Unobstructedness of deformations for a d-semistable central corank one boundary point

Abstract In this paper, we prove the unobstructedness of the functor of log smooth deformations for an irreducible type II degeneration of complex abelian surfaces — shifted gluing of a $$\mathbb {P}^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> </mml:math> -bundle over an elliptic curve along two disjoint sections — using the $$T^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>T</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> -lifting technique and the degeneration on the first page of a Hodge–de Rham like spectral sequence for the sheaf of torsion-free differentials. From this result and by comparing the log smooth and ordinary flat deformations at first order of such surfaces, we deduce that the functors of locally trivial and flat deformations of such surfaces are unobstructed; we deduce a smoothability result for such surfaces as well.