# Hodge conjecture

+ 1 like - 0 dislike
784 views

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.

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOv$\varnothing$rflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). Please complete the anti-spam verification