ERRATUM/CORRIGENDUM, OCTOBER 2020, FOR ’A BOGOMOLOV UNOBSTRUCTEDNESS THEOREM FOR LOG-SYMPLECTIC MANIFOLDS IN GENERAL POSITION’ (J. INST. MATH. JUSSIEU 19 (2018), 1509–1519)

Abstract The general position hypothesis needs strengthening.

I am grateful to Brent Pym (personal communication) for showing me by an example due jointly to him, Mykola Matviichuk and Travis Schedler -and related to their article in arxiv:math 201008692 -that the general position hypothesis in the paper is not strong enough. As their example shows, it is possible even with the 2-general position hypothesis for the inclusion of the log complex to the log-plus complex to not induce a surjection on local and even global cohomology, and for the Poisson structure to admit deformations where the Pfaffian divisor deforms non locally-trivially.
We are thus led to introduce a stronger general position condition that we call 'very general position' on a Poisson structure. Throughout this note, we keep the notations and assumptions of the original paper.
In other words, for any set of integers r 1 ,...,r m , not all zero, we have The following is a partial substitute for Lemma 4 of the original paper: Proof. Continuing with the notations of the paper, let us further set The complex denoted Q I is generated by subgroups with the differential where the first summand acts on the dlog x I factor where d acts solely on the O DI factor. Note that the two summands commute. LetÔ DI be the formal completion of O DI at the maximal ideal m of the origin. As usual, the flatness ofÔ DI over O DI yields that it will suffice to prove that the formal completion Q I ⊗Ô DI is exact in degrees < t -that is, that the sequence is exact in the middle in degrees < t. Now assume temporarily that the matrix B = (b ij ) is constant -that is, Φ has constant coefficients in the dlog(x i ), and hence so do the forms Note that thanks to our t-very general hypothesis, η is a 'primitive' or cotorsion-free element of Q 1 I -that is, it maps to a nonzero element of the fibre of Q 1 I at the origin. As is well known and easy to prove (e.g., by making η part of a basis), the wedge product with such an element η yields an exact complex ∧ • Q 1,M In the general case we use semicontinuity: let α s be multiplication by s ∈ C defined locally near the origin, and set Φ s = α * s Φ. For s = 0, Φ s has constant coefficients, hence the corresponding complex is exact in degree < t. Then by semicontinuity, the corresponding Erratum/Corrigendum 277 complex remains exact in degrees < t for all small enough s. But since the complexes are equivalent for all s = 0, it follows that the original complex is exact in degrees < t.
As in the original paper, we deduce the following: Theorem (Theorem 8 corrected). The conclusions of Theorem 8 hold under the additional hypothesis that (X,Π) is in 2-very general position.
Corollaries 9 and 10 in the paper should be similarly corrected to replace 2-general position by 2-very general position.
The following example, communicated by B. Pym, is due to M. Matviichuk, B. Pym and T. Schedler and is related to their paper in arxiv 2010.08692, though not explicitly contained in it.

Example 2 (Matviichuk, Pym and Schedler). Consider the matrix
and the corresponding log-symplectic form on C 4 , Φ = i<j b ij dzi zi ∧ dzj zj and corresponding Poisson structure Π = Φ −1 , both of which extend to P 4 with Pfaffian divisor D = (z 0 z 1 z 2 z 3 z 4 ), z 0 = hyperplane at infinity. Then Π admits the first-order Poisson deformation with bivector z 3 z 4 ∂ z1 ∂ z2 , which in fact extends to a Poisson deformation of P 4 ,Π over the affine line C, and the Pfaffian divisor deforms as (z 3 z 4 z 0 (z 1 z 2 − tz 3 z 4 )), hence non locally-trivially. Correspondingly, the log-plus form z 3 z 4 φ 1 φ 2 is closed (and not exact). Then d(z 3 z 4 φ 1 φ 2 ) = 0 corresponds to the integral column relation k 1 − k 2 + (e 1 + e 2 ) − (e 3 + e 4 ) = 0, where the k i and e j are the columns of the B matrix and the identity, respectively, showing that Π, although 2-general is not 2-very general.
Matviichuk, Pym and Schedler (arxiv:math 201008692) consider deformations of the 'generalised geometry', in Hitchin's sense, associated to (X,Π) and give an analogue of the theorem in this note for the corresponding deformations (see their Remark 5.24).