , NUMBER =. doi:10.4310/jsg.250820165748 , URL =

2025 doi
[28]

, TITLE =

Friedman, Robert and Morgan, John W. , TITLE =. 1994 , PAGES =. doi:10.1007/978-3-662-03028-8 , URL =

1994 doi
[29]

Topics in symplectic

McDuff, Dusa , TITLE =. Topics in symplectic. 1998 , ISBN =

1998
[30]

Chen, Shuwen and Zheng, Fangyang , TITLE =. J. Geom. Phys. , FJOURNAL =. 2025 , PAGES =. doi:10.1016/j.geomphys.2024.105390 , URL =

2025
[31]

Guo, Yuqin and Zheng, Fangyang , TITLE =. Ann. Mat. Pura Appl. (4) , FJOURNAL =. 2026 , NUMBER =. doi:10.1007/s10231-025-01597-6 , URL =

2026 doi
[32]

Acta Math

Salamon, Dietmar , TITLE =. Acta Math. Vietnam. , FJOURNAL =. 2013 , NUMBER =. doi:10.1007/s40306-012-0004-x , URL =

2013 doi
[33]

2017 , PAGES =

McDuff, Dusa and Salamon, Dietmar , TITLE =. 2017 , PAGES =. doi:10.1093/oso/9780198794899.001.0001 , URL =

2017
[34]

2026 , journal=

Dax invariants, light bulbs, and isotopies of symplectic structures , author=. 2026 , journal=

2026
[35]

Manuscripta Math

Fricke, Jan and Habermann, Lutz , TITLE =. Manuscripta Math. , FJOURNAL =. 2002 , NUMBER =. doi:10.1007/s00229-002-0319-3 , URL =

2002 doi
[36]

Wilson, P. M. H. , TITLE =. Manuscripta Math. , FJOURNAL =. 2005 , NUMBER =. doi:10.1007/s00229-004-0518-1 , URL =

2005 doi
[37]

Differential Geom

Fricke, Jan and Habermann, Katharina and Habermann, Lutz , TITLE =. Differential Geom. Appl. , FJOURNAL =. 2005 , NUMBER =. doi:10.1016/j.difgeo.2005.03.002 , URL =

2005 doi
[38]

, TITLE =

Catanese, F. , TITLE =. Handbook of moduli. 2013 , ISBN =

2013
[39]

Tohoku Math

Tsuchihashi, Hiroyasu , TITLE =. Tohoku Math. J. (2) , FJOURNAL =. 1979 , NUMBER =. doi:10.2748/tmj/1178229800 , URL =

1979
[40]

Suwa, Tatsuo , TITLE =. J. Fac. Sci. Univ. Tokyo Sect. I , FJOURNAL =. 1969 , PAGES =

1969
[41]

Thesis (Ph.D.) Technische Universit\"at M\"unchen , YEAR =

Boada De Narvaez, Daniel , TITLE =. Thesis (Ph.D.) Technische Universit\"at M\"unchen , YEAR =
[42]

Catanese, Fabrizio , TITLE =. Commun. Contemp. Math. , FJOURNAL =. 2009 , NUMBER =. doi:10.1142/S0219199709003478 , URL =

2009 doi
[43]

Manetti, Marco , TITLE =. Invent. Math. , FJOURNAL =. 2001 , NUMBER =. doi:10.1007/s002220000101 , URL =

2001 doi
[44]

Ning, Shengzhen , TITLE =. J. Lond. Math. Soc. (2) , FJOURNAL =. 2025 , NUMBER =. doi:10.1112/jlms.70117 , URL =

2025 doi
[45]

Li, T. J. and Liu, A. , TITLE =. Math. Res. Lett. , FJOURNAL =. 1995 , NUMBER =. doi:10.4310/MRL.1995.v2.n4.a6 , URL =

1995 doi
[46]

Li, T. J. and Liu, A. , TITLE =. Math. Res. Lett. , FJOURNAL =. 1995 , NUMBER =. doi:10.4310/MRL.1995.v2.n6.a11 , URL =

1995 doi
[47]

Li, Tian-Jun and Liu, Ai-Ko , TITLE =. J. Differential Geom. , FJOURNAL =. 2001 , NUMBER =

2001
[48]

, TITLE =

Lamari, A. , TITLE =. J. Math. Pures Appl. (9) , FJOURNAL =. 1999 , NUMBER =. doi:10.1016/S0021-7824(98)00005-1 , URL =

1999 doi
[49]

Buchdahl, Nicholas , TITLE =. Ann. Inst. Fourier (Grenoble) , FJOURNAL =. 1999 , NUMBER =. doi:10.5802/aif.1674 , URL =

1999 doi
[50]

and Koziarz, Vincent and Yeung, Sai-Kee , TITLE =

Cartwright, Donald I. and Koziarz, Vincent and Yeung, Sai-Kee , TITLE =. J. Algebraic Geom. , FJOURNAL =. 2017 , NUMBER =. doi:10.1090/jag/696 , URL =

2017 doi
[51]

Chen, Bing-Long and Yang, Xiaokui , TITLE =. Math. Ann. , FJOURNAL =. 2018 , NUMBER =. doi:10.1007/s00208-017-1521-7 , URL =

2018 doi
[52]

Yau, Shing Tung , TITLE =. Proc. Nat. Acad. Sci. U.S.A. , FJOURNAL =. 1977 , NUMBER =. doi:10.1073/pnas.74.5.1798 , URL =

1977 doi
[53]

and Spencer, D

Kodaira, K. and Spencer, D. C. , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1960 , PAGES =. doi:10.2307/1969879 , URL =

1960 doi
[54]

Latschev, Janko and McDuff, Dusa and Schlenk, Felix , TITLE =. Geom. Topol. , FJOURNAL =. 2013 , NUMBER =. doi:10.2140/gt.2013.17.2813 , URL =

2013 doi
[55]

Manuscripta Math

Siu, Yum Tong , TITLE =. Manuscripta Math. , FJOURNAL =. 1981 , NUMBER =. doi:10.1007/BF01263265 , URL =

1981 doi
[56]

Shioda, Tetsuji , TITLE =. J. Fac. Sci. Univ. Tokyo Sect. IA Math. , FJOURNAL =. 1978 , NUMBER =

1978
[57]

, TITLE =

Todorov, Andrei N. , TITLE =. Invent. Math. , FJOURNAL =. 1980 , NUMBER =. doi:10.1007/BF01390067 , URL =

1980 doi
[58]

Burns, Jr., Dan and Rapoport, Michael , TITLE =. Ann. Sci. \'Ecole Norm. Sup. (4) , FJOURNAL =. 1975 , NUMBER =

1975
[59]

Streets-

Kexiang Cao and Fangyang Zheng , year=. Streets-. arXiv:2410.04791 , archivePrefix=

arXiv
[60]

, TITLE =

Morgan, John W. , TITLE =. 1996 , PAGES =

1996
[61]

Donaldson, S. K. , TITLE =. Topology , FJOURNAL =. 1990 , NUMBER =. doi:10.1016/0040-9383(90)90001-Z , URL =

1990 doi
[62]

Friedman, Robert and Qin, Zhenbo , TITLE =. Math. Res. Lett. , FJOURNAL =. 1994 , NUMBER =. doi:10.4310/MRL.1994.v1.n3.a8 , URL =

1994 doi
[63]

Hind, Richard and Medori, Costantino and Tomassini, Adriano , TITLE =. J. Geom. Anal. , FJOURNAL =. 2023 , NUMBER =. doi:10.1007/s12220-023-01391-x , URL =

2023 doi
[64]

Forum Math

Migliorini, Massimiliano and Tomassini, Adriano , TITLE =. Forum Math. , FJOURNAL =. 2000 , NUMBER =. doi:10.1515/form.2000.025 , URL =

2000 doi
[65]

, TITLE =

Friedman, Robert and Morgan, John W. , TITLE =. J. Algebraic Geom. , FJOURNAL =. 1997 , NUMBER =

1997
[66]

Namikawa, Yukihiko , TITLE =. Math. Ann. , FJOURNAL =. 1985 , NUMBER =

1985
[67]

Raman, Sidhanth , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2025 , PAGES =. doi:10.1515/crelle-2025-0007 , URL =

2025 doi
[68]

Complex and symplectic geometry , SERIES =

Catanese, Fabrizio and Corvaja, Pietro , TITLE =. Complex and symplectic geometry , SERIES =. 2017 , ISBN =

2017
[69]

Memorie di Matematica e di Fisica della Societ

Bagnera, Giuseppe and de Franchis, Michele , title =. Memorie di Matematica e di Fisica della Societ
[70]

Acta Math

Enriques, Federigo and Severi, Francesco , TITLE =. Acta Math. , FJOURNAL =. 1910 , NUMBER =. doi:10.1007/BF02393217 , URL =

1910 doi
[71]

2025 , PAGES =

Cossec, François and Dolgachev, Igor and Liedtke, Christian , TITLE =. 2025 , PAGES =. doi:10.1007/978-981-96-1214-7 , URL =

2025 doi
[72]

and Hulek, Klaus and Peters, Chris A

Barth, Wolf P. and Hulek, Klaus and Peters, Chris A. M. and Van de Ven, Antonius , TITLE =. 2004 , PAGES =. doi:10.1007/978-3-642-57739-0 , URL =

2004 doi
[73]

Tohoku Math

Fino, Anna and Kasuya, Hisashi and Vezzoni, Luigi , TITLE =. Tohoku Math. J. (2) , FJOURNAL =. 2015 , NUMBER =. doi:10.2748/tmj/1429549577 , URL =

2015
[74]

Lalonde, Francois and McDuff, Dusa , TITLE =. Math. Res. Lett. , FJOURNAL =. 1996 , NUMBER =. doi:10.4310/MRL.1996.v3.n6.a5 , URL =

1996 doi
[75]

Siu, Yum Tong , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1980 , NUMBER =. doi:10.2307/1971321 , URL =

1980 doi
[76]

Zheng, Fangyang , TITLE =. Math. Z. , FJOURNAL =. 1995 , NUMBER =. doi:10.1007/BF02572607 , URL =

1995 doi
[77]

Duke Math

Abouzaid, Mohammed and Smith, Ivan , TITLE =. Duke Math. J. , FJOURNAL =. 2010 , NUMBER =. doi:10.1215/00127094-2010-015 , URL =

2010 doi
[78]

Homological mirror symmetry for projective

Paul Hacking and Ailsa Keating , year=. Homological mirror symmetry for projective. arXiv:2503.05680 , archivePrefix=

arXiv
[79]

Cascini, Paolo and Panov, Dmitri , TITLE =. J. Symplectic Geom. , FJOURNAL =. 2012 , NUMBER =. doi:10.4310/jsg.2012.v10.n4.a1 , URL =

2012 doi
[80]

and Kharlamov, V

Kulikov, Vik.\ S. and Kharlamov, V. M. , TITLE =. Izv. Ross. Akad. Nauk Ser. Mat. , FJOURNAL =. 2002 , NUMBER =. doi:10.1070/IM2002v066n01ABEH000374 , URL =

2002 doi
[81]

, TITLE =

Mumford, D. , TITLE =. Amer. J. Math. , FJOURNAL =. 1979 , NUMBER =. doi:10.2307/2373947 , URL =

1979 doi

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

Li, Tian-Jun , TITLE =. Third. 2008 , ISBN =

2008

[2] [2]

Wilson, P. M. H. , TITLE =. Invent. Math. , FJOURNAL =. 1992 , NUMBER =. doi:10.1007/BF01231902 , URL =

work page doi:10.1007/bf01231902 1992

[3] [3]

Entov, Michael and Verbitsky, Misha , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2023 , NUMBER =. doi:10.1093/imrn/rnac107 , URL =

work page doi:10.1093/imrn/rnac107 2023

[4] [4]

Shevchishin, Vsevolod and Smirnov, Gleb , TITLE =. J. Symplectic Geom. , FJOURNAL =. 2020 , NUMBER =. doi:10.4310/JSG.2020.v18.n5.a3 , URL =

work page doi:10.4310/jsg.2020.v18.n5.a3 2020

[5] [5]

2025 , journal=

Infinite connected components of the space of symplectic forms on ruled surfaces , author=. 2025 , journal=

2025

[6] [6]

Donaldson, S. K. , TITLE =. Inspired by. 2006 , ISBN =. doi:10.1142/9789812772688\_0007 , URL =

work page doi:10.1142/9789812772688 2006

[7] [7]

Smirnov, Gleb , TITLE =. Geom. Funct. Anal. , FJOURNAL =. 2022 , NUMBER =. doi:10.1007/s00039-022-00600-z , URL =

work page doi:10.1007/s00039-022-00600-z 2022

[8] [8]

Smirnov, Gleb , TITLE =. J. Topol. , FJOURNAL =. 2023 , NUMBER =. doi:10.1112/topo.12304 , URL =

work page doi:10.1112/topo.12304 2023