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Christian Pauly

Publications and source records attributed to Christian Pauly.

At least 37 records · Page 2Linked to original sources
arXiv2015

Hitchin-Mochizuki morphism, Opers and Frobenius-destabilized vector bundles over curves

Let X be a smooth projective curve of genus g \textgreater{}1 defined over an algebraically closed field k of characteristic p \textgreater{}0. For p sufficiently large (explicitly given in terms of r,g) we construct an atlas for the locus of all Frobenius-destabilized bundles (i.e. we construct all Frobenius-destabilized bundles of degree zero up to isomorphism). This is done by exhibiting a surjective morphism from a certain Quot-scheme onto the locus of stable Frobenius-destabilized bundles. Further we show that there is a bijective correspondence between the set of stable vector bundles E over X such that the pull-back F^*(E) under the Frobenius morphism of X has maximal Harder-Narasimhan polygon and the set of opers having zero p-curvature. We also show that, after fixing the determinant, these sets are finite, which enables us to derive the dimension of certain Quot-schemes and certain loci of stable Frobenius-destabilized vector bundles over X. The finiteness is proved by studying the properties of the Hitchin-Mochizuki morphism; an alternative approach to finiteness has been realized in a recent preprint by Chen and Zhu. In particular we prove a generalization of a result of Mochizuki to higher ranks.

math.AG↗
arXiv2013

Evidence for topological band inversion of the phase change material Ge2Sb2Te5

We present an angle-resolved photoemission study of a ternary phase change material, namely Ge2Sb2Te5, epitaxially grown on Si(111) in the metastable cubic phase. The observed upper bulk valence band shows a minimum at Gamma-bar being 0.3 eV below the Fermi level E_F and a circular Fermi contour around Gamma-bar with a dispersing diameter of 0.27-0.36 Anstroms^-1. This is in agreement with density functional theory calculations of the Petrov stacking sequence in the cubic phase which exhibits a topological surface state. The topologically trivial cubic KH stacking shows a valence band maximum at Gamma in line with all previous calculations of the hexagonal stable phase exhibiting the valence band maximum at Gamma for a trivial Z_2 topological invariant nu_0 and away from Gamma for non-trivial nu_0. Scanning tunneling spectroscopy exhibits a band gap of 0.4 eV around E_F.

cond-mat.mes-hall↗
arXiv2012

The space of generalised G2-theta functions of level one

Let C be a smooth projective complex curve of genus at least 2. For a simply-connected complex Lie group G the vector space of global sections H^0(M(G), L^l) of the l-th power of the ample generator L of the Picard group of the moduli stack of principal G-bundles over C is commonly called the space of generalized G-theta functions or Verlinde space of level l. In the case G = G_2, the exceptional Lie group of automorphisms of the complex Cayley algebra, we study natural linear maps between the Verlinde space H^0(M(G_2), L) of level one and some Verlinde spaces for SL_2 and SL_3. We deduce that the image of the monodromy representation of the WZW-connection for G = G_2 and l=1 is infinite.

math.AG↗
arXiv2012

Strange duality revisited

We give a proof of the strange duality or rank-level duality of the WZW models of conformal blocks by extending the genus-0 result, obtained by Nakanishi-Tsuchiya in 1992, to higher genus curves via the sewing procedure. The new ingredient of the proof is an explicit use of the branching rules of the conformal embedding of the affine Lie algebras sl(r) x sl(l) in sl(rl). We recover the strange duality of spaces of generalized theta functions obtained by Belkale, Marian-Oprea, as well as by Oudompheng in the parabolic case.

math.AG↗
arXiv2012

On the monodromy of the Hitchin connection

For any genus g > 1 we give an example of a family of smooth complex projective curves of genus g such that the image of the monodromy representation of the Hitchin connection on the sheaf of generalized SL(2)-theta functions of level l different from 1,2,4 and 8 contains an element of infinite order.

math.AG↗
arXiv2011

Prym varieties of spectral covers

Given a possibly reducible and non-reduced spectral cover X over a smooth projective complex curve C we determine the group of connected components of the Prym variety Prym(X/C). As an immediate application we show that the finite group of n-torsion points of the Jacobian of C acts trivially on the cohomology of the twisted SL_n-Higgs moduli space up to the degree which is predicted by topological mirror symmetry. In particular this yields a new proof of a result of Harder--Narasimhan, showing that this finite group acts trivially on the cohomology of the twisted SL_n stable bundle moduli space.

math.AG↗
arXiv2009

Strange duality for Verlinde spaces of exceptional groups at level one

The moduli stack M_X(E_8) of principal E_8-bundles over a smooth projective curve X carries a natural divisor Delta. We study the pull-back of the divisor Delta to the moduli stack M_X(P), where P is a semi-simple and simply connected group such that its Lie algebra Lie(P) is a maximal conformal subalgebra of Lie(E_8). We show that the divisor Delta induces "Strange Duality"-type isomorphisms between the Verlinde spaces at level one of the following pairs of groups (SL(5), SL(5)), (Spin(8), Spin(8)), (SL(3), E_6) and (SL(2), E_7).

math.AG↗
arXiv2009

On theta functions of order four

We prove that the fourth powers of theta functions with even characteristics form a basis of the space of even theta functions of order four on a principally polarized Abelian variety without vanishing theta-null.

math.AG↗
arXiv2008

Orthogonal bundles over curves in characteristic two

Let X be a smooth projective curve of genus g \geq 2 defined over a field of characteristic two. We give examples of stable orthogonal bundles with unstable underlying vector bundles and use them to give counterexamples to Behrend's conjecture on the canonical reduction of principal G-bundles for G= SO(n) with n \geq 7.

math.AG↗
arXiv2008

Rank four vector bundles without theta divisor over a curve of genus two

We show that the locus of stable rank four vector bundles without theta divisor over a smooth projective curve of genus two is in canonical bijection with the set of theta-characteristics. We give several descriptions of these bundles and compute the degree of the rational theta map.

math.AG↗
arXiv2008

On the base locus of the linear system of generalized theta functions

Let $\cM_r$ denote the moduli space of semi-stable rank-$r$ vector bundles with trivial determinant over a smooth projective curve $C$ of genus $g$. In this paper we study the base locus $\cB_r \subset \cM_r$ of the linear system of the determinant line bundle $\cL$ over $\cM_r$, i.e., the set of semi-stable rank-$r$ vector bundles without theta divisor. We construct base points in $\cB_{g+2}$ over any curve $C$, and base points in $\cB_4$ over any hyperelliptic curve.

math.AG↗
arXiv2007

Polarizations of Prym varieties for Weyl groups via abelianization

Let $π: Z \ra X$ be a Galois covering of smooth projective curves with Galois group the Weyl group of a simple and simply-connected Lie group $G$. For any dominant weight $λ$ consider the curve $Y = Z/\Stab(λ)$. The Kanev correspondence defines an abelian subvariety $P_λ$ of the Jacobian of $Y$. We compute the type of the polarization of the restriction of the canonical principal polarization of $\Jac(Y)$ to $P_λ$ in some cases. In particular, in the case of the group $E_8$ we obtain families of Prym-Tyurin varieties. The main idea is the use of an abelianization map of the Donagi-Prym variety to the moduli stack of principal $G$-bundles on the curve $X$.

math.AG↗
arXiv2006

Semistability of Frobenius direct images over curves

Let $X$ be a smooth projective curve of genus $g \geq 2$ defined over an algebraically closed field $k$ of characteristic $p>0$. Given a semistable vector bundle $E$ over $X$, we show that its direct image $F\_*E$ under the Frobenius map $F$ of $X$ is again semistable. We deduce a numerical characterization of the stable rank-$p$ vector bundles $F\_*L$, where $L$ is a line bundle over $X$.

math.AG↗
arXiv2005

On Frobenius-destabilized rank-2 vector bundles over curves

Let X be a smooth projective curve of genus g \geq 2 over an algebraically closed field k of characteristic p > 0. Let M_X be the moduli space of semistable rank-2 vector bundles over X with trivial determinant. The relative Frobenius map F: X \to X_1 induces by pull-back a rational map V: M_{X_1} \to M_{X}. In this paper we show the following results. 1) For any line bundle L over X, the rank-p vector bundle F_*L is stable. 2) The rational map V has base points, i.e., there exist stable bundles E over X_1 such that F^* E is not semistable. 3) Let B \subset M_{X_1} denote the scheme-theoretical base locus of V. If g=2, p>2 and X ordinary, then B is a 0-dimensional local complete intersection of length {2/3}p(p^2 -1) and the degree of V equals {1/3}p(p^2 +2).

math.AG↗
arXiv2003

On cubics and quartics through a canonical curve

We construct families of quartic and cubic hypersurfaces through a canonical curve, which are parametrized by an open subset in a Grassmannian and a Flag variety respectively. Using G. Kempf's cohomological obstruction theory, we show that these families cut out the canonical curve and that the quartics are birational (via a blowing-up of a linear subspace) to quadric bundles over the projective plane, whose Steinerian curve equals the canonical curve.

math.AG↗
arXiv2002

The Frobenius map, rank 2 vector bundles and Kummer's quartic surface in characteristic 2 and 3

Let X be a smooth projective curve of genus g>1 defined over an algebraically closed field k of characteristic p>0. Let M_X(r) be the moduli space of semi-stable rank r vector bundles with fixed trivial determinant. The relative Frobenius map F : X \to X_1 induces by pull-back a rational map V: M_{X_1}(r) \to M_X(r). We determine the equations of V in the following two cases (1) (g,r,p) = (2,2,2) and X non-ordinary with Hasse-Witt invariant equal to 1 (see math.AG/0005044 for the case X ordinary), and (2) (g,r,p) = (2,2,3). We also show, for any triple (g,r,p), the existence of base points of V, i.e., semi-stable bundles E such that F^* E is not semi-stable.

math.AG↗
arXiv2001

Self-duality of Coble's quartic hypersurface and applications

The moduli space M_0 of semi-stable rank 2 vector bundles with fixed trivial determinant over a non-hyperelliptic curve C of genus 3 is isomorphic to a quartic hypersurface in P^7 (Coble's quartic). We show that M_0 is self-dual and that its polar map associates to a stable bundle E \in M_0 a bundle F which is characterized by dim H^0(C, E \otimes F) = 4. The projective space PH^0(C, E \otimes F) is equipped with a net of quadrics Πand it is shown that the map which associates to E \in M_0 the isomorphism class of the plane quartic Hessian curve of Πis a dominant map to the moduli space of genus 3 curves.

math.AG↗