Narrow Results

Let $X$ be an abelian surface, and $D$ be the sum of $N(\ge 2)$ distinct theta divisors having normal crossings. We set $M=X-D$. We study the structure of the nonvanishing twisted cohomology group $H^2(M,{ℒ})$, where ${ℒ}$ denotes a locally constant sheaf over $M$ defined by a multiplicative meromorphic function on $M$ infinitely ramified just along the divisor $D$ (as will be seen below, we will take as such a function a product of complex powers of theta functions). The de Rham complex on $X$ with logarithmic poles along $D$, associated to the twisted cohomology groups $H^p(M,{ℒ})$, is $P$-valued, where $P$ denotes a topologically trivial (i.e. Chern class zero) line bundle over $X$ determined by the locally constant sheaf ${ℒ}$. Therefore the main results of this paper, which give us information on the order of poles of meromorphic 2-forms on $X$ generating the cohomology group $H^2(M,{ℒ})$, are divided into Theorems 4.5 and 4.6, according as the de Rham complex on $X$ with logarithmic poles along $D$ takes the values in a holomorphically nontrivial line bundle $P\ne 1$ or a holomorphically trivial one $P=1$ ($1$ denoting the holomorphically trivial line bundle $C\times X$). Such a phenomenon does not occur in the case of the twisted cohomology of complex projective space with hyperplane arrangement.

We show that the fundamental groups of normal complex algebraic varieties share many properties of the fundamental groups of smooth varieties. The jump loci of rank one local systems on a normal variety are related to the jump loci of a resolution and of a smoothing of this variety.