Gravitational instantons from rational elliptic surfaces

Probably not a substantial claim: may not pass S3 (load-bearing).

Köhler-Kühnel [32] proves uū = O(r^{-N}) for all N for the Calabi-Yau manifolds in [53].

“While the present paper was under review, I learned of a recent paper by Köhler-Kühnel [32], which proves uū = O(r -N ) for all N for the Calabi-Yau manifolds in [53].”

• cites [32] Koehler, Bert, 2010, On asymptotics of complete Ricci-flat Kähler metrics on open manifolds, Manuscripta Math. sourceDisputed

  The Introduction (p. 2) says the decay exponent obtained is 1/(6n), calls it too small for the reducible-divisor case, and hopes to improve it. The proof (p. 10) likewise concludes with a decay rate of 1/(6n), not O(r^{-N}) for every N. The arbitrary N mentioned in the construction of the initial metric (p. 4) does not establish the claimed rate for the Ricci-flat metric. (LLM, conf=0.87)

  Checked by a reasoning model — may be wrong.

  Experimental — accuracy not yet measured
• cites [53] Tian, G., 1990, Complete Kähler manifolds with zero Ricci curvature. I, J. Amer. Math. Soc. sourceNot checked — no full text of the source

May combine two claims.

The restriction map H^2(X) -> H^2(M) has a 1-dimensional kernel spanned by the Poincaré dual of the fibers, i.e. c1(X), and maps onto the 9-dimensional subspace H^{1,1}(M) ⊂ H^2(M) of classes containing a closed form which is (1,1) with respect to the complex structure of X.

“More precisely, the restriction map H 2 (X) → H 2 (M ) has a 1-dimensional kernel (spanned by the Poincaré dual of the fibers, i.e. c 1 (X)) and maps onto the 9-dimensional subspace H 1,1 (M ) ⊂ H 2 (M ) of classes containing a closed form which is (1, 1) with respect to the complex structure of X.”

• cites [47] Nordström, Johannes, 2008, Deformations of asymptotically cylindrical 𝐺₂-manifolds, Math. Proc. Cambridge Philos. Soc. sourcePartly supported

  Split into 2 atoms — 0 confirmed, 1 partially supported, 0 disputed, 0 misattributed, 1 unverifiable. Expand the claim to see per-atom verdicts.

  Experimental — accuracy not yet measured

May combine two claims.

One way of proving the unobstructedness in Theorem 1.10 is by taking products with T^3 and appealing to Nordström's work on G_2-structures, and the parameter count is consistent with Nordström's Proposition 3.5.

“One way of proving the unobstructedness in Theorem 1.10 is by taking products with T 3 and appealing to Nordström's work [47] on G 2 -structures.Our parameter count is consistent with [47, Proposition 3.5], though our proof provides a specific bit of extra information concerning Lie derivatives; see also Remark 4.6.”

• cites [47] Nordström, Johannes, 2008, Deformations of asymptotically cylindrical 𝐺₂-manifolds, Math. Proc. Cambridge Philos. Soc. sourceSupported

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Kovalev has sketched a general deformation theory for asymptotically cylindrical Calabi-Yau manifolds, but his parameter count is not entirely accurate and he needs to assume an appropriate analog of the Tian-Todorov theorem, which seems still unknown.

“Indeed, Kovalev [35] has sketched a general deformation theory for asymptotically cylindrical Calabi-Yau manifolds, similar to Koiso's approach [33] in the compact case.However, his parameter count is not entirely accurate, and he needs to assume an appropriate analog of the Tian-Todorov theorem [51,55], which seems to be still unknown; cf.Corollary 4.3.”

• cites [35] Kovalev, Alexei, 2006, Ricci-flat deformations of asymptotically cylindrical Calabi-Yau manifolds sourcePartly supported

  Split into 2 atoms — 1 confirmed, 1 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.

  Experimental — accuracy not yet measured
• cites [33] Koiso, N., 1983, Einstein metrics and complex structures, Invent. Math. sourceNot checked — no full text of the source
• cites [51] Tian, Gang, 1987, Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric sourceNot checked — no full text of the source
• cites [55] Todorov, Andrey N., 1989, The Weil-Petersson geometry of the moduli space of 𝑆𝑈(𝑛≥3) (Calabi-Yau) manifolds. I, Comm. Math. Phys. sourceNot checked — no full text of the source

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

According to Table 1, the estimate (3.17) in Song-Tian [50] is incorrect, but the error is irrelevant for the rest of that paper.

“According to Table 1, the estimate (3.17) in Song-Tian [50] is incorrect, but the error is irrelevant for the rest of that paper.”

• cites [50] Song, Jian, 2007, The Kähler-Ricci flow on surfaces of positive Kodaira dimension, Invent. Math. sourceCould not verify

  The copy’s Section 3 (pp. 7–8) contains estimates numbered (3.1)–(3.5), but searches for (3.17) did not locate the cited estimate. The copy also does not address whether an error in that estimate would be irrelevant to the rest of the paper. The citing authors attribute their assessment to their own Table 1, which is not evidence available from this source. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

Probably not a substantial claim: may not pass S3 (load-bearing).

The method employed in the proof of Lemma 2.2 is due to Grigoryan and Saloff-Coste in much greater generality.

“The method employed in the proof of Lemma 2.2 is due to Grigoryan and Saloff-Coste [19] in much greater generality.”

• cites [19] Grigor’yan, Alexander, 2005, Stability results for Harnack inequalities, Ann. Inst. Fourier (Grenoble) sourceCould not verify

  The Introduction (page unavailable) says Grigor’yan and Saloff-Coste provide methods for proving Harnack inequalities, and the overview describes a discretization technique. But the claim does not identify the method used in the citing paper’s Lemma 2.2, so our copy cannot establish that it is the same method or that this paper treats it in greater generality. Searched for the lemma number and descriptions of the proof method. (LLM, conf=0.70)

  Experimental — accuracy not yet measured

May combine two claims.

h^1(X, TX) = 8 and h^1(X, TX(log D)) = 9.

“Moreover, h 1 (X, T X ) = 8 and h 1 (X, T X (log D)) = 9.”

• cites [35] Kovalev, Alexei, 2006, Ricci-flat deformations of asymptotically cylindrical Calabi-Yau manifolds sourceCould not verify

  The discussion of logarithmic tangent sheaves (p. 13) explains the role of H¹(TW(log D)) but gives neither claimed dimension. Section 6.1 (p. 15) calculates a dimension of 10 for a different example. Searches for the values 8 and 9 alongside the tangent-sheaf terms found no passage establishing the claimed values for X. (LLM, conf=0.73)

  Experimental — accuracy not yet measured

May combine two claims.

If D is singular with finite monodromy, then ω_CY is ALG(δ, [θ, ε, τ]) with type I*, and generically the decay rate δ is 2 for I*_0, II, III, IV, and δ is 4/5, 2/3, 1/2 for II*, III*, IV*, though nongeneric ALG spaces may exhibit faster decay.

“(ii) If D is singular with finite monodromy, then ω CY is ALG(δ, [θ, ε, τ ]) with type I * Generically, the decay rate δ = 2 for I * 0 , II, III, IV, and δ = 4 5 , 2 3 , 1 2 for II * , III * , IV * , but nongeneric ALG spaces may exhibit faster decay.”

May combine two claims.

If the elliptic fibration is isotrivial, then ω_0 is flat and the ALG(δ) decay can be improved to ALG(2 + 1/θ).

“However, if the elliptic fibration is isotrivial, then ω 0 is indeed flat and we can improve ALG(δ) to ALG(2 + 1 θ ); see also [5].”

• cites [5] O. Biquard, V. Minerbe, A Kummer construction for gravitational instantons, Comm. Math. Phys., to appear.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S4 (based).

Up to orientation, there are two bad cycles if the monodromy is trivial, one if the generator of the monodromy is conjugate to (1 b; 0 1), b ∈ N, and none otherwise.

“Up to orientation, there are two bad cycles if the monodromy is trivial, one if the generator of the monodromy is conjugate to ( 1 b 0 1 ), b ∈ N, and none at all otherwise.”

May combine two claims.

If D is singular with infinite monodromy of Kodaira type I*_b, with monodromy conjugate to -(1 b; 0 1), b ∈ {1,...,4}, then ω_CY satisfies |B(x0,s)| ∼ s^2, inj(x) ∼ |B(x,1)| ∼ (log r(x))^{-1/2}, |Rm| ∼ r^{-2}(log r)^{-1}, |∇^k Rm| ≲ r^{-2-k}(log r)^{-1}, and has a unique tangent cone at infinity given by R^2/Z_2.

“• Kodaira type I * b , with monodromy conjugate to -( 1 b 0 1 ), where b ∈ {1, ..., 4}: In this case, |B(x 0 , s)| ∼ s 2 for s 1, inj(x) ∼ |B(x, 1)| ∼ (log r(x)) -1/2 for all x, |Rm| ∼ r -2 (log r) -1 , and |∇ k Rm| k r -2-k (log r) -1 for all k ∈ N.Moreover, ω CY has a unique tangent cone at infinity given by R 2 /Z 2 .”

May combine two claims.

If D is singular with infinite monodromy of Kodaira type I_b, with monodromy conjugate to (1 b; 0 1), b ∈ {1,...,9}, then ω_CY satisfies |B(x0,s)| ∼ s^{4/3}, inj(x) ∼ |B(x,1)| ∼ r(x)^{-1/3}, |Rm| ∼ r^{-2}, |∇^k Rm| ≲ r^{-2-k}, and has a unique tangent cone at infinity given by the half-line R_+.

“• Kodaira type I b , with monodromy conjugate to ( 1 b 0 1 ), where b ∈ {1, ..., 9}: Then ω CY satisfies |B(x 0 , s)| ∼ s 4/3 for s 1, inj(x) ∼ |B(x, 1)| ∼ r(x) -1/3 for all x, |Rm| ∼ r -2 , and |∇ k Rm| k r -2-k for all k ∈ N.Moreover, ω CY has a unique tangent cone at infinity given by the half-line, R + .”

May combine two claims.

For the complete I*b metrics, inj(x) ∼ |B(x,1)| ∼ (log r_x)^{-1/2}.

“For the same general reason as before, inj(x) ∼ |B(x, 1)| ∼ (log r x ) -1 2 .”

May combine two claims.

For Γ=Z2, setting (T,X,Y)=(t^2,x,ty) induces a birational map from (P1_t×E)/Z2 to the surface S: Y^2=T g(X), which is an isomorphism of elliptic fibrations away from t=0,∞.

“Setting (T, X, Y ) = (t 2 , x, ty) yields a rational map P 1 t × E P 1 T × P 2 XY , which factors through the globally defined Z 2 -action (t, x, y) → (-t, x, -y) and induces a birational map from (P 1 t × E)/Z 2 to the surface S : Y 2 = T g(X), which is in fact an isomorphism of elliptic fibrations away from t = 0, ∞.”

May combine two claims.

There are two assumptions under which the required Neumann eigenvalue lower bound can be proved: (1) A_i ⊂ B(x_i, ρ_i) with x_i ∈ A_i and ρ_i ∼ r_{i+1}/r_i = o(r_i), and (2) r_{i+1}/r_i ∼ r_i.

“There are two different assumptions under which we are able to prove (by using Lemmas 2.1-2.2) that this is indeed the case: (1) A i ⊂ B(x i , ρ i ) with x i ∈ A i , ρ i ∼ r i+1r i = o(r i ), and (2) r i+1r i ∼ r i .”

May combine two claims.

The main underlying idea works for β ≤ 2 only and strongly relies on the fact that ∫|∇u|^2 < ∞ in this case.

“The main underlying idea works for β ≤ 2 only and strongly relies on the fact that |∇u| 2 < ∞ in this case.”

May combine two claims.

In Lemma 3.3(ii), the metrics g_{S,ε} on the base are Kähler, and their Ricci forms are ρ(ω_{S,ε})=-i∂∂ log det(Im Z)=i/4((Im Z)_{ab} dZ_{bc}∧(Im Z)_{cd} dZ_{da}), the pullback of an invariant Kähler metric on H_m under the period map Z.

“(ii) The metrics g S,ε on the base are Kähler as well.Their Ricci forms ρ(ω S,ε ) = -i∂ ∂ log det(Im Z) = i 4 ((Im Z) ab dZ bc ∧ (Im Z) cd d Zda ), the pullback of an invariant Kähler metric on H m under the period map Z.”

May combine two claims.

The semi-flat metric is given by (3.8), with Im(τ1τ2)=√3/2 |z|^{5/3}(1-|z|...).

“ω sf,ε = i|k| 2 Im(τ 1 τ 2 ) ε dz ∧ dz |z| 4 + i 2 ε Im(τ 1 τ 2 ) (dw -Γdz) ∧ (d w -Γdz), (3.8) Im(τ 1 τ 2 ) = √ 3 2 |z| 5 3 (1 -|z|”

May combine two claims.

For the complete I*b metrics, as |u_x| tends to 1, r_x is asymptotic to dist_g(u_x,u_x0) and to (1/|u_x|)|log |u_x||^{1/2}.

“Using notation analogous to our discussion of the I b metrics, (3.18) yields |u x | 1 =⇒ r x ∼ dist g (u x , u x 0 ) ∼ 1 |u x | |log |u x || 1 2”

Probably not a substantial claim: may not pass S3 (load-bearing).

Table 1 lists τ1, τ2, N, and the tangent cones of the metrics on the base: incomplete (div(Ω)=0), and complete (div(Ω)=-D).

“Table 1 lists τ 1 , τ 2 , N, and the tangent cones of the metrics on the base: incomplete (div(Ω) = 0), and complete (div(Ω) = -D).”

May combine two claims.

In these coordinates, ω_sf,ε = i/2 (du∧dū + ε Im(ζ3) η∧η)(1+O(|u|^{-4m})), with η=dv-Πdu, Π(u,v)=1/u(Pv+Qv), P,Q=O(|u|^{-2m}).

“ω sf,ε = i 2 (du ∧ dū + ε Im(ζ 3 ) η ∧ η)(1 + O(|u| -4m )), (3.10) η = dv -Π du, Π(u, v) = 1 u ( P v + Qv), P , Q = O(|u| -2m ). (3.11)”

Classified as probably not a substantial claim.

Theorems 1.9 and 1.10 follow from Hodge theory on manifolds with cylindrical ends.

“Theorems 1.9-1.10follow from Hodge theory on manifolds with cylindrical ends, which is of course a very well-understood subject.”

May combine two claims.

For a hyper-Kähler ALH manifold (M^4, g), the kernel E of the linearized Einstein operator inside H satisfies E = D ⊕ L_bd, where D = ker Δ_Lich has dimension 34 and L_bd = {L_X g : X = parallel + ED}.

“Prove that E = D ⊕ L bd , where D = ker Δ Lich has dimension 34 and admits a clear geometric description, whereas L bd = {L X g : X = parallel + ED}.”

• Also stated in: § Step 2: The kernel of the linearized Einstein operator. · plain

  “The first consequence of this is that D ⊂ ker B| H ; hence E = D ⊕ L bd by (4.2), because wedging a scalar form with ω preserves harmonicity, so that tr h is constant for all h ∈ D, and div h = 0 because bounded harmonic forms are d * -closed.”

May combine two claims.

For an elliptic fibration U→Δ with a section, U|Δ* ≅ L/Λ for a holomorphic line bundle L and lattice bundle Λ; since H1(Δ*,O*)=0, L≅O and U|Δ* ≅ (Δ*×C)/(Zτ1+Zτ2) for multivalued holomorphic functions τ1,τ2 on Δ*.

“To explain the notion of monodromy, we begin by noting that U | Δ * ∼ = L/Λ, due to the existence of σ, where L → Δ * is a holomorphic line bundle and Λ ⊂ L is a lattice bundle.Now H 1 (Δ * , O * ) = 0, so L ∼ = O and U | Δ * ∼ = (Δ * × C)/(Zτ 1 + Zτ 2 ) for multivalued holomorphic functions τ 1 , τ 2 on Δ * .”

May combine two claims.

To close the slit, one twists by the monodromy, which is an isometry in this case.

“To close the slit, we then simply twist by the monodromy, which is an isometry in this case.”

May combine two claims.

For a degeneration f:U→Δ with I*_b monodromy, the pullback of U|Δ* under z=u^2 extends as an I_{2b} fibration U→Δ_u, and Z2 acts naturally on U such that the quotient has four ordinary double points on its central fiber.

“For a degeneration f : U → Δ with I * b monodromy, the pullback of U | Δ * under z = u 2 extends as an I 2b fibration U → Δ u , and Z 2 acts naturally on U such that the quotient has four ordinary double points on its central fiber.”

May combine two claims.

As |z_x| tends to 1, r_x is asymptotic to dist_g(z_x,z_x0) and to |log |z_x||^{3/2}.

“|z x | 1 =⇒ r x ∼ dist g (z x , z x 0 ) ∼ |log |z x || 3 2 .”

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

Δd is Fredholm of index -(b_r(T^3) + b_{r-1}(T^3)) when acting on weighted Hölder spaces of r-forms with a small enough exponential decay rate, and there are elliptic estimates for Δd on such spaces.

“Finally, we will need to know that Δ d is Fredholm of index -(b r (T 3 ) + b r-1 (T 3 )) when acting on weighted Hölder spaces of r-forms with a small enough exponential decay rate; also, there are elliptic estimates for Δ d acting on such spaces.”

May combine two claims.

For every z in Δ*, the intrinsic diameter of the fiber f^{-1}(z) and the g-length of the Euclidean circle of radius |z| in Δ* are both asymptotic to |log |z||^{1/2}.

“For every z ∈ Δ * , the intrinsic diameter of the fiber f -1 (z) and the g-length of the Euclidean circle of radius |z| in Δ * are both ∼ |log |z|| 1 2 .”

May combine two claims.

The pullback U of U|Δ* under z=u^6 can be written as (Δ*_u × C_v)/(Z+Zτ(u^6)) and extends to the whole disk Δ_u with central fiber C/(Z+Zζ3).

“The pullback U of U | Δ * under z = u 6 can be written as (Δ * u × C v )/(Z + Zτ (u 6 )) and thus extends to the whole disk Δ u with central fiber C/(Z + Zζ 3 ).”

May combine two claims.

If g is Kähler with Ric ≥ 0, then the L^2-harmonics are primitive (1,1); if g is even Kähler with respect to the complex structure of X, then H^{1,1}(M) = R[ω] ⊕ L^2-harmonics.

“If g is in addition Kähler with Ric ≥ 0, then the L 2 -harmonics are primitive (1, 1) since [Δ d , ω ∧ -] = 0 and because harmonic ED (2, 0)-forms are trivial by the Bochner formula; if g is even Kähler with respect to the complex structure of X, then H 1,1 (M ) = R[ω] ⊕ L 2 -harmonics.”

May combine two claims.

As r_x tends to infinity, |Θ(x)|^2 = 6π^2 ε^2 b^2 |k(0)|^4 |log |z_x||^{-6} (1+o(1)) and is asymptotic to r_x^{-4}.

“As r x → ∞, |Θ(x)| 2 = 6π 2 ε 2 b 2 |k(0)| 4 |log |z x || -6 (1 + o(1)) ∼ r -4 x .”

Classified as probably not a substantial claim. May combine two claims.

In the Calabi-Yau setting with dim S=1 and X admitting a holomorphic volume form Ω, Ω defines a faithful pairing ∧^m E⊗T^{1,0}S→C, and using the metric on ∧^m E induced by h_ε one obtains a Riemannian metric g_{S,ε} as before.

“Calabi-Yau setting.Let dim S = 1 and suppose that X admits a holomorphic volume form Ω. Then Ω defines a faithful pairing ∧ m E ⊗ T 1,0 S → C. Using the metric on ∧ m E induced by h ε , we obtain a Riemannian metric g S,ε as before.”

• AssumptionThe authors state this as an assumption: In the Calabi-Yau setting, dim S=1 and X admits a holomorphic volume form Ω.

May combine two claims.

Solving the Monge-Ampère equation changes the metric by O(r^{-2-δ}), by Proposition 2.9, and in actuality by no more than O(r^{-8}).

“Solving the Monge-Ampère equation changes the metric by O(r -2-δ ), by Proposition 2.9, and in actuality by no more than O(r -8 ); cf.Example 2.10.”

• cites Proposition 2.9Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Example 2.10Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

In the Gross-Wilson setting where div(Ω)=0, one replaces |z|^{-4} dz∧dz in (3.8) by |z|^{-2} dz∧dz; the semi-flat metric is incomplete, and the metric on the base has a conical singularity with angle 5/6.

“To conclude, we briefly look at the Gross-Wilson setting, where div(Ω) = 0. We only need to replace |z| -4 dz ∧ dz in (3.8) by |z| -2 dz ∧ dz.The semi-flat metric is incomplete, and the metric on the base has a conical singularity with angle 5 6 .”

May combine two claims.

A holomorphic lattice bundle Λ⊂E induces a flat R-linear connection on E, hence an integrable horizontal distribution H on X, and this is independent of the chosen fiber-preserving biholomorphism X≅E/Λ.

“Choose a fiber-preserving biholomorphism X ∼ = E/Λ for some holomorphic lattice bundle Λ ⊂ E. Then Λ induces a flat R-linear connection on E, hence an integrable horizontal distribution H on X, and this is independent of how we made X ∼ = E/Λ since any two such isomorphisms only differ by a holomorphic section S → Aut(E).”

May combine two claims.

Blowing down E turns E3 into a (-1)-curve, blowing down E3 turns E2 into a (-1)-curve, and after blowing down E2, the remaining curve E1 becomes a rational cusp curve of self-intersection zero.

“Blowing down E turns E 3 into a (-1)-curve, blowing down E 3 turns E 2 into a (-1)-curve, and after blowing down E 2 , the remaining curve E 1 becomes a rational cusp curve of self-intersection zero.”

May combine two claims.

In each case, if |B(x,1)| does not decay too fast, Moser iteration over B(x,1) gives L∞ decay, and derivative bounds follow from Schauder theory.

“In each case, if |B(x, 1)| does not decay too fast, we then apply Moser iteration over B(x, 1) to get L ∞ decay, based on Lemma 2.1 with α > 1. Derivative bounds follow from Schauder theory, in the setting of Definitions 2.6 and 2.8.”

May combine two claims.

The background manifolds (M,ω0) used in constructing the I_b spaces in Theorem 1.5(iii) are SOB(4/3) and CYL(1/3), with |B(x,1)| decaying only polynomially.

“We will see in Section 3.3 that these are SOB( 43 ) and CYL( 13 ), with |B(x, 1)| decaying only polynomially.”

May combine two claims.

The injectivity radius satisfies inj(x) ∼ |B(x,1)| ∼ r_x^{-1/3}.

“As for the injectivity radius, we have inj(x) ∼ |B(x, 1)| ∼ r -1 3 x .”

May combine two claims.

If J(0)∈C∖{0,1} and A=-1, then after base change z=u^2 the lifted fibration extends to Δ_u with a smooth central fiber, the quotient has four A1 singularities over z=0, and resolving them gives the normal form with a type I*_0 central fiber.

“If A = -1, we first make a base change z = u 2 over Δ * z .The lifted fibration U → Δ * u extends to Δ u with a smooth central fiber, D .The free Z 2 -action on U over Δ * u also extends, but with four fixed points on D .The quotient is then an elliptic fibration over Δ z , isomorphic to U over Δ * z , but with four surface singularities of type A 1 over z = 0. Resolving these singularities gives”

Probably not a substantial claim: may not pass S3 (load-bearing).

As a corollary of the proof, N in (2.2) could be replaced by the largest number of balls in a shortest ball chain connecting any two points in A.

“As a corollary of the proof, it is worth noting that N in (2.2) could be replaced by the largest number of balls in a shortest ball chain connecting any two points in A.”

May combine two claims.

Under the coordinate change z=(u/u0)^{-6}, w=(u/u0)^{-5}v, with u0:=6 4√3 ε^{-1/2}|k(0)|, u ranges over the truncated sector |u|>u0, 0<arg u<2π/6, and v belongs to a fundamental cell in {u}×C that converges to the one spanned by 1, ζ3 as |u|→∞.

“Here, u ranges over the truncated sector |u| > u 0 , 0 < arg u < 2π 6 , and v belongs to a fundamental cell in {u} × C that converges to the one spanned by 1, ζ 3 as |u| → ∞.”

The quote only partly states this claim.

For I*_b, U|Δ* = (Δ*×C_v)/(Zτ1+Zτ2) with τ1=1 and τ2=b/(πi) log u.

“Now, according to what we said about the I 2b case, U | Δ * = (Δ * × C v )/(Zτ 1 + Zτ 2 ) with τ 1 = 1 and τ 2 = b πi log u.”

May combine two claims.

For Γ=Z2, the resulting elliptic fibration has four sections, given by the (-1)-curves E^2_i and E^3_0, and two singular fibers, both of Kodaira type I*_0.

“The resulting elliptic fibration then has four sections, given by the (-1)-curves E 2 i and E 3 0 , and two singular fibers.The one at T = 0 consists of a”

• Also stated in: § 3.1. · plain

  “Both fibers are of Kodaira type I * 0 .”

May combine two claims.

The multiplicities of z•Φ along E1,E2,E3,E are 1,2,3,6, whereas the multiplicities of Φ*(dz∧dw) are 1,3,5,10.

“One checks that the multiplicities of z •Φ along these are 1, 2, 3, 6, whereas the multiplicities of Φ * (dz ∧dw) are 1, 3, 5, 10.”

May combine two claims.

On manifolds asymptotic to R+ × Y, the space of bounded harmonic r-forms modulo ED has dimension ≤ b_r(Y) + b_{r-1}(Y); equality holds for r = 2 in the setting of this paper.

“Remark 4.2.On manifolds asymptotic to R + × Y , the space of bounded harmonic r-forms modulo ED always has dimension ≤ b r (Y ) + b r-1 (Y ).It is a key point for us (and rather a coincidence) that we have equality for r = 2 in our setting.”

May combine two claims.

If β>2 is an integer and (M,ω0) is asymptotically a bundle with bounded fibers over a β-dimensional cone with radius function r, then Δ(r^{2-μ}) ≈ (2-μ)(β-μ)r^{-μ}<0 for μ∈(2,β), so |u_ε| ≤ C_μ r^{2-μ} for every μ<β if f∈C∞_0(M).

“To illustrate the barrier method, suppose that β > 2 is an integer and (M, ω 0 ) is asymptotically a bundle with bounded fibers over a β-dimensional cone with radius function r.Then Δ(r 2-μ ) ≈ (2 -μ)(β -μ)r -μ < 0 if μ ∈ (2, β), so |u ε | ≤ C μ r 2-μ for every μ < β if f ∈ C ∞ 0 (M ).”

• AssumptionThe authors state this as an assumption: In the barrier method illustration, β>2 is an integer and (M,ω0) is asymptotically a bundle with bounded fibers over a β-dimensional cone with radius function r.

May combine two claims.

Existence of a section is necessary to construct g_{sf,ε}; g_{sf,ε} as a tensor field depends on the choice of section σ; for every smooth section σ there is a vertical translation map T(x)=x+σ(f(x)), and g_{sf,ε}[σ]=T*g_{sf,ε}[σ'] if σ' is holomorphic; T preserves g_{sf,ε} iff σ is tangent to H, which implies σ is holomorphic.

“Existence of a section is necessary to construct g sf,ε , and g sf,ε as a tensor field depends on the particular choice of a section σ.However, σ also defines a complex Lie group structure on each fiber X s with unit σ(s), so for every smooth section σ there is an associated vertical translation map, T (x) = x + σ (f (x)), and g sf,ε [σ] = T * g sf,ε [σ ] if σ is holomorphic.Also, T preserves g sf”

May combine two claims.

If J(0)∈C∖{0,1} and A=1, then τ1,τ2 extend as single-valued functions on Δ, and U≅(Δ×C)/(Zτ1+Zτ2) with D smooth (I0).

“If A = 1, then τ 1 , τ 2 extend as single-valued functions on Δ, and U ∼ = (Δ × C)/(Zτ 1 + Zτ 2 ) with D smooth (I 0 ).”

May combine two claims.

inj(x) is bounded below by |B(x,1)| from [9, Theorem 4.3] because |Rm| ≤ C, and one easily finds a subset of B(x,1) with sufficient volume.

“To make this rigorous, note that inj(x) |B(x, 1)| from [9, Theorem 4.3] because |Rm| ≤ C, and one easily finds a subset of B(x, 1) with sufficient volume.”

• cites [9, Theorem 4.3]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

The topological obstruction vanishes; H^2(X|Δ*,Z) is generated by the class of the fibers and by the bad cycles C, for which <ω_sf(α), C> = 0 by computation.

“We have in fact already set things up in such a way that the topological obstruction vanishes; as we will see, H 2 (X| Δ * , Z) is generated by the class of the fibers and by the bad cycles, C, for which ω sf (α), C = 0 by computation.”

May combine two claims.

Since N=1, Ω = g dz∧dw with g(z)= z^{-2} k(z) or z^{-1} k(z), and k(0)=0, if div(Ω)=-D or 0, respectively.

“Since N = 1, we have Ω = g dz ∧ dw with g(z) = z -2 k(z) or z -1 k(z), and k(0) = 0, if div(Ω) = -D or 0, respectively.”

May combine two claims.

The semi-flat metric g_{sf,ε} and the base metric g_{S,ε} are Kähler, and the volume form of g_{sf,ε} is a constant multiple of Υ^m∧Ῡ^m or Ω∧Ω, so that Hol(X,g_{sf,ε})⊂Sp(m) or SU(m+1).

“We will see that g sf,ε and g S,ε are in fact Kähler and that the volume form of g sf,ε is a constant multiple of Υ m ∧ Ῡm or Ω ∧ Ω, so that Hol(X, g sf,ε ) ⊂ Sp(m) or SU(m + 1).”

May combine two claims.

For Γ=Z2, the 2-forms Ω0=dt∧dw and Ω∞=ds∧dw are invariant under the Z2-action; hence they lift to the rational elliptic surface with pole divisor the singular fiber at T=∞,0, respectively.

“The key fact is that both Ω 0 and Ω ∞ are invariant under the Z 2 -action; hence they lift to the rational elliptic surface with pole divisor the singular fiber at T = ∞, 0, respectively.”

Probably not a substantial claim: may not pass S4 (based).

In the hyper-Kähler setting with dim S=m and X admitting a holomorphic symplectic form Υ such that all fibers X_s are complex Lagrangians, there exists a unique Riemannian metric g_{S,ε} on S, compatible with the complex structure on S, such that the faithful pairing E⊗T^{1,0}S→C given by Υ is isometric with respect to h_ε and the Hermitian metric induced by g_{S,ε}.

“Hyper-Kähler setting.Let dim S = m and assume that X admits a holomorphic symplectic form Υ such that all X s are complex Lagrangians.Then there exists a unique Riemannian metric g S,ε on S, compatible with the complex structure on S, such that the faithful pairing E ⊗ T 1,0 S → C given by Υ is isometric with respect to h ε and the Hermitian metric induced by g S,ε .”

• AssumptionThe authors state this as an assumption: In the hyper-Kähler setting, X admits a holomorphic symplectic form Υ such that all fibers X_s are complex Lagrangians.

May combine two claims.

In the main application to Theorem 1.3, f = log((αΩ∧Ω)/ω_0^2) is in C∞_0(M), and β ≤ 2 holds.

“In our main application (Theorem 1.3), f = log((αΩ ∧ Ω)/ω 2 0 ) ∈ C ∞ 0 (M ), but indeed β ≤ 2.”

Probably not a substantial claim: may not pass S3 (load-bearing).

This recovers the optimal Neumann-Poincaré constant r^2 for annuli A(r, r+1) in R^n up to some bounded factor.

“In particular, this then recovers the optimal Neumann-Poincaré constant r 2 for annuli of the form A(r, r + 1) in R n up to some bounded factor.”

May combine two claims.

For Γ=Z3, the resulting singular fiber over T=0 has spine X=0 and three two-component ribs E^1_i∪E^2_i (Kodaira type IV*), while the fiber over T=∞ consists of L1,L2,L3 intersecting in (1:0:0)∈L3 (Kodaira type IV).

“The resulting singular fiber over T = 0 has a spine-rib structure again, with X = 0 as spine and three two-component ribs E 1 i ∪ E 2 i (Kodaira type IV * ).The fiber over T = ∞ consists of L 1 , L 2 , L 3 , intersecting in (1 : 0 : 0) ∈ L 3 (Kodaira type IV).”

May combine two claims.

In Corollary 3.4, for an elliptic fibration over a Riemann surface with a holomorphic section and no singular fibers, with Ω a holomorphic symplectic form and ω_{sf,ε} the semi-flat hyper-Kähler metric constructed from σ and Ω/√2 with fiber area ε, one has Ω=g dz∧dw and explicit formulas for ω_{sf,ε} and Γ; the Ricci form of the induced base metric is the pullback of the hyperbolic metric of Gauß曲

“Corollary 3.4.Let f : X → S be an elliptic fibration over a Riemann surface with a holomorphic section σ and no singular fibers.Let Ω be a holomorphic symplectic form on X.Let ω sf,ε be semi-flat hyper-Kähler constructed from σ, Ω/ √ 2, such that the fibers of f have area ε with respect to ω sf,ε .Thus, in particular, ω 2 sf,ε = Ω ∧ Ω.”

• Also stated in: § Lemma 3.3. · plain

  “On U , the Ricci form of the induced metric ω S,ε on the base is given by ρ(ω S,ε ) = -i∂ ∂ log Im(τ ) = i 4 dτ ∧ dτ Im(τ ) 2 , the pullback of the hyperbolic metric of Gauß curvature -2 on the upper half-plane H = {Im(τ ) > 0} under the period map τ = τ 2 τ 1 : U → H.”

May combine two claims.

T*Ω = Ω, so the integrand in ∫(ω0(α,t)^2 - αΩ∧Ω) has compact support.

“(Notice that T * Ω = Ω, so the integrand here has compact support.)”

May combine two claims.

To determine the multiplicity N of Φ*(dz∧dw) along D, it suffices to compute Ψ*Θ=2πi dz∧dw, where Θ:=dz∧(dx/y) has multiplicity zero along D; thus N=0.

“To determine the multiplicity N of Φ * (dz ∧ dw) along D, it suffices to compute that Ψ * Θ = 2πi dz ∧ dw, where Θ := dz ∧ (dx/y) obviously has multiplicity zero along D. Thus, N = 0.”

May combine two claims.

The fiber over u in Δ+ has diameter asymptotic to |log |u||^{1/2}, and the length of the half-circle of radius |u| in Δ+ is asymptotic to |u|^{-1}|log |u||^{1/2}.

“since the fiber over u ∈ Δ + has diameter ∼ |log |u|| 1/2 , and the length of the half- circle of radius |u| in Δ + is ∼ |u| -1 |log |u|| 1/2 .”

May combine two claims.

In Section 3.3, for each model fibration in Kodaira's list, one constructs an explicit isomorphism Φ: U|Δ*→(Δ*×C)/(Zτ1+Zτ2), determines the multiplicity N of Φ*(dz∧dw) along D, and obtains g(z)=z^{-N-1}k(z) or g(z)=z^{-N}k(z), k(0)=0, according to whether div(Ω)=-D or div(Ω)=0.

“For each member of this list, we then construct an explicit isomorphism Φ : U | Δ * → (Δ * × C)/(Zτ 1 + Zτ 2 ), meaning that both the map Φ and the functions τ 1 , τ 2 will be explicit.This allows us to determine the multiplicity N of Φ * (dz ∧ dw) along D, so that Φ * (dz ∧ dw) = f N h dx ∧ dy with h(p) = 0 in local coordinates (x, y) on U near each point p ∈ D. Thus, we either have g(z) = z -N -”

• cites [29] Kodaira, K., 1963, On compact analytic surfaces. II, III, Ann. of Math. (2) sourceNot checked — no full text of the source

• AssumptionThe authors state this as an assumption: In Section 3.3, D does not contain (-1)-curves and is reduced, equivalently f admits a holomorphic section, equivalently U|Δ*≅(Δ*×C)/(Zτ1+Zτ2).
• AssumptionThe authors state this as an assumption: In Section 3.3, div(Ω) is either -D (the interesting case on a rational elliptic surface) or 0 (the Gross-Wilson setting).

May combine two claims.

For the complete I*b metrics, the fiber has diameter asymptotic to (log r_x)^{1/2}, whereas the g-length of the half-circle is asymptotic to r_x, i.e. precisely linear in the distance from any fixed base point rather than sublinear.

“In particular, the fiber has diameter ∼ (log r x ) 1/2 , whereas the g-length of the half-circle is ∼ r x , i.e. precisely linear in the distance from any fixed base point rather than sublinear.”

May combine two claims.

If J(0)=0, then τ is possibly multivalued but still regular at 0, with τ(0) in the orbit of ζ3; [A] fixes τ(0), yielding six possibilities for A∈SL(2,Z) up to conjugation, including A=±1.

“If J (0) = 0, then τ is possibly multivalued, but still regular at 0, with τ (0) in the orbit of ζ 3 .[A] fixes τ (0), which in total yields six possibilities for A ∈ SL(2, Z) up to conjugation, including A = ±1.”

May combine two claims.

Conversely, every linear vector field X0 on C^2/Λ defines a unique linear growth harmonic vector field X on M, and then h = L_X g ∈ D.

“Conversely, every linear vector field X 0 on C 2 /Λ defines a unique linear growth harmonic vector field X on M , and then indeed h = L X g ∈ D because we trivially have Eh = 0, and Bh = -∇ * ∇X = 0 by the Bochner formula from Step 2.”

May combine two claims.

If the ALH coordinate system is such that t is a Busemann function, and X is a vector field with L_X g = parallel + ED, then |X| ≤ Ct and |∇_{∂t} X| ≤ C.

“Claim. Assume our ALH coordinate system is such that t is a Busemann function.If X is a vector field with L X g = parallel + ED, then |X| ≤ Ct and |∇ ∂ t X| ≤ C.”

• AssumptionThe authors state this as an assumption: The ALH coordinate system is such that t is a Busemann function.

May combine two claims.

D = Sym^2 R^4 has dimension 10 and hence overcounts the number of moduli of the flat metric g on C^2/Λ by 4; D contains a 4-dimensional space L_lin = {h ∈ D : H(Λ) = 0} of Lie derivatives of g by linear vector fields on C^2/Λ.

“(i) D has dimension 10 and hence overcounts the number of moduli of the flat metric g on C 2 /Λ by 4. This is because D contains a 4-dimensional space L lin := {h ∈ D : H(Λ) = 0} of Lie derivatives of g by linear vector fields on C 2 /Λ.”

May combine two claims.

In all the constructed cases, the underlying complex surfaces are complements of fibers in rational elliptic surfaces, and the metrics form one connected moduli space.

“In all of these cases, the underlying complex surfaces are complements of fibers in rational elliptic surfaces, and so our metrics form one connected moduli space.”

May combine two claims.

A complete Kähler manifold with |Rm| + Σ_{i=1}^{k-1} r^{iλ}|∇^i Scal| ≤ C r^{-2λ} for some k ∈ N and λ ∈ [0,1] is HMG(λ,k,α) for every α ∈ (0,1), and the maps Φ_x can be taken holomorphic.

“A complete Kähler manifold with |Rm| + k-1 i=1 r iλ |∇ i Scal| ≤ Cr -2λ for some k ∈ N and λ ∈ [0, 1] is HMG(λ, k, α) for every α ∈ (0, 1).Moreover, the maps Φ x : C m ⊃ B → M in Definition 2.6 can be taken to be holomorphic.”

• cites [53] Tian, G., 1990, Complete Kähler manifolds with zero Ricci curvature. I, J. Amer. Math. Soc. sourceNot checked — no full text of the source

May combine two claims.

All finite monodromy types can occur on rational elliptic surfaces, as well as I_b for b∈{1,...,9} and I*_b for b∈{1,...,4}.

“All finite monodromy types can occur, as well as I b for b ∈ {1, ..., 9} and I * b for b ∈ {1, ..., 4}.”

May combine two claims.

Persson [45] determined all possible configurations of singular fibers on rational elliptic surfaces; there are 279 in total, the generic one being 12×I1.

“Persson [45] has determined all possible configurations of singular fibers; there are 279 in total, the generic one being 12 × I 1 .”

• cites [45] Miranda, Rick, 1990, Persson’s list of singular fibers for a rational elliptic surface, Math. Z. sourceNot checked — no full text of the source

The quote only partly states this claim.

The negative dz∧dz components are balanced by adding tf*β with a bump form β ≥ 0 on P1 and t large.

“We balance these by adding tf * β with a bump form β ≥ 0 on P 1 , t 1.”

May combine two claims.

For a rational elliptic surface X, the natural fibration f: X → P1 is the complete linear system |-K_X|, and for each fiber D=f^{-1}(p) there exists a rational 2-form Ω on X, unique up to scale, such that div(Ω)=-D; all rational 2-forms on X arise this way.

“Consider a rational elliptic surface X as in Definition 1.1.The natural fibration f : X → P 1 has an intrinsic description as the complete linear system |-K X |, so for each fiber D = f -1 (p) there exists a rational 2-form Ω on X, which is unique up to a scale, such that div(Ω) = -D, and all rational 2-forms on X are obtained in this way.”

May combine two claims.

The I*b geometry still suffices for Proposition 2.9(ii), but M is then certainly not CYL(γ) for any γ.

“Notice that this still suffices for Proposition 2.9(ii), but M is then certainly not CYL(γ) for any γ.”

Probably not a substantial claim: may not pass S4 (based).

If E=C/Λ is a complex torus and Γ is a finite subgroup of Aut(E), then Γ is generated by a root of unity and is either Z2 (for any Λ), Z4 (for Λ=Z[i]), or Z3, Z6 (for Λ=Z[ζ3]).

“Let E = C/Λ be a complex torus and let Γ be a finite subgroup of Aut(E), so Γ is generated by a root of unity and either Γ = Z 2 (for any Λ), Γ = Z 4 (for Λ = Z[i]), or Γ = Z 3 , Z 6 (for Λ = Z[ζ 3 ]), ζ m := exp(2πi/m).”

May combine two claims.

The T2-bundles over S1 obtained by restricting U to a loop around 0 are orientable; they are flat (Bieberbach) manifolds if the monodromy is trivial or finite, nilmanifolds for I_b (b≥1), and infranil for I*_b (b≥1).

“The T 2 -bundles over S 1 obtained by restricting U to a loop around 0 are orientable.They are flat (Bieberbach) manifolds if the monodromy is trivial or finite, nilmanifolds for I b (b ≥ 1), and infranil for I * b (b ≥ 1).”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

A fiber f^{-1}(p) in a general elliptic fibration is reduced if and only if f admits a section near p.

“A fiber f -1 (p) in a general elliptic fibration is reduced if and only if f admits a section near p.”

May combine two claims.

In the ALH and Ib cases, derivatives of u decay at the same rate as uū, and in the ALG and I*b cases one gains a factor r^{-1} per derivative, by Proposition 2.9(ii).

“thus, in the ALH and I b cases, the derivatives of u decay at the same rate as uū, and we even gain a factor of r -1 per derivative in the ALG and I * b cases, by Proposition 2.9(ii).”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.

The topological type of a central fiber determines the monodromy, whose order as an element of SL(2,Z) is finite except in the I_b and I*_b cases for b≥1.

“Here, we only note that the topological type determines the monodromy, whose order as an element of SL(2, Z) is finite except in the I b and I * b cases for b ≥ 1.”

May combine two claims.

The volume growth is at most 2; as discussed after Proposition 2.4, it was necessary to take care of the integrability condition.

“The volume growth is at most 2; as discussed after Proposition 2.4, it therefore was indeed necessary to take care of the integrability condition.”

Classified as probably not a substantial claim. May combine two claims.

Theorem 1.3 follows from Propositions 3.7 and 2.4, using the SOB properties of ω_sf(α) from Section 3.3.

“Theorem 1.3 now follows from Propositions 3.7 and 2.4, using the SOB properties of ω sf (α) from Section 3.3.”

May combine two claims.

The asymptotic cones of the complete Ib and I*b metrics are R+ and R2/Z2, respectively, as for the complete I0 and I*0 metrics.

“so in particular the asymptotic cones are R + , R 2 /Z 2 , as for the complete I 0 , I * 0 metrics”

May combine two claims.

M satisfies CYL(1/3) according to Definition 2.5, as well as the diameter growth condition from Definition 2.8.

“Taken together, these properties already imply that M satisfies CYL( 13 ) according to Definition 2.5, as well as the diameter growth condition from Definition 2.8.”

May combine two claims.

In the setting of Theorem 1.3, besides deforming S^1 and T^2 separately, there are 2 more moduli to break the S^1 × T^2 splitting, and these can be realized inside every fixed Kähler class with respect to the given complex structure.

“Thus, in the setting of Theorem 1.3, besides deforming the S 1 and T 2 separately, we have 2 more moduli to break the S 1 × T 2 splitting.In fact, these can be realized inside every fixed Kähler class with respect to the given complex structure.”

May combine two claims.

The metric g on the base has positive injectivity radius, while the fibers are almost totally geodesic flat tori of constant area whose shortest closed geodesic is given by the monodromy invariant generator τ1.

“The point here is that the metric g on the base has positive injectivity radius, while the fibers are almost totally geodesic flat tori of constant area whose shortest closed geodesic is given by the monodromy invariant generator τ 1 .”

Probably not a substantial claim: may not pass S3 (load-bearing).

Example 3.1 explains why the complete cone angle when removing one fiber equals the incomplete cone angle when removing its dual.

“Example 3.1 also explains why the complete cone angle when removing one fiber is equal to the incomplete cone angle when removing its dual.”

• cites Example 3.1Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S3 (load-bearing).

The calculation of |Θ(x)|^2 was performed on a computer, proceeding from (3.16).

“This was calculated on a computer, proceeding from (3.16).”

Classified as probably not a substantial claim.

Proposition 3.7 follows immediately from what is established in Claim 4.

“Proposition 3.7 follows immediately from what is established in Claim 4.”

Classified as probably not a substantial claim.

The asymptotics in Theorem 1.5 follow from Proposition 2.9.

“The asymptotics in Theorem 1.5 follow from Proposition 2.9.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.

All finite monodromies can be treated in this way and lead to semi-flat metrics with similar properties.

“All finite monodromies can be treated in this way, and lead to semi-flat metrics with similar properties.”