Logarithmic Projective Connections

Logarithmic Heat Projective Operators

Let C → S be a proper flat family of stable curves, smooth over S0 = S r ∆. One can associate a flat family f : SUC(r, d) → S0 of moduli spaces SUCs(r, d) of semistable vector bundles of rank r and degree d with fixed determinant over the curves Cs, and also a line bundle Θ on SUC(r.d) such that its restriction to each fibre is the line bundle on SUCs(r, d) defined by the theta divisors. It is ...

Flat Projective Connections

Complete Projective Connections

The first examples of complete projective connections are uncovered: normal projective connections on surfaces whose geodesics are all closed and embedded are complete, as are normal projective connections induced from complete affine connections with slowly decaying positive Ricci curvature. Moreover, I explain the relationship between projective and affine connections.

Moduli of Semistable Logarithmic Connections

Under the Riemann Hilbert correspondence, which is an equivalence of categories, local systems on a nonsingular complex projective variety X correspond to pairs E = (g' , V) where g' is a locally free sheaf on X , and V: g' -t n ~ 0g' is an integrable connection on g'. Generalizing this correspondence to noncomplete quasi-projective varieties, Oeligne (see [0]) showed that the correct algebro-g...

On the logarithmic comparison theorem for integrable logarithmic connections

LetX be a complex analytic manifold, D ⊂ X a Koszul free divisor with jacobian ideal of linear type (e.g. a locally quasi-homogeneous free divisor), j : U = X −D →֒ X the corresponding open inclusion, E an integrable logarithmic connection with respect to D and L the local system of the horizontal sections of E on U . In this paper we prove that the canonical morphisms ΩX(logD)(E(kD)) −→ Rj∗L, j...