The logarithmic Bogomolov–Tian–Todorov theorem

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...

A Characterisation Theorem of the Logarithmic

The original idea of such characterisations is that of [2]. He proved that if a real-valued additive function is non-decreasing, or satisfies f(n + 1) − f(n) → 0 (as n → ∞), then it must have the form c log n for some constant c. He had a separate argument for each case. In [1] Elliott solves the problem in a generalised form: Let a > 0, b, A > 0 and B be integers with ∆ = aB − Ab non-zero. If ...

Linear Free Divisors and the Global Logarithmic Comparison Theorem

A complex hypersurface D in Cn is a linear free divisor (LFD) if its module of logarithmic vector elds has a global basis of linear vector elds. We classify all LFDs for n at most 4. By analogy with Grothendieck's comparison theorem, we say that the global logarithmic comparison theorem (GLCT) holds for D if the complex of global logarithmic di erential forms computes the complex cohomology of ...

Testing the Logarithmic Comparison Theorem for Free Divisors

A vanishing theorem for a class of logarithmic D-modules

Let OX (resp. DX) be the sheaf of holomorphic functions (resp. the sheaf of linear differential operators with holomorphic coefficients) on X = C. Let D ⊂ X be a locally weakly quasi-homogeneous free divisor defined by a polynomial f . In this paper we prove that, locally, the annihilating ideal of 1/f over DX is generated by linear differential operators of order 1 (for k big enough). For this...