Related papers: On almost holomorphic Lagrangian fibrations

We investigate the positivity properties of the direct image $f_{\ast}(K_{X/Y} \otimes L)$ of the adjoint line bundle associated with a big and nef line bundle $L$, under a smooth fibration $f: X\to Y$ between projective varieties. We show…

We construct a universal partial compactification of the relative moduli space of semistable meromorphic Higgs bundles over the stack of stable pointed curves. It parametrizes meromorphic Gieseker Higgs bundles, and is equipped with a flat…

We show that an exact Lagrangian cobordism $L\subset \mathbb R \times P \times \mathbb R$ from a Legendrian submanifold $\Lambda\subset P\times \mathbb R$ to itself satisfies $H_i(L;\mathbb F)=H_i(\Lambda;\mathbb F)$ for any field $\mathbb…

Berndtsson's famous theorem asserts that, for a compact K\"ahler fibration $p:X\to Y$, the direct image bundle $p_*(K_{X/Y}\otimes L)$ of a semi-positive Hermitian holomorphic line bundle $L\to X$ is Nakano semi-positive. As a continuation…

We study piecewise linear knot diagrams in the base of almost toric fibrations of symplectic four-manifolds. These diagrams translate to deformations of the almost toric fibration. We give several applications to symplectic topology, among…

Let $M$ be a holomorphic symplectic K\"ahler manifold equipped with a Lagrangian fibration $\pi$ with compact fibers. The base of this manifold is equipped with a special K\"ahler structure, that is, a K\"ahler structure $(I, g, \omega)$…

Let $L$ be a line bundle on a scheme $X$, proper over a field. The property of $L$ being nef can sometimes be "thickened", allowing reductions to positive characteristic. We call such line bundles arithmetically nef. It is known that a line…

We study restriction of logarithmic Higgs bundles to the boundary divisor and we construct the corresponding nearby-cycles functor in positive characteristic. As applications we prove some strong semipositivity theorems for analogs of…

For a Lagrangian torus A in a simply-connected projective symplectic manifold M, we prove that M has a hypersurface disjoint from a deformation of A. This implies that a Lagrangian torus in a compact hyperk\"ahler manifold is a fiber of an…

In this note, we prove effective semi-ampleness conjecture due to Prokhorov and Shokurov for a special case, more concretely, for Q-Gorenstein klt-trivial fibrations over smooth projective curves whose fibers are all klt log Calabi-Yau…

We calculate curvature tensors of metrics on the total spaces of holomorphic fibrations. Our main tool is a theory of Chern connections and curvature forms for possibly degenerate Hermitian forms on holomorphic vector bundles. We prove a…

We look at natural foliations on the Painlev\'e VI moduli space of regular connections of rank 2 on $\pp ^1 -{t_1,t_2,t_3,t_4}$. These foliations are fibrations, and are interpreted in terms of the nonabelian Hodge filtration, giving a…

Let $(M, \omega)$ be a K\"ahler manifold and let $(L, \nabla)$ be a prequantum line bundle over $M$. Let $X \subset M$ be a Bohr-Sommerfeld Lagrangian submanifold of $(L, \nabla)$. In this paper, we study an asymptotic behaviour of…

We prove an existence result for exact lagrangian cobordisms between closed legendrians.

We determine all possible multiplicities of general singular fibers of a holomorphic Lagrangian fibration, under the assumption that all components of the fibers are of Fujiki class. The multiplicities are at most 6 and the possible values…

In this note we examine the question of expressing the determinant of the push forward of a symmetric line bundle on an abelian fibration in terms of the pull back of the relative dualizing sheaf via the zero section.

Let $k$ be a number field and $X$ a smooth integral affine variety equipped with a morphism $f : X \to A^1_k$ to the affine line. Assume that all fibres of $f$ are split, for instance that they are geometrically integral. Assume that the…

The cotangent bundle $T^*X$ of a smooth intersection $X$ of two quadrics admits a Lagrangian fibration determined by the intrinsic geometry of $X$. We show that this fibration is actually the Hitchin morphism if we endow $X$ with a…

We prove conditions under which the total space of the pullback of a sphere fibration over a connected sum is homotopy equivalent to a connected sum with a gyration. Existing results of this type often depend on geometric methods. We…

We construct, for a second-order homogeneous Lagrangian in two independent variables, a differential 2-form with the property that it is closed precisely when the Lagrangian is null. This is similar to the property of the 'fundamental…