Trivial Witt groups of flag varieties

Let G be a split semi-simple linear algebraic group over a field, let P be a parabolic subgroup and let L be a line bundle on the projective homogeneous variety G/P. We give a simple condition on the class of L in Pic(G/P)/2 in terms of Dynkin diagrams implying that the Witt groups W^i(G/P,L) are zero for all integers i. In particular, if B is a Borel subgroup, then W^i(G/B,L) is zero unless L is trivial in Pic(G/B)/2.Comment: 3 pages, 1 figur

Schemes over \F_1 and zeta functions

We determine the {\em real} counting function $N(q)$ ($q\in [1,\infty)$) for the hypothetical "curve" C=\overline {\Sp \Z} over \F_1, whose corresponding zeta function is the complete Riemann zeta function. Then, we develop a theory of functorial \F_1-schemes which reconciles the previous attempts by C. Soul\'e and A. Deitmar. Our construction fits with the geometry of monoids of K. Kato, is no longer limited to toric varieties and it covers the case of schemes associated to Chevalley groups. Finally we show, using the monoid of ad\`ele classes over an arbitrary global field, how to apply our functorial theory of \Mo-schemes to interpret conceptually the spectral realization of zeros of $L$-functions.Comment: 1 figure, 32 page

Quantifying residual finiteness of arithmetic groups

The normal Farb growth of a group quantifies how well-approximated the group is by its finite quotients. We show that any S-arithmetic subgroup of a higher rank Chevalley group G has normal Farb growth n^dim(G).Comment: 18 page

Hilbert schemes of points on a locally planar curve and the Severi strata of its versal deformation

Let C be a locally planar curve. Its versal deformation admits a stratification by the genera of the fibres. The strata are singular; we show that their multiplicities at the central point are determined by the Euler numbers of the Hilbert schemes of points on C. These Euler numbers have made two prior appearances. First, in certain simple cases, they control the contribution of C to the Pandharipande-Thomas curve counting invariants of three-folds. In this context, our result identifies the strata multiplicities as the local contributions to the Gopakumar-Vafa BPS invariants. Second, when C is smooth away from a unique singular point, a special case of a conjecture of Oblomkov and Shende identifies the Euler numbers of the Hilbert schemes with the "U(infinity)" invariant of the link of the singularity. We make contact with combinatorial ideas of Jaeger, and suggest an approach to the conjecture.Comment: 16 page

On the centralizer of the sum of commuting nilpotent elements

Let X and Y be commuting nilpotent K-endomorphisms of a vector space V, where K is a field of characteristic p >= 0. If F=K(t) is the field of rational functions on the projective line, consider the K(t)-endomorphism A=X+tY of V. If p=0, or if the (p-1)-st power of A is 0, we show here that X and Y are tangent to the unipotent radical of the centralizer of A in GL(V). For all geometric points (a:b) of a suitable open subset of the projective line, it follows that X and Y are tangent to the unipotent radical of the centralizer of aX+bY. This answers a question of J. Pevtsova.Comment: 12 pages. To appear in the Friedlander birthday volume of J. Pure and Applied Algebr

Vector bundles of rank four and A_3 = D_3

Over a scheme with 2 invertible, we show that a vector bundle of rank four has a sub or quotient line bundle if and only if the canonical symmetric bilinear form on its exterior square has a lagrangian subspace. For this, we exploit a version of "Pascal's rule" for vector bundles that provides an explicit isomorphism between the moduli functors represented by projective homogeneous bundles for reductive group schemes of type A_3 and D_3. Under additional hypotheses on the scheme (e.g. proper over a field), we show that the existence of sub or quotient line bundles of a rank four vector bundle is equivalent to the vanishing of its Witt-theoretic Euler class.Comment: 16 pages, final version; IMRN 2012 rns14

Decomposition of splitting invariants in split real groups

To a maximal torus in a quasi-split semi-simple simply-connected group over a local field of characteristic 0, Langlands and Shelstad construct a cohomological invariant called the splitting invariant, which is an important component of their endoscopic transfer factors. We study this invariant in the case of a split real group and prove a decomposition theorem which expresses this invariant for a general torus as a product of the corresponding invariants for simple tori. We also show how this reduction formula allows for the comparison of splitting invariants between different tori in the given real group.Comment: 22 page

Deformations of Calabi-Yau hypersurfaces arising from deformations of toric varieties

There are easy "polynomial" deformations of Calabi-Yau hypersurfaces in toric varieties performed by changing the coefficients of the defining polynomial of the hypersurface. In this paper, we explicitly constructed the ``non-polynomial'' deformations of Calabi-Yau hypersurfaces, which arise from deformations of the ambient toric variety

Connected Hopf Algebras of Dimension p2p^2

Let $H$ be a finite-dimensional connected Hopf algebra over an algebraically closed field \field of characteristic $p>0$. We provide the algebra structure of the associated graded Hopf algebra \gr H. Then, we study the case when $H$ is generated by a Hopf subalgebra $K$ and another element and the case when $H$ is cocommutative. When $H$ is a restricted universal enveloping algebra, we give a specific basis for the second term of the Hochschild cohomology of the coalgebra $H$ with coefficients in the trivial $H$-bicomodule \field. Finally, we classify all connected Hopf algebras of dimension $p^2$ over \field.Comment: Accepted by Journal of Algebra, 29 page