9 problems

Corti–Hanamura's motivic decomposition conjecture. Every such morphism admits a motivic decomposition.

Lagrangian-fibration conjecture. The morphism $f$ admits a multiplicative decomposition theorem

Let $\pi_M:(M,\sigma)\to B$ be an integrable system, with partial Fourier–Mukai transform $\phi_{\mathrm{FM}}$, sheaf of Kähler differentials $\Omega_M^k$, and decomposition-theore…

The decomposition theorem concerns the decomposition of a direct image of an intersection cohomology complex into shifted intersection cohomology complexes. Strong decomposition-th…

K-theoretic decomposition conjecture.

Let $K/\mathbf{Q}_p$ be a finite extension, and let $f:X\to Y$ be a projective map of rigid spaces over $K$ with $Y$ qcqs. Let $\mathrm{IC}_{X,\mathbf{Q}_\ell}$ be the intersection…

Let $f:X\to S$ be the projective morphism in the paper, let ${^p\!R}^i f_*\mathbb Q_\ell$ denote its perverse direct-image complex in degree $i$, and let…

Let $X$ be a smooth complex algebraic variety, and let $\mathcal{M}_B(X)$ be the moduli space of semisimple local systems of all ranks. An absolute $K$-constructible set is a subse…

Let $f:X\to Y$ be a proper algebraic map, and let $\mathcal P$ be a simple perverse sheaf on $X$. The decomposition theorem concerns the decomposition of $Rf_*\mathcal P$ into shif…