The Virasoro conjecture for smooth complete intersections in projective space (and its transfer to projective bundles)

Principal theorem: for a smooth complete intersection $X\subset\mathbb P^N_{\mathbb C}$, $L_k^XZ_X=0$ for all $k\ge-1$ in Getzler's first-Hodge-degree superspace convention, coefficientwise in every genus and curve class, with arbitrary insertions from $H^*(X;\mathbb C)$. Companion: if the full constraints hold for a smooth connected projective base $B$, they hold for $\mathbb P_B(E)$ for any algebraic vector bundle $E$ of rank at least 2 (no splitting or positivity), hence for projective-bundle towers. The conjecture for general smooth projective varieties (for example general hypersurfaces in other ambients, or varieties with no degeneration to known pieces) remains open.

Principal theorem: for a smooth complete intersection $X\subset\mathbb P^N_{\mathbb C}$, $L_k^XZ_X=0$ for all $k\ge-1$ in Getzler's first-Hodge-degree superspace convention, coefficientwise in every genus and curve class, with arbitrary insertions from $H^*(X;\mathbb C)$. Companion: if the full constraints hold for a smooth connected projective base $B$, they hold for $\mathbb P_B(E)$ for any algebraic vector bundle $E$ of rank at least 2 (no splitting or positivity), hence for projective-bundle towers. The conjecture for general smooth projective varieties (for example general hypersurfaces in other ambients, or varieties with no degeneration to known pieces) remains open.