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In this paper, we develop the theory of singular Hermitian metrics on vector bundles. As an application, we give a structure theorem of a projective manifold X with pseudo-effective tangent bundle; X admits a smooth fibration $X \\to Y$ to a flat projective manifold Y such that its general fibre is rationally connected. Moreover, by applying this structure theorem, we classify all the minimal surfaces with pseudo-effective tangent bundle and study general nonminimal surfaces, which provide examples of (possibly singular) positively curved tangent bundles.

We construct a partial compactification of the moduli space,

We bring in Landau-Lifshitz-Bloch equation on m-dimensional closed Riemannian manifold and prove that it admits a unique local solution. When m ⩾ 3 and the initial data in L∞-norm is suffciently small, the solution can be extended globally. Moreover, for m = 2, we can prove that the unique solution is global without assuming small initial data.

In this paper the authors introduce the notion of a geometric associative $r$-matrix attached to a genus one fibration with a section and irreducible fibres. It allows them to study degenerations of solutions of the classical Yang-Baxter equation using the approach of Polishchuk. They also calculate certain solutions of the classical, quantum and associative Yang-Baxter equations obtained from moduli spaces of (semi-)stable vector bundles on Weierstrass cubic curves.

This book studies generalized Donaldson-Thomas invariants $\\bar{DT}{}^\\alpha(\\tau)$. They are rational numbers which `count' both $\\tau$-stable and $\\tau$-semistable coherent sheaves with Chern character $\\alpha$ on $X$; strictly $\\tau$-semistable sheaves must be counted with complicated rational weights. The $\\bar{DT}{}^\\alpha(\\tau)$ are defined for all classes $\\alpha$, and are equal to $DT^\\alpha(\\tau)$ when it is defined. They are unchanged under deformations of $X$, and transform by a wall-crossing formula under change of stability condition $\\tau$. To prove all this, the authors study the local structure of the moduli stack $\\mathfrak M$ of coherent sheaves on $X$. They show that an atlas for $\\mathfrak M$ may be written locally as $\\mathrm{Crit}(f)$ for $f:U\\to{\\mathbb C}$ holomorphic and $U$ smooth, and use this to deduce identities on the Behrend function $\\nu_\\mathfrak M$. They compute the invariants $\\bar{DT}{}^\\alpha(\\tau)$ in examples, and make a conjecture about their integrality properties. They also extend the theory to abelian categories $\\mathrm{mod}$-$\\mathbb{C}Q\\backslash I$ of representations of a quiver $Q$ with relations $I$ coming from a superpotential $W$ on $Q$.

We study some canonical differential operators on vector bundles over smooth, complete Riemannian manifolds. Under very general assumptions, we show that smooth, compactly supported sections are dense in the domains of these operators. Furthermore, we show that smooth, compactly supported functions are dense in second order Sobolev spaces on such manifolds under the sole additional assumption that the Ricci curvature is uniformly bounded from below.

For two-dimensional manifold with locally symmetric connection ∇ and with ∇-parallel volume element vol one can construct a flat connection on the vector bundle ⊕ , where is a trivial bundle. The metrizable case, when is a Riemannian manifold of constant curvature, together with its higher dimension generalizations, was studied by A.V. Shchepetilov [J. Phys. A: (2003), 3893-3898]. This paper deals with the case of non-metrizable locally symmetric connection. Two flat connections on ⊕ (ℝ × ) and two on ⊕ (ℝ × ) are constructed. It is shown that two of those connections – one from each pair – may be identified with the standard flat connection in ℝ , after suitable local affine embedding of ( ∇) into ℝ

Let X be a compact Riemann surface of genus g ≥ 2 and let n ≥ 3 be an integer number. The group of automorphisms of the moduli space of vector bundles over X with rank n and trivial determinant is isomorphic to H 1 ( X , Z / ( n ) ) ⋊ ( Out ( SL ( n , C ) ) × Aut ( X ) ) . Several papers have studied the subvarieties of fixed points for the action of the unique outer involution of SL ( n , C ) on this moduli space. In this paper, explicit descriptions of the fixed points for the actions of the elements of H 1 ( X , Z / ( n ) ) , H 1 ( X , Z / ( n ) ) ⋊ Out ( SL ( n , C ) ) , and Out ( SL ( n , C ) ) × Aut ( X ) on the moduli space of rank n and trivial determinant vector bundles over X are provided. For the description of the fixed points for the action of the elements of Out ( SL ( n , C ) ) × Aut ( X ) , the notion of Galois bundle is introduced. Specifically, Galois bundles over X admitting a nontrivial automorphism which commutes with the Galois structure are constructed associated with an involution σ X of X . Finally, it is discussed how the description of fixed points for the action of the elements of is covered by the descriptions above.

Let 𝒞 be an admissible category over manifolds and 𝓥𝓑𝒞 be the category of vector bundles with bases being 𝒞-objects and vector bundle maps with base maps being 𝒞-maps. Assume that any 𝒞-morphism is a local isomorphism. We describe all jet like functors (i.e. fiber product preserving gauge bundle functors) of order r on 𝓥𝓑𝒞 by means of Jr (−, R)-module bundle functors on 𝒞. Then we describe all jet like functors of vertical type of order r on 𝓥𝓑𝒞 by means of vector bundle functors on 𝒞 of order r. As an application we classify jet like functors of some type on 𝓥𝓑 m . Finally, we determine all natural Jr (M, R)-module bundle structures on the vector bundles JrE → M and J υ r E → M , where E → M is a vector bundle with m-dimensional basis and m ≥ 2.

We introduce a collection of convex polytopes associated to a torus-equivariant vector bundle on a smooth complete toric variety. We show that the lattice points in these polytopes correspond to generators for the space of global sections and relate edges to jets. Using the polytopes, we also exhibit vector bundles that are ample but not globally generated, and vector bundles that are ample and globally generated but not very ample.