On S n -invariant conformal blocks vector bundles of rank one on {\overline{M}_{0,n}}

For any simple Lie algebra, a positive integer, and n-tuple of compatible weights, the conformal blocks bundle is a globally generated vector bundle on the moduli space of pointed rational curves. We classify all vector bundles of conformal blocks for \({\mathfrak{sl}_n}\), with S _n -invariant weights, which have rank one. We show that the cone generated by their base point free first Chern classes is polyhedral, generated by level one divisors.     

Vector bundles and regulous maps

Let $X$ be a compact nonsingular affine real algebraic variety. We prove that every pre-algebraic vector bundle on $X$ becomes algebraic after finitely many blowing ups. Using this theorem, we then prove that the Stiefel-Whitney classes of any pre-algebraic $\mathbb{R }$ -vector bundle on $X$ are algebraic. We also derive that the Chern classes of any pre-algebraic $\mathbb{C }$ -vector bundles and the Pontryagin classes of any pre-algebraic $\mathbb{R }$ -vector bundle are blow- $\mathbb{C }$ -algebraic. We also provide several results on line bundles on $X$ .     

K-Orbit Closures on G/B as Universal Degeneracy Loci for Flagged Vector Bundles with Symmetric or Skew-Symmetric Bilinear Form

We use equivariant localization and divided difference operators to determine formulas for the torus-equivariant fundamental cohomology classes of K-orbit closures on the flag variety G/B, where G = GL(n, $ \mathbb{C} $ ), and where K is one of the symmetric subgroups O(n, $ \mathbb{C} $ ) or Sp(n, $ \mathbb{C} $ ). We realize these orbit closures as universal degeneracy loci for a vector bundle over a variety equipped with a single flag of subbundles and a nondegenerate symmetric or skew-symmetric bilinear form taking values in the trivial bundle. We describe how our equivariant formulas can be interpreted as giving formulas for the classes of such loci in terms of the Chern classes of the various bundles.     

Determinants of parabolic bundles on Riemann surfaces

LetX be a compact Riemann surface andM _s ^p (X) the moduli space of stable parabolic vector bundles with fixed rank, degree, rational weights and multiplicities. There is a natural Kähler metric onM _s ^p (X). We obtain a natural metrized holomorphic line bundle onM _s ^p (X) whose Chern form equalsmr times the Kähler form, wherem is the common denominator of the weights andr the rank.     

Stability of quadric bundles

A decorated vector bundle on a smooth projective curve \(X\) is a pair \((E,\varphi )\) consisting of a vector bundle and a morphism \(\varphi :(E^{\otimes a})^{\oplus b}\rightarrow (\det E)^{\otimes c}\otimes \mathsf {N}\) , where \(\mathsf {N}\in \text {Pic}(X)\) . There is a suitable semistability condition for such objects which has to be checked for any weighted filtration of \(E\) . We prove, at least when \(a=2\) , that it is enough to consider filtrations of length \(\le \) 2. In this case decorated bundles are very close to quadric bundles and to check semistability condition one can just consider the former. A similar result for L-twisted Higgs bundles and quadric bundles was already proved (García-Prada et al. in The Hitchin–Kobayashi correspondence, Higgs pairs and surface group representations, 2012; Schmitt in Geometric Invariant Theory and Decorated Principal Bundles, Zurich Lectures in Advanced Mathematics. European Mathematical Society, Zurich, 2008). Our proof provides an explicit algorithm which requires a destabilizing filtration and ensures a destabilizing subfiltration of length at most two. Quadric bundles can be thought as a generalization of orthogonal bundles. We show that the simplified semistability condition for decorated bundles coincides with the usual semistability condition for orthogonal bundles. Finally we note that our proof can be easily generalized to decorated vector bundles on nodal curves.     

Minimizers of Higher Order Gauge Invariant Functionals

We introduce higher order variants of the Yang–Mills functional that involve \((n-2)\)-th order derivatives of the curvature. We prove coercivity and smoothness of critical points in Uhlenbeck gauge in dimensions \(\mathrm {dim}M\le 2n\). These results are then used to establish the existence of smooth minimizers on a given principal bundle \(P\rightarrow M\) for subcritical dimensions \(\mathrm {dim}M<2n\). In the case of critical dimension \(\mathrm {dim}M=2n\) we construct a minimizer on a bundle which might differ from the prescribed one, but has the same Chern classes \(c_1,\ldots ,c_{n-1}\). A key result is a removable singularity theorem for bundles carrying a \(W^{n-1,2}\)-connection. This generalizes a recent result by Petrache and Rivière.     

Trace Formulae for Curvature of Jet Bundles over Planar Domains

For a domain \(\varOmega \) in \(\mathbb {C}\) and an operator \(T\) in \({\mathcal {B}}_n(\varOmega )\) , Cowen and Douglas construct a Hermitian holomorphic vector bundle \(E_T\) over \(\varOmega \) corresponding to \(T\) . The Hermitian holomorphic vector bundle \(E_T\) is obtained as a pull-back of the tautological bundle \(S(n,{\mathcal {H}})\) defined over \({\mathcal {G}}r(n,{\mathcal {H}})\) by a nondegenerate holomorphic map \(z\mapsto {\mathrm{ker}}(T-z),\;z\in \varOmega \) . To find the answer to the converse, Cowen and Douglas studied the jet bundle in their foundational paper. The computations in this paper for the curvature of the jet bundle are rather intricate. They have given a set of invariants to determine if two rank \(n\) Hermitian holomorphic vector bundle are equivalent. These invariants are complicated and not easy to compute. It is natural to expect that the equivalence of Hermitian holomorphic jet bundles should be easier to characterize. In fact, in the case of the Hermitian holomorphic jet bundle \({\mathcal {J}}_k({\mathcal {L}}_f)\) , we have shown that the curvature of the line bundle \({\mathcal {L}}_f\) completely determines the class of \({\mathcal {J}}_k({\mathcal {L}}_f)\) . In case of rank \(n\) Hermitian holomorphic vector bundle \(E_f\) , We have calculated the curvature of jet bundle \({\mathcal {J}}_k(E_f)\) and also obtained a trace formula for jet bundle \({\mathcal {J}}_k(E_f)\) .     

Transferred characteristic classes and generalized cohomology rings

In this paper, we study the interaction between transferred Chern classes and Chern classes of transferred bundles. We calculate the algebra $ B{P^{*}}\left( {X_{{h\varSigma p}}^p} \right) $ and show that its multiplicative structure is completely determined by the Frobenius reciprocity. We also give some tables of the initial segments of the formal group law in the Morava K-theory which are often useful in calculations.     

Scaling of conformal blocks and generalized theta functions over $$\overline{\mathcal {M}}_{g,n}$$

By way of intersection theory on \(\overline{\mathcal {M}}_{g,n}\), we show that geometric interpretations for conformal blocks, as sections of ample line bundles over projective varieties, do not have to hold at points on the boundary. We show such a translation would imply certain recursion relations for first Chern classes of these bundles. While recursions can fail, geometric interpretations are shown to hold under certain conditions.     

Cheeger–Chern–Simons Theory and Differential String Classes

We construct new concrete examples of relative differential characters, which we call Cheeger–Chern–Simons characters. They combine the well-known Cheeger–Simons characters with Chern–Simons forms. In the same way as Cheeger–Simons characters generalize Chern–Simons invariants of oriented closed manifolds, Cheeger–Chern–Simons characters generalize Chern–Simons invariants of oriented manifolds with boundary. We study the differential cohomology of compact Lie groups G and their classifying spaces BG. We show that the even degree differential cohomology of BG canonically splits into Cheeger–Simons characters and topologically trivial characters. We discuss the transgression in principal G-bundles and in the universal bundle. We introduce two methods to lift the universal transgression to a differential cohomology valued map. They generalize the Dijkgraaf–Witten correspondence between 3-dimensional Chern–Simons theories and Wess–Zumino–Witten terms to fully extended higher-order Chern–Simons theories. Using these lifts, we also prove two versions of a differential Hopf theorem. Using Cheeger–Chern–Simons characters and transgression, we introduce the notion of differential trivializations of universal characteristic classes. It generalizes well-established notions of differential String classes to arbitrary degree. Specializing to the class \({\frac{1}{2} p_1 \in H^4(B{\rm Spin}_n;\mathbb{Z})}\), we recover isomorphism classes of geometric string structures on Spin _n -bundles with connection and the corresponding spin structures on the free loop space. The Cheeger–Chern–Simons character associated with the class \({\frac{1}{2} p_1}\) together with its transgressions to loop space and higher mapping spaces defines a Chern–Simons theory, extended down to points. Differential String classes provide trivializations of this extended Chern–Simons theory. This setting immediately generalizes to arbitrary degree: for any universal characteristic class of principal G-bundles, we have an associated Cheeger–Chern–Simons character and extended Chern–Simons theory. Differential trivialization classes yield trivializations of this extended Chern–Simons theory.     

An upper bound of the heat kernel along the harmonic-Ricci flow

Let \({c : C \rightarrow X \times X}\) be a correspondence with C and X quasi-projective schemes over an algebraically closed field k. We show that if \({u_\ell : c_1^*\mathbb{Q}_\ell \rightarrow c_2^!\mathbb{Q}_\ell}\) is an action defined by the localized Chern classes of a c ₂-perfect complex of vector bundles on C, where ? is a prime invertible in k, then the local terms of u _? are given by the class of an algebraic cycle independent of ?. We also prove some related results for quasi-finite correspondences. The proofs are based on the work of Cisinski and Deglise on triangulated categories of motives.     

Pseudo-effective cone of Grassmann bundles over a curve

Let \(E\) be a vector bundle over a smooth projective curve \(X\) defined over an algebraically closed field \(k\) . For any integer \(1\,\le \, r\, <\, \mathrm{rank}(E)\) , let \(\mathrm{Gr}_r(E)\,\longrightarrow \, X\) be a Grassmann bundle parametrizing all \(r\) dimensional quotients of the fibers of \(E\) . We compute the pseudo-effective cone in the real Néron–Severi group \(\mathrm{NS}(\mathrm{Gr}_r(E))_\mathbb{R }\) . We prove that this cone coincides with the nef cone in \(\mathrm{NS}(\mathrm{Gr}_r(E))_\mathbb{R }\) if and only if the vector bundle \(E\) is semistable (respectively, strongly semistable) when the characteristic of \(k\) is zero (respectively, positive). Examples are given to show that this characterization of (strong) semistability is not true for vector bundles on higher dimensional projective varieties.     

Schubert polynomials and Arakelov theory of orthogonal flag varieties

We propose a theory of combinatorially explicit Schubert polynomials which represent the Schubert classes in the Borel presentation of the cohomology ring of the orthogonal flag variety ${\mathfrak X={\rm SO}_N/B}$ . We use these polynomials to describe the arithmetic Schubert calculus on ${\mathfrak X}$ . Moreover, we give a method to compute the natural arithmetic Chern numbers on ${\mathfrak X}$ , and show that they are all rational numbers.     

Another proof of Grothendieck’s theorem on the splitting of vector bundles on the projective line

This note contains another proof of Grothendieck‘s theorem on the splitting of vector bundles on the projective line over a field k. Actually the proof is formulated entirely in the classical terms of a lattice \(\Lambda \cong k[T]^d\), discretely embedded into the vector space \(V \cong K_\infty ^d\), where \(K_\infty \cong k((1/T))\) is the completion of the field of rational functions k(T) at the place \(\infty \) with the usual valuation.     

Unitary representations of the fundamental group of orbifolds

Let X be a smooth complex projective variety of dimension n and \(\mathcal {L}\) an ample line bundle on it. There is a well known bijective correspondence between the isomorphism classes of polystable vector bundles E on X with \(c_{1}(E) = 0 = c_{2} (E) \cdot c_{1} (\mathcal {L})^{n-2}\) and the equivalence classes of unitary representations of π₁(X). We show that this bijective correspondence extends to smooth orbifolds.     

On vector bundles over surfaces and Hilbert schemes

Let X be an irreducible smooth projective surface over ${{\mathbb{C}}}$ and Hilb ^d (X) the Hilbert scheme parametrizing the zero-dimensional subschemes of X of length d. Given a vector bundle E on X, there is a naturally associated vector bundle ${{\mathcal{F}}_d(E)}$ over Hilb ^d (X). If E and V are semistable vector bundles on X such that ${{\mathcal{F}}_d(E)}$ and ${{\mathcal{F}}_d(V)}$ are isomorphic, we prove that E is isomorphic to V. A key input in the proof is provided by Biswas and Nagaraj (see [1]).     

Koppelman formulas on flag manifolds and harmonic forms

We construct Koppelman formulas on manifolds of flags in ${\mathbb{C}^N}$ for forms with values in any holomorphic line bundle as well as in the tautological vector bundles and their duals. As an application we obtain new explicit proofs of some vanishing theorems of the Bott–Borel–Weil type by solving the corresponding ${\bar{\partial}}$ -equation. We also construct reproducing kernels for harmonic (p, q)-forms in the case of Grassmannians.     

Characteristic rank of vector bundles over Stiefel manifolds

The characteristic rank of a vector bundle ξ over a finite connected CW-complex X is by definition the largest integer ${k, 0 \leq k \leq \mathrm{dim}(X)}$ , such that every cohomology class ${x \in H^{j}(X;\mathbb{Z}_2), 0 \leq j \leq k}$ , is a polynomial in the Stiefel–Whitney classes w _i (ξ). In this note we compute the characteristic rank of vector bundles over the Stiefel manifold ${V_k(\mathbb{F}^n), \mathbb{F} = \mathbb{R}, \mathbb{C}, \mathbb{H}}$ .     

L’espace symétrique de Drinfeld et correspondance de Langlands locale I

We give an upper bound of a Hamiltonian displacement energy of a unit disk cotangent bundle $D^*M$ in a cotangent bundle $T^*M$ , when the base manifold $M$ is an open Riemannian manifold. Our main result is that the displacement energy is not greater than $C r(M)$ , where $r(M)$ is the inner radius of $M$ , and $C$ is a dimensional constant. As an immediate application, we study symplectic embedding problems of unit disk cotangent bundles. Moreover, combined with results in symplectic geometry, our main result shows the existence of short periodic billiard trajectories and short geodesic loops.