FROM COMPLETE TO PARTIAL FLAGS IN GEOMETRIC EXTENSION ALGEBRAS

A geometric extension algebra is an extension algebra of a semi-simple perverse sheaf (allowing shifts), e.g., a push-forward of the constant sheaf under a projective map. Particular nice situations arise for collapsings of homogeneous vector bundles over homogeneous spaces. In this paper, we study the relationship between partial flag and complete flag cases. Our main result is that the locally finite modules over the geometric extension algebras are related by a recollement. As examples, we investigate parabolic affine nil Hecke algebras, geometric extension algebras associated with parabolic Springer maps and an example of Reineke of a parabolic quiver-graded Hecke algebra.

Twisted cohomology of the complement of theta divisors in an abelian surface

Let [Formula: see text] be an abelian surface, and [Formula: see text] be the sum of [Formula: see text] distinct theta divisors having normal crossings. We set [Formula: see text]. We study the structure of the nonvanishing twisted cohomology group [Formula: see text], where [Formula: see text] denotes a locally constant sheaf over [Formula: see text] defined by a multiplicative meromorphic function on [Formula: see text] infinitely ramified just along the divisor [Formula: see text] (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 [Formula: see text] with logarithmic poles along [Formula: see text], associated to the twisted cohomology groups [Formula: see text], is [Formula: see text]-valued, where [Formula: see text] denotes a topologically trivial (i.e. Chern class zero) line bundle over [Formula: see text] determined by the locally constant sheaf [Formula: see text]. Therefore the main results of this paper, which give us information on the order of poles of meromorphic 2-forms on [Formula: see text] generating the cohomology group [Formula: see text], are divided into Theorems 4.5 and 4.6, according as the de Rham complex on [Formula: see text] with logarithmic poles along [Formula: see text] takes the values in a holomorphically nontrivial line bundle [Formula: see text] or a holomorphically trivial one [Formula: see text] ([Formula: see text] denoting the holomorphically trivial line bundle [Formula: see text]). Such a phenomenon does not occur in the case of the twisted cohomology of complex projective space with hyperplane arrangement.