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The main goal of this short paper is to prove the existence of rank 2 simple and special Ulrich bundles on a wide class of elliptic surfaces: namely, on regular Weierstrass fibrations \(\pi : S\rightarrow \mathbb {P}^1\). Alongside we also show the existence of rank 2 weakly Ulrich sheaves on arbitrary Weierstrass fibrations \(S\rightarrow C_0\) and we deal with the (non-)existence of rank one Ulrich bundles on them.

In this paper, we introduce the associated geodesic-Einstein flow for a relative ample line bundle L over the total space $$\mathcal {X}$$ of a holomorphic fibration and obtain a few properties of that flow. In particular, we prove that the pair $$(\mathcal {X}, L)$$ is nonlinear semistable if the associated Donaldson type functional is bounded from below and the geodesic-Einstein flow has long-time existence property. We also define the associated S-classes and C-classes for $$(\mathcal {X}, L)$$ and obtain two inequalities between them when L admits a geodesic-Einstein metric. Finally, in the appendix of this paper, we prove that a relative ample line bundle is geodesic-Einstein if and only if an associated infinite rank bundle is Hermitian–Einstein.

From string theory, the notion of deformed Hermitian Yang–Mills connections has been introduced by Marino et al. (J High Energy Phys Paper 5, 2000). After that, Leung et al. (Adv Theor Math Phys 4(6):1319–1341, 2000) proved that it naturally appears as mirror objects of special Lagrangian submanifolds via Fourier–Mukai transform between dual torus fibrations. In their paper, some conditions are imposed for simplicity. In this paper, data to glue their construction on tropical manifolds are proposed and a generalization of the correspondence is proved without the assumption that the Lagrangian submanifold is a section of the torus fibration.

In this paper we prove several related results concerning smooth \(\mathbb{Z }_p\) or \(\mathbb{S }^1\) actions on \(4\)-manifolds. We show that there exists an infinite sequence of smooth \(4\)-manifolds \(X_n\), \(n\ge 2\), which have the same integral homology and intersection form and the same Seiberg-Witten invariant, such that each \(X_n\) supports no smooth \(\mathbb{S }^1\)-actions but admits a smooth \(\mathbb{Z }_n\)-action. In order to construct such manifolds, we devise a method for annihilating smooth \(\mathbb{S }^1\)-actions on \(4\)-manifolds using Fintushel-Stern knot surgery, and apply it to the Kodaira-Thurston manifold in an equivariant setting. Finally, the method for annihilating smooth \(\mathbb{S }^1\)-actions relies on a new obstruction we derived in this paper for existence of smooth \(\mathbb{S }^1\)-actions on a \(4\)-manifold: the fundamental group of a smooth \(\mathbb{S }^1\)-four-manifold with nonzero Seiberg-Witten invariant must have infinite center. We also include a discussion on various analogous or related results in the literature, including locally linear actions or smooth actions in dimensions other than four.

In [25], higher genus Gromov–Witten invariants of the stack of r-th roots of a smooth projective variety X along a smooth divisor D are shown to be polynomials in r. In this paper we study the degrees and coefficients of these polynomials.

In this paper we study the problem of extension of holomorphic sections of adjoint line bundles/vector bundles from reduced unions of strata of divisors. We prove a qualitative extension theorem for adjoint bundles. The main technical result is an extension theorem of the Ohsawa–Takegoshi type.

In this paper, we study Steinness of nonpositively or nonnegatively curved strongly convex Kahler Finsler manifolds. In particular, we prove that every strongly convex Kahler Finsler manifold with a pole and nonpositive bisectional curvature must be Stein. In addition, every complete noncompact and strongly convex Kahler Berwald manifold is Stein if it has positive flag curvature everywhere, or it has nonnegative flag curvature and everywhere positive holomorphic bisectional curvature.

Let $(X,L_{X})$ be an $n$-dimensional polarized manifold. Let $D$ be a smooth hypersurface defined by a holomorphic section of $L_{X}$. We prove that if $D$ has a constant positive scalar curvature Kähler metric, $X \setminus D$ admits a complete scalar-flat Kähler metric, under the following three conditions: (i) $n \geq 6$ and there is no nonzero holomorphic vector field on $X$ vanishing on $D$, (ii) an average of a scalar curvature on $D$ denoted by $\hat{S}_{D}$ satisfies the inequality $0 < 3 \hat{S}_{D} < n(n-1)$, (iii) there are positive integers $l(>n),m$ such that the line bundle $K_{X}^{-l} \otimes L_{X}^{m}$ is very ample and the ratio $m/l$ is sufficiently small., 18 pages, no figures, This paper is merged with arXiv:1907.09780 and arXiv:1908.05583 and appear in Math. Z./

Let $X_w$ be a Schubert subvariety of a cominuscule Grassmannian $X$, and let $\mu:T^*X\rightarrow\mathcal N$ be the Springer map from the cotangent bundle of $X$ to the nilpotent cone $\mathcal N$. In this paper, we construct a resolution of singularities for the conormal variety $T^*_XX_w$ of $X_w$ in $X$. Further, for $X$ the usual or symplectic Grassmannian, we compute a system of equations defining $T^*_XX_w$ as a subvariety of the cotangent bundle $T^*X$ set-theoretically. This also yields a system of defining equations for the corresponding orbital varieties $\mu(T^*_XX_w)$. Inspired by the system of defining equations, we conjecture a type-independent equality, namely $T^*_XX_w=\pi^{-1}(X_w)\cap\mu^{-1}(\mu(T^*_XX_w))$. The set-theoretic version of this conjecture follows from this work and previous work for any cominuscule Grassmannian of type A, B, or C., Comment: Version 1 included similar results in type D. However, there were serious errors in the proof. All claims to results in type D have been removed in this version. We are grateful to Anna Melnikov for pointing us to these errors

In the present paper, we prove an infinite dimensional reversible Kolmogorov-Arnold-Moser (KAM) theorem. As an application, we study the existence of KAM tori for a class of two dimensional (2D) non-Hamiltonian completely resonant beam equations with derivative nonlinearities. The Birkhoff normal form theory is also used since there are no external parameters in the equations.

The purpose of this paper is to study holomorphic approximation and approximation of $$\overline{\partial }$$ -closed forms in complex manifolds of complex dimension $$n\ge 1$$ . We consider extensions of the classical Runge theorem and the Mergelyan property to domains in complex manifolds for the $${{\mathcal {C}}}^\infty $$ -smooth and the $$L^2$$ topology. We characterize the Runge or Mergelyan property in terms of certain Dolbeault cohomology groups and some geometric sufficient conditions are given.

In this paper, motivated by Chen–Ruan’s stringy orbifold theory on almost complex orbifolds, we construct a new equivariant cohomology ring $${\mathscr {H}}^*_G(X)$$ for an equivariant almost complex pair (X, G), where X is a compact connected almost complex manifold, G is a connected compact Lie group which acts on X and preserves the almost complex structure.

In this paper, we first give several comparison theorems and their applications in Finsler geometry. Moreover, by means of the Hessian comparison theorems of a real Finsler manifold, Cartan connection, radial flag curvature and tangent curvature, we give Wu’s theorem on a strongly convex weakly Kahler Finsler manifold with a pole. Finally, by using the complex Rund connection and definition of radial flag curvature on a strongly pseudoconvex complex Finsler manifold, we further discuss Wu’s theorem on a strongly convex Kahler Finsler manifold.

Parreau compactified the Hitchin component of a closed surface $S$ of negative Euler characteristic in such a way that a boundary point corresponds to the projectivized length spectrum of an action of $\pi_1(S)$ on an $\mathbb R$-Euclidean building. In this paper, we use the positivity properties of Hitchin representations introduced by Fock and Goncharov to explicitly describe the geometry of a preferred collection of apartments in the limiting building., Comment: 26 pages, 4 figures, Minor corrections (typos, etc.)

Let the circle act on a compact almost complex manifold M. In this paper, we classify the fixed point data of the action if there are 4 fixed points and the dimension of the manifold is at most 6. By the fixed point data we mean a collection of the multisets of the weights at the fixed points. First, if $$\dim M=2$$, then M is a disjoint union of rotations on two 2-spheres. Second, if $$\dim M=4$$, we prove that the action alikes a circle action on a Hirzebruch surface. Finally, if $$\dim M=6$$, we prove that six types occur for the fixed point data; $$\mathbb {CP}^3$$ type, complex quadric in $$\mathbb {CP}^4$$ type, Fano threefold type, $$S^6 \cup S^6$$ type, blow up of a fixed point of a rotation on $$S^6$$ type, and unknown type that might possibly be realized as a blow up of $$S^2$$ inside a manifold like $$S^6$$. When $$\dim M=6$$, we recover the result by Ahara (J Fac Sci Univ Tokyo Sect IA Math 38(1):47–72, 1991) in which the fixed point data is determined if furthermore $$\mathrm {Todd}(M)=1$$ and $$c_1^3(M)[M] \ne 0$$, and the result by Tolman (Trans Am Math Soc 362(8):3963–3996, 2010) in which the fixed point data is determined if furthermore the base manifold admits a symplectic structure and the action is Hamiltonian.

Let (M, g) be a compact Kahler manifold and f a positive smooth function such that its Hamiltonian vector field $$K = J\mathrm {grad}_g f$$ for the Kahler form $$\omega _g$$ is a holomorphic Killing vector field. We say that the pair (g, f) is conformally Einstein–Maxwell Kahler metric if the conformal metric $$\tilde{g} = f^{-2}g$$ has constant scalar curvature. In this paper we prove a reductiveness result of the reduced Lie algebra of holomorphic vector fields for conformally Einstein–Maxwell Kahler manifolds, extending the Lichnerowicz–Matsushima Theorem for constant scalar curvature Kahler manifolds. More generally we consider extensions of Calabi functional and extremal Kahler metrics, and prove an extension of Calabi’s theorem on the structure of the Lie algebra of holomorphic vector fields for extremal Kahler manifolds. The proof uses a Hessian formula for the Calabi functional under the set up of Donaldson-Fujiki picture.

The twistor construction for Riemannian manifolds is extended to the case of manifolds endowed with generalized metrics (in the sense of generalized geometry \`a la Hitchin). The generalized twistor space associated to such a manifold is defined as the bundle of generalized complex structures on the tangent spaces of the manifold compatible with the given generalized metric. This space admits natural generalized almost complex structures whose integrability conditions are found in the paper. An interesting feature of the generalized twistor spaces discussed in it is the existence of intrinsic isomorphisms., Comment: typos corrected, minor changes, to appear in Math. Z

A classical question in spectral geometry is, for each pair of nonnegative integers $(p,n)$ such that $p\leq 2n$, if the eigenvalues of Laplacian on $p$-forms of a compact K\"{a}hler manifold are the same as those of $\mathbb{C}P^n$ equipped with the Fubini-Study metric, then whether or not this K\"{a}hler manifold is holomorphically isometric to $\mathbb{C}P^n$. For every positive even number $p$, we affirmatively solve this problem in all dimensions $n$ with at most two possible exceptions. We also clarify in this paper some gaps in previous literature concerned with this question, among which one is related to the volume estimate of Fano K\"{a}hler-Einstein manifolds., Comment: 28 pages, version 2, some typos corrected

In this paper we introduce the concept of inflexibleCR submanifolds. These are CR submanifolds of some complex Euclidean space such that any compactly supported CR deformation is again globally CR embeddable into some complex Euclidean space. Our main result is that any 2-pseudoconcave quadratic CR submanifold of type (n, d) in \(\mathbb {C}^{n+d}\) is inflexible.

In this paper, we establish an equality between the analytic torsion introduced by Dar (Math Z 194(2): 193–216, 1987) and the orbifold analytic torsion defined by Ma (Trans Am Math Soc 357(6): 2205–2233, 2005) on an even dimensional manifold with isolated conical singularities which in addition has an orbifold structure. We assume the orbifold flat vector bundle is an honest vector bundle, although the metric on the flat bundle may not be flat.

Let V be the even part of the vertex operator super-algebra of r pairs of symplectic fermions. Up to two conjectures, we show that V admits a unique holomorphic extension if r is a multiple of 8, and no holomorphic extension otherwise. This is implied by two results obtained in this paper: (1) If r is a multiple of 8, one possible holomorphic extension is given by the lattice vertex operator algebra for the even self dual lattice \(D_{r}^+\) with shifted stress tensor. (2) We classify Lagrangian algebras in \(\mathcal {S}\mathcal {F}(\mathfrak {h})\), a ribbon category associated to symplectic fermions. The classification of holomorphic extensions of V follows from (1) and (2) if one assumes that \(\mathcal {S}\mathcal {F}(\mathfrak {h})\) is ribbon equivalent to \({\mathrm {Rep}}(V)\), and that simple modules of extensions of V are in one-to-one relation with simple local modules of the corresponding commutative algebra in \(\mathcal {S}\mathcal {F}(\mathfrak {h})\).

In this paper we find sufficient conditions for the vanishing of the Morse–Novikov cohomology on Riemannian foliations. We work out a Bochner technique for twisted cohomological complexes, obtaining corresponding vanishing results. Also, we generalize for our setting vanishing results from the case of closed Riemannian manifolds. Several examples are presented, along with applications in the context of l.c.s. and l.c.K. foliations.

In this paper we study the deformation theory of submanifolds characterized by a system of differential forms and provide a criterion for deformations of such submanifolds to be unobstructed. We apply this deformation theory to special Legendrian submanifolds in Sasaki–Einstein manifolds. In general, special Legendrian deformations have the obstruction. However, we show that the deformation space of special Legendrian submanifolds is the intersection of two larger smooth deformation spaces of different types. We also prove that any special Legendrian submanifold admits smooth deformations, which are not special Legendrian deformations, given by harmonic 1-forms.

This paper is concerned with non-symplectic involutions of irreducible symplectic manifolds of $$K3^{[n]}$$ -type. We will give a criterion for deformation equivalence and use this to give a lattice-theoretic description of all deformation types. While moduli spaces of $$K3^{[n]}$$ -type manifolds with non-symplectic involutions are not necessarily Hausdorff, we will construct quasi-projective moduli spaces for a certain well-behaved class of such pairs.

This paper consists of two parts. In the first part, we use symplectic homology to distinguish the contact structures on the Brieskorn manifolds $\Sigma(2l,2,2,2)$, which contact homology cannot distinguish. This answers a question from [22]. In the second part, we prove the existence of infinitely many exotic but homotopically trivial exotic contact structures on $S^7$, distinguished by the mean Euler characteristic of $S^1$-equivariant symplectic homology. Apart from various connected sum constructions, these contact structures can be taken from the Brieskorn manifolds $\Sigma(78k+1,13,6,3,3)$. We end with some considerations about extending this result to higher dimensions., Comment: 33 pages; several small corrections. To appear in Mathematische Zeitschrift. The final publication is available at http://link.springer.com/article/10.1007/s00209-015-1596-3

The present paper is devoted to the classification of symplectic automorphisms of some hyperkahlermanifolds. The result presented here is a proof that all finite groups of symplectic automorphisms of manifolds of \(K3^{[n]}\) typeare contained in Conway’s group \(Co_1\).

In 2006 Masuda and Suh asked if two compact non-singular toric varieties having isomorphic cohomology rings are homeomorphic. In the first part of this paper we discuss this question for topological generalizations of toric varieties, so-called torus manifolds. For example we show that there are homotopy equivalent torus manifolds which are not homeomorphic. Moreover, we characterize those groups which appear as the fundamental groups of locally standard torus manifolds. In the second part we give a classification of quasitoric manifolds and certain six-dimensional torus manifolds up to equivariant diffeomorphism. In the third part we enumerate the number of conjugacy classes of tori in the diffeomorphism group of torus manifolds. For torus manifolds of dimension greater than six there are always infinitely many conjugacy classes. We give examples which show that this does not hold for six-dimensional torus manifolds., 21 pages, 2 figures, results about quasitoric manifolds added