By John von Neumann

Measures and integrals

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**Additional info for Functional Operators.The geometry of orthogonal spaces**

**Example text**

A basic result in commutative algebra tells us, that any prime ideal of height one in a UFD is principal. Therefore, p = (/) for some irreducible / € Oc,o- In other words, any irreducible analytic germ of codimension one is defined by one irreducible holomorphic function. More generally, any analytic germ of codimension one is the zero set of a single holomorphic function. In the theory of functions of one variable a meromorphic function / on an open subset U C C is a holomorphic function defined on the complement of a discrete set of points S C U such that / has poles of finite order in all points of S.

E. Aa is uniquely determined by the condition (Aa, j3) = (a, L/3) for all /? € / \ V*. The C-linear extension /\* V£ —> /\* V^? of the dual Lefschetz operator will also be denoted by A. 3). Thus, the Hodge ^-operator is well-defined. Using an orthonormal basis xi,j/i = I(x\),... ,xn,yn = I(xn) as above, a straightforward calculation yields n\ • ujn = vol, where ui is the associated fundamental form. 9 for a far reaching generalization of this. e. A{/\ V*) C /\ ~ V*. Moreover, one has A = *~1 o L o *.

Az t Thus, / is holomorphic if and only if df = 0. 10) the operators d and d can be expressed explicitly as follows: d(fdzilA.. AdzipAdSj1 .. Ad2jq) = V^ --—dz^Adz^A.. AdzipAdzj1 .. fe=i k k 44 1 Local Theory and n fit B(fdzi1A.. AdzipAd2j1 .. Adzjq) = \^ -—dziAdz^A.. AdzipAd2j1 .. AdEjq. 6 For the differential operators 8 and B one has: i) d = 8 + B. ii) d2 = B2 = 0 and 3d = -3d. e. d{a Af3) = d(a) A /3 + (-l)p+qa B(a A S3) = 3(a) Af3 + (-l)p+qa A d{(3) A 3{(3) for a G Ap'q{U) and S3 G Ar's{U).

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Categories: Algebraic Geometry