## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### Infinitesimal locally trivial deformation spaces of compact complex surfaces with ordinary singularities

Shoji Tsuboi

#### Abstract

Let $S$ be a compact complex surface with ordinary singularities. We denote by $\Theta_S$ the sheaf of germs of holomorphic tangent vector fields on $S$. In this paper we shall give a description of the cohomology $H^1(S, \Theta_S)$, which is called the infinitesimal locally trivial deformation space of $S$, using a 2-cubic hyper-resolution of $S$ in the sense of F. Guillén, V. Navarro Aznar et al. ([1]). As a by-product, we shall show that the natural homomorphisim $H^1(S, \Theta_S)\rightarrow H^1(X, \Theta_X(-\log D_X))$ is injective under some condition, where $X$ is the (non-singular) normal model of $S$, $D_X$ the inverse image of the double curve $D_S$ of $S$ by the normalization map $f\colon X\rightarrow S$, and $\Theta_X(-\log D_X)$ the sheaf of germs of logarithmic tangent vector fields along $D_X$ on $X$. Note that the cohomology $H^1(X, \Theta_X(-\log D_X))$ is nothing but the infinitesimal locally trivial deformation space of a pair $(X, D_X)$.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 75, Number 7 (1999), 99-102.

Dates
First available in Project Euclid: 23 May 2006

https://projecteuclid.org/euclid.pja/1148393857

Digital Object Identifier
doi:10.3792/pjaa.75.99

Mathematical Reviews number (MathSciNet)
MR1729852

Zentralblatt MATH identifier
0964.32026

#### Citation

Tsuboi, Shoji. Infinitesimal locally trivial deformation spaces of compact complex surfaces with ordinary singularities. Proc. Japan Acad. Ser. A Math. Sci. 75 (1999), no. 7, 99--102. doi:10.3792/pjaa.75.99. https://projecteuclid.org/euclid.pja/1148393857

#### References

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