Abstract
We prove the Lipman–Zariski conjecture for complex surface singularities with$p_{g}-g-b\leqslant 2$. Here$p_{g}$is the geometric genus,$g$is the sum of the genera of exceptional curves and$b$is the first Betti number of the dual graph. This improves on a previous result of the second author. As an application, we show that a compact complex surface with a locally free tangent sheaf is smooth as soon as it admits two generically linearly independent twisted vector fields and its canonical sheaf has at most two global sections.
Highlights
The Lipman–Zariski conjecture asserts that a complex algebraic variety X with a locally free tangent sheaf TX is necessarily smooth
By the combined work of Lipman [Lip[65], Theorem 3], Becker [Bec[78], Section 8, page 519] and Flenner [Fle[88], Corollary], it is known that it suffices to prove the conjecture for normal surface singularities
In a previous paper [Gra19], the second author dealt with surface singularities that are ‘not too far’ from being rational
Summary
The Lipman–Zariski conjecture asserts that a complex algebraic variety (or complex space) X with a locally free tangent sheaf TX is necessarily smooth. In [Gra19], the second author used his (local) main result to study compact complex surfaces whose tangent sheaf satisfies some global triviality properties. This result generalizes [Gra[19], Corollary 1.2], where X was assumed to be almost homogeneous. Note that this is nothing but the special case where both Li ∼= OX. A more interesting example would be a primary Kodaira surface, or more generally any elliptic fibre bundle S → C that is not topologically trivial In this case, H2(S, R) can be arbitrarily large (depending on C), but φ∗ : H2(C, R) → H2(S, R) always is the zero map [BHPV04, Proposition V.5.3]. The proof of Corollary 2 shows the following: Assume that for some integer C, we knew the Lipman–Zariski conjecture for surface singularities satisfying pg − g − b C. The additional assumption in Corollary 2 can be weakened to “dimC H0(X, ωX ) C”
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