### ag.algebraic geometry - Extension of Kollár's vanishing theorem to singular varieties?

A theorem by Kollár asserts that if $X$ and $Y$ are projective varieties with $X$ smooth, and $f : X \to Y$ is a surjective map, then the higher direct images $R^if_*\omega_X$ vanish for $i$ greater than the generic fiber dimension. I'd like to know if one can weaken the assumption that $X$ is smooth, for example, to $X$ having quotient singularities, or something even weaker (whereby the sheaf $\omega_X$ represents an appropriate dualizing sheaf in such a case). Kollár's original proof uses the fact that $X$ is smooth to prove that the higher ...Read more

### ag.algebraic geometry - Is SL_n/S(GL_k x GL_n-k) symmetric?

Background: a symmetric variety is a homogeneous space $G/H$ associated to an involution $\theta$ of a semisimple algebraic group $G$ and $\{g | \theta(g) = g\} = G^\theta \subset H \subset N_G(G^\theta) = \{h| hG^\theta h^{-1} = G^\theta \}$.A spherical variety is a homogeneous space $G/H$ which contains an open orbit under the action of a Borel subgroup $B$. It is known that symmetric varieties give examples of spherical varieties.Let $G = SL_n(\mathbb{C})$ and $H = S(GL_k \times GL_{n-k}) = \{(g,g') | \det g \det g' = 1 \}$. It is known that...Read more

### ag.algebraic geometry - Is an algebraic space group always a scheme?

Suppose G is a group object in the category of algebraic spaces (over a field, if you like, or even over ℂ if you really want). Is G necessarily a scheme?My feeling is that the answer is "yes" because an algebraic space group which is not a scheme would be too awesome. Any group homomorphism from such a G to an algebraic group (a scheme group) would have to have infinite kernel since an algebraic space which is quasi-finite over a scheme is itself a scheme. In particular, G would have no faithful representations or faithful actions on pro...Read more

### ag.algebraic geometry - Moduli space of semistable bundles

It is well-known that the space of $S$-equivalence classes of rank 2 semistable holomorphic vector bundles with trivial determinant on a genus 2 Riemann surface $M$ is $CP^3$ (more concretely $PH^0(Jac(M),L(2\theta)$). Especially, the points corresponding to semistable (and not stable) bundles are smooth points. On the open dense subspace consisting of points corresponding to stable bundles, there is a natural symplectic structure, compatible with the natural Riemannian metric. It defines a Kaehler structure. It should be true that this Kaehler...Read more