The Dolbeault operator on Hermitian spin surfaces

Alexandrov, Bogdan; Grantcharov, Gueo; Ivanov, Stefan

The Dolbeault operator on Hermitian spin surfaces

Authors

Alexandrov, Bogdan

Grantcharov, Gueo

Ivanov, Stefan

Description

We consider the Dolbeault operator of $K^{1/2}$ -- the square root of the canonical line bundle which determines the spin structure of a compact Hermitian spin surface (M,g,J). We prove that the Dolbeault cohomology groups of $K^{1/2}$ vanish if the scalar curvature of g is non-negative and non-identically zero. Moreover, we estimate the first eigenvalue of the Dolbeault operator when the conformal scalar curvature k is non-negative and when k is positive. In the first case we give a complete list of limiting manifolds and in the second one we give non-Kähler examples of limiting manifolds.
11 pages, Latex format, no figures