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Register a new userDeformations of minimal cohomology classes on abelian varieties
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We show that the infinitesimal deformations of the Brill--Noether locus $W_d$ attached to a smooth non-hyperelliptic curve $C$ are in one-to-one correspondence with the deformations of $C$. As an application, we prove that if a Jacobian $J$ deforms together with a minimal cohomology class out the Jacobian locus, then $J$ is hyperelliptic. In particular, this provides an evidence to a conjecture of Debarre on the classification of ppavs carrying a minimal cohomology class. Finally, we also study simultaneous deformations of Fano surfaces of lines and intermediate Jacobians.
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We deal with $g$-dimensional abelian varieties $X$ over finite fields. We prove that there is an universal constant (positive integer) $N=N(g)$ that depends only on $g$ that enjoys the following properties. If a certain self-product of $X$ carries an exotic Tate class then the self-product $X^{2N}$of $X$ also carries an exotic Tate class. This gives a positive answer to a question of Kiran Kedlaya.
We reproduce the quantum cohomology of toric varieties (and of some hypersurfaces in projective spaces) as the cohomology of certain vertex algebras with differential. The deformation technique allows us to compute the cohomology of the chiral de Rham complex over the projective space.
We construct non-isogenous simple ordinary abelian varieties over an algebraic closure of a finite field with isomorphic endomorphism algebras.
Given a smooth quasiprojective variety $Y$ over $mathbb C$ that is not projective, consider its unordered configuration spaces $mathrm{Conf}^n(Y)$ for $ngeq 0$. Remove a point $P$ of $Y$ and obtain a one-puncture $Y-P$ of $Y$. We give a decomposition formula that computes the singular cohomology groups of $mathrm{Conf}^n(Y-P)$ in terms of those of $mathrm{Conf}^m(Y); (0leq mleq n)$, and prove it for several families of examples of $Y$, including the case where $Y$ is obtained from a smooth projective variety by puncturing one or more points. This formula keeps track of the mixed Hodge structures of the cohomology groups as well. This result simultaneously implies a result of Kallel involving Betti numbers and a consequence of a combinatorial property of configuration spaces due to Vakil and Wood. We also obtain intermediate results involving ordered configuration spaces that potentially work for more examples of $Y$.
This is an expository lecture, for the Abel bicentennial (Oslo, 2002), describing some recent work on the (small) quantum cohomology ring of Grassmannians and other homogeneous varieties.
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