In this article we obtain criteria for the splitting and triviality of vector bundles by restricting them to partially ample divisors. This allows us to study the problem of splitting on the total space of fibre bundles. The statements are illustrated with examples.

For products of minuscule homogeneous varieties, we show that the splitting of vector bundles can be tested by restricting them to subproducts of Schubert $2$-planes. By using known cohomological criteria for multiprojective spaces, we deduce necessary and sufficient conditions for the splitting of vector bundles on products of minuscule varieties.

The triviality criteria are particularly suited to Frobenius split varieties. We prove that a vector bundle on a smooth toric variety, whose anticanonical bundle has stable base locus of codimension at least three, is trivial precisely when its restrictions to the invariant divisors are trivial, with trivializations compatible along the various intersections.

In this note, we show that if $T$ is a multilinear singular integral operator associated with a kernel satisfies the so-called multilinear $L^r$-Hörmander condition, then $T$ can be dominated by multilinear sparse operators.

In this article, we derive meromorphic continuation of multiple Lerch zeta functions by generalizing an elegant identity of Ramanujan. Further, we describe the set of all possible singularities of these functions. Finally, for the multiple Hurwitz zeta functions, we list the exact set of singularities.

We consider the previously introduced notion of the $K$-quadrilateral cosine, which is the cosine under parallel transport in model $K$-space, and which is denoted by ${cosq}_K$. In $K$-space, $\left|{cosq}_K\right|\le 1$ is equivalent to the Cauchy–Schwarz inequality for tangent vectors under parallel transport. Our principal result states that a geodesically connected metric space (of diameter not greater than $\pi /\left(2\sqrt{K}\right)$ if $K>0$) is an ${\Re }_K$ domain (otherwise known as a $CAT\left(K\right)$ space) if and only if always ${cosq}_K\le 1$ or always ${cosq}_K\ge -1$. (We prove that in such spaces always ${cosq}_K\le 1$ is equivalent to always ${cosq}_K\ge -1$.) The case of $K=0$ was treated in our previous paper on quasilinearization. We show that in our theorem the diameter hypothesis for positive $K$ is sharp, and we prove an extremal theorem—isometry with a section of $K$-plane—when $\left|{cosq}_K\right|$ attains an upper bound of $1$, the case of equality in the metric Cauchy–Schwarz inequality. We derive from our main theorem and our previous result for $K=0$ a complete solution of Gromov’s curvature problem in the context of Aleksandrov spaces of curvature bounded above.

We consider the orbifold curve that is a quotient of an elliptic curve $\mathcal{E}$ by a cyclic group of order 4. We develop a systematic way to obtain a Givental-type reconstruction of Gromov–Witten theory of the orbifold curve via the product of the Gromov–Witten theories of a point. This is done by employing mirror symmetry and certain results in FJRW theory. In particular, we present the particular Givental’s action giving the CY/LG correspondence between the Gromov–Witten theory of the orbifold curve $\mathcal{E}/{\mathbb{Z}}_4$ and FJRW theory of the pair defined by the polynomial $x^4+y^4+z^2$ and the maximal group of diagonal symmetries. The methods we have developed can easily be applied to other finite quotients of an elliptic curve. Using Givental’s action, we also recover this FJRW theory via the product of the Gromov–Witten theories of a point. Combined with the CY/LG action, we get a result in “pure” Gromov–Witten theory with the help of modern mirror symmetry conjectures.

By analogy with the program of McKinnon and Roth [10], we define and study approximation constants for points of a projective variety $X$ defined over $\mathbf{K}$, the function field of an irreducible and nonsingular in codimension $1$ projective variety defined over an algebraically closed field of characteristic zero. In this setting, we use Wang’s theorem, which is an effective version of Schmidt’s subspace theorem, to give a sufficient condition for such approximation constants to be computed on a proper $\mathbf{K}$-subvariety of $X$. We also indicate how our approximation constants are related to volume functions and Seshadri constants.

We show that every tight contact structure on any of the lens spaces $L(n{s}^2-s+1,s^2)$ with $n\ge 2$ and $s\ge 1$ can be obtained by a single Legendrian surgery along a suitable Legendrian realisation of the negative torus knot $T(s,-(sn-1\left)\right)$ in the tight or an overtwisted contact structure on the $3$-sphere.

We compute the $Pin\left(2\right)$-equivariant monopole Floer homology for the class of plumbed 3-manifolds considered by Ozsváth and Szabó [18]. We show that for these manifolds, the $Pin\left(2\right)$-equivariant monopole Floer homology can be calculated in terms of the Heegaard Floer/monopole Floer lattice complex defined by Némethi [15]. Moreover, we prove that in such cases the ranks of the usual monopole Floer homology groups suffice to determine both the Manolescu correction terms and the $Pin\left(2\right)$-homology as an Abelian group. As an application, we show that $\beta (-Y,s)=\overline{\mu }(Y,s)$ for all plumbed 3-manifolds with at most one “bad” vertex, proving (an analogue of) a conjecture posed by Manolescu [12]. Our proof also generalizes results by Stipsicz [21] and Ue [26] relating $\overline{\mu }$ with the Ozsváth–Szabó $d$-invariant. Some observations aimed at extending our computations to manifolds with more than one bad vertex are included at the end of the paper.