# Algebraic geometric measure theory

Suppose I have $V\subset \mathbb{C}^n$ be the zero set of a polynomial $P(z_1, \dotsc, z_n),$ with bounded height of coefficients (where height is, to fix something, $|\log|a||$) and degree $d.$ Suppose I now have a ball $B=B(z_0, r) \subseteq \mathbb{C}^n.$ Is there an upper bound on $2n-2$ dimensional measure of $B\cap V?$

EDIT A quasi-answer: Wirtinger's formula (see Griffiths and Harris, p. 31) seems to indicate that the Fubini-Study volume of a $k$-dimensional sub variety $V$ of $\mathbb{P}^n$ equals $\deg(V) \mathrm{vol}(\mathbb{P}^k).$ For real algebraic varieties, there seems to be only a Cauchy-Crofton derived inequality, as suggested in the answer.

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sorry, what is a height of coefficient? –  Fedor Petrov Aug 5 '11 at 17:42
I actually define it in the question in parentheses as the absolute value of the log of the modulus -- I am not sure if this is the most natural definition... –  Igor Rivin Aug 5 '11 at 18:14

There is an explicit upper bound based on a 2-d version of the Crofton formula. Namely, the area of $B \cap V$ is the integral of the number of points of intersection $W \cap (B \cap V)$ over the space of all affine 2-planes $W \subseteq \mathbb{R}^{2n}$. Since the real algebraic variety $V$ has degree $\leq d^2$ the number of points of intersection is at most $d^2$. So an upper bound is $d^2$ times the measure of the space of affine $2$-planes meeting $B$. It seems to me that, unless I have misunderstood, the bound on the coefficients is unnecssary.
Well, it seems to me that the fundamental difference is not between real and complex varieties, but between affine and projective ones. Do you know of a relation between the volumes of intersection $B \cap V$ as in your question and the FS volume of the projective closure of $V$? –  S123 Aug 7 '11 at 3:37