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

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

Let $G$ be a complex simple reductive group. Then the set of isomorphy classes $Irr G$ is isomorphic to the set of dominant weights $\Lambda_+$ in the weight lattice of the maximal torus of a Borel subgroup of $G$. Further, $\Lambda_+$ is isomorphic to $\mathbb{N}_0^\ell$ for some integer $\ell$. Thus we can write $\rho = n_1\lambda_1 + ... + n_{\ell}\lambda_{\ell} = \underline{n}$ for any irreducible representation $\rho\colon G \to End(V_{\rho})$.Now let $R$ be a ring with $G$-action, $M$ an $R$-$G$-module with isotypic decomposition $M \cong...Read more

I found some absurd observation which I could not fix by myself. For an elliptic curve $E$ over $\mathbb{Q}$, let $\overline{E}=E\otimes\overline{\mathbb{Q}}$. Every multiplication-by-$n$ map $\overline{E}\to \overline{E}:P\mapsto nP$ defines an étale covering of $E$. It means that for any $n$ and $n$-torsion point $P$, the morphism $\mathcal{O}_{\overline{E},O}\to\mathcal{O}_{\overline{E},P}$ sends a local parameter $\varpi_O$ at $O$ to $u_P\cdot\varpi_P$ where $u_P$ is a unit in $\mathcal{O}_{\overline{E},P}$ and $\varpi_P$ is a local paramet...Read more

There is elliptic curve $C (y^2 = x^3 + Ax + B)$ over $GF(q)$. There is algebraic function f on C. We have div(f). How calculate f as rational function ( $f = (f_1(x) + yf_2(x)) / (g_1(x) + yg_2(x))$)?What is bound $deg(f_i)$ and $deg(g_i)$ in depend of $deg(f)$?...Read more

Let $X\rightarrow T$ be a fibre bundles with smooth projective fibre $F$ and $X$ and $T$ are also smooth. Let $D$ is relative effective Weil divisor. Suppose $W_1 $ and $W_2$ are relative subvarieties which are isomorphic(by a relative map say $\phi$). Let $L:=\mathcal{O}(D)$. Let $L_t|_{W_{1,t}}\cong L_t|_{W_{2,t}}$(via $\phi_t$). Is it true that $L|_{W_{1}}\cong L|_{W_{2}}$ (via $\phi$)?...Read more

In an article of Robert Friedman, I came up with a comment:There are finitely many deformation types of Calabi-Yau threefolds for a given diffeomorhpic type if $b_2 =1$.And it is said that this is a special case of a result due to Kollar.(p. 113, Friedman, Robert{On threefolds with trivial canonical bundle}. Complex geometry and Lie theory)Can anyone explain why or give some references for it?...Read more

The question is related to this MO question. From the answer of the above question, we know T. Shioda in "Some remarks on Abelian varieties" found counter-examples of the "cancellation law" of abelian varieties. From the mathscinet review I found that in particular Shioda found elliptic curves $E$, $E^{\prime}$ and $E^{\prime\prime}$ such that $E\times E^{\prime\prime}$ is isomorphic to $E^{\prime}\times E^{\prime\prime}$ but $E$ is not isomorphic to $E^{\prime}$.I don't have the access to the above paper. Is there anyone who has some ideas on ...Read more

Let $E$ be a smooth elliptic curve over a field $k$. Let $$i : E \to \mathbb{P}^2 \times \mathbb{P}^2,$$be an embedding. How one can find an explicit canonical forms of equations cutting $E$ in $\mathbb{P}^2 \times \mathbb{P}^2$? Are there analogs of Weierstrass, Hesse and Legendre forms of equations?...Read more

as a warm up for his thesis I would like a student of mine to read something on parabolic bundles. He is reading the famous Atiyah paper on vector bundles on elliptic curves, so I think it would be nice if he could proceed studying parabolics on elliptic curve. Do you know a specific reference? If not, what would you suggest?...Read more

Where can I find a proof of the following fact:If $X_1$ and $X_2$ are subvarieties of $\mathbb{G}(k,n)$ of codimension $c_1$ and $c_2$ satisfying $c_1+c_2<n+1-2k$, then the intersection $X_1\cap X_2$ is nonempty.(This is Lemma 3.13 in the paper Hypersurfaces of Low Degree, and it was said that it is a standard fact and stated without proof or reference.)...Read more

For rings such as $\mathbb{Z}[x,y]$ is there software to compute any of:1.) The integral closure of $\mathbb{Z}[x,y]/(f)$. de Jong has a very general algorithm that works in this context (http://arxiv.org/abs/alg-geom/9704017)2.) The radical of an ideal $I$.3.) Minimal primes of an ideal/the primary decomposition.4.) The dimension of $\mathbb{Z}[x,y]/I$5.) Any invariants of a singularity on such a scheme.Over fields $\mathbb{Q}, \mathbb{F}_p$ I think Sage, Macaulay 2 etc., implement 1-4. Are 1-4 difficult to compute for such rings, or are there...Read more

in continuation of my previous post:Question Regarding Riemann-Hurwitz Formula ProofI need another help in understanding how to compute Cohomology of Sheaves (and right derived functors):Given a compact Riemann Surface $X$ , and the sheaf $O_X(U):= \{f:U \to \mathbb{C} |f -holomorphic \} $ , I want to compute the cohomology groups $H^n (X, O_X) = R^n \Gamma (O_X) $ where $\Gamma$ is the global section functor: $ \Gamma(O_X) = O_X(X) $ . As far as I know, this computation should give: $H^0 (X,O_X) = \mathbb{C} $ , $H^1(X,O_X) = \mathbb{C} ^ {2g}...Read more

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

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