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Let $M$ be a projective nonsingular complex variety. Let $I$ and $J$ be two complex structures on M. We then have the corresponding Kälher classes $\omega_I$ and $\omega_J$ in $H^2(M, \mathbb{R})$, an …

In Weil II, Deligne proves that the six functors preserve the category of mixed $l$-adic complexes. Most difficulties are encountered when proving that if $f\colon X\to Y$ is a finite type separated m …

Let $X$ and $Y$ be two algebraic varieties over $\mathbb{C}$ and $f\colon X\rightarrow Y$ be a proper map. Assume that $Y$ is smooth. I am interested in sufficient and necessary conditions for $f$ to …

For a variety over a finite field Deligne define weights by looking at the eigenvalues of the Frobenius. On the other hand, if we take a variety over $\mathbb{Q}$, at least for its constant sheaf we h …

Let $\mathcal{E}$ be a projective bundle of rank $r$ over a smooth complex quasi projective variety $B$, and form its associated projective bundle $\chi :=\mathbb{P}(\mathcal{E})$. Let $\pi : \chi \ri …