15 problems

Let $f: X \to D$ be a Kähler deformation of a Kähler manifold $X_{o}$ over $D$, and identify $\pi_{1}(X)$ with $\pi_{1}(X_{o})$. Let $H \subset \pi_{1}(X)$ be a subgroup for which…

Let $X$ be a compact Kähler manifold whose universal covering is Stein. The large-fundamental-group conjecture. Small Kähler deformations of $X$ have large fundamental group. This…

Let $X$ be a Kähler manifold and let $\rho: \pi_{1}(X) \to GL(n, {\mathbb C})$ be an infinite linear representation. The deformation-stability conjecture. Having a generically larg…

Let $A$ be a domain equipped with an injective well-ordered valuation $\nu$, and let $gr A$ denote the associated graded algebra of the filtration on $A$ induced by $\nu$. A family…

No coalescence conjecture. The coalescence problem does not occur: a one-parameter deformation $f_t$ cannot have more than one instability and satisfy

Contractible stable perturbation conjecture. There are such germs for which

Houston's conjecture. Excellency in families can be controlled by means of the image Milnor number.

Let $(X,L)$ be a polarized complex variety, and let $\eta\in H^1(X,\mathcal{E}_L)$ be a deformation class of $(X,L)$. A test configuration is compatible with $\eta$ if its first-or…

Let $X$ be a conic affine symplectic variety carrying a compatible hyperkähler cone metric on its smooth part $X_{reg}$, and suppose that $X$ admits a symplectic resolution…

Let the moduli spaces under consideration be moduli spaces of low-dimensional complex Lie and associative algebras, and let a jump deformation be a deformation in which a generic a…

Let a family of nilpotent algebras mean a family in the moduli space of complex Lie algebras whose members are nilpotent, and similarly for a family of solvable algebras. Nilpotent…

Generic Milnor number sequence conjecture.

Generic Milnor number conjecture. For generic homogeneous singularities of degree $d$,

Let $f$ be an isolated hypersurface singularity and consider a deformation of $f$ that is $\mu$-constant, meaning that the Milnor number $\mu$ remains constant throughout the defor…

Let $\pi:X^\mathrm{tot}\to S$ be a homogeneous deformation with special fiber determined by a polyhedral subdivision $\Xi_0$, and let $s\in S$. Write…