# Hodge conjecture

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Let $M$ be a projectiv variety with a rational $(p,p)$ cycle $c$, then :

$$c=\sum_i q_i [M_i]$$

with $M_i$, $(p,p)$ sub-varieties and $q_i$, rational numbers. We try to show an homotopy with algebraic sub-varieties $M_i \cong A_i$. At this aim, we construct a flow over sub-varieties $X$:

$$\frac{\partial X}{\partial t}= - grad (F)_X$$

with $F$ the following functional:

$$F(X)=\int_X ||J^*||^2$$

where $J^*$ is the non-diagonal part of the complex structure $J$ when we decompose the tangent space following the normal and tangent bundle of $X$.

We have :

$F(X)=0$ iff $J^*=0$ iff $X$ is complex iff $X$ is algebraic (following the GAGA theorem).

All the difficulty is to show that the flow is well defined and converges to an algebraic sub-variety, the flow will have certainly singularities which will have to be studied.

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