Related papers: An integrable system of K3-Fano flags

We consider a semistable degeneration of K3 surfaces, equipped with an effective divisor that defines a polarisation of degree two on a general fibre. We show that the map to the relative log canonical model of the degeneration maps every…

We gather evidence for a conjecture of Galkin predicting the derived category of the Fano variety of lines contained in a smooth cubic fourfold to be equivalent to the Hilbert square of the Kuznetsov component of the derived category of the…

Let $f:S \fr B$ be a surface fibration with fibres of genus 5. We find a linear relation between the fundamental invariants of the surface. Namely $K_f^2=\chi_f+N$ where $N$ is the number of trigonal fibres. Our proof is based on the…

Even if there are too many elliptic fibrations to investigate and describe on the singular $K3$ surface $Y_{10}$ of discriminant 72 and belonging to the Ap\'ery-Fermi pencil $(Y_k)$, we find on it many interesting properties. For example…

Associated to every complete affine 3-manifold M with nonsolvable fundamental group is a noncompact hyperbolic surface S. We classify such complete affine structures when Sigma is homeomorphic to a three-holed sphere. In particular, for…

In this article we discuss some numerical parts of the mirror conjecture. For any 3 - dimensional Calabi - Yau manifold author introduces a generalization of the Casson invariant known in 3 - dimensional geometry, which is called Casson -…

Among smooth non-rational Fano 3-folds, the non-hyperelliptic Fano 3-fold X(10) of genus 6 has the unique property to admit a non-trivial orbit of birationally isomorphic 3-folds, inside its moduli space. Here we prove that these orbits…

In this paper the author discuss the relation between Lagrangian Floer homology and Gauge-theory (Donaldson theory) Floer homology. It can be regarded as a version of Atiyah-Floer type conjecture in the case of $SO(3)$-bundle with…

We show that odd order transcendental elements of the Brauer group of a K3 surface can obstruct the Hasse principle. We exhibit a general K3 surface $Y$ of degree 2 over $\mathbb{Q}$ together with a three torsion Brauer class $\alpha$ that…

Let us consider the rank 14 lattice $P=D_4^3\oplus < -2> \oplus < 2>$. We define a K3 surface S of type P with the property that $P\subset {\rm Pic}(S) $, where ${\rm Pic}(S) $ indicates the Picard lattice of S. In this article we study the…

Using the bulk-deformation of Floer cohomology by Schubert cycles and non-Archimedean analysis of Fukaya--Oh--Ohta--Ono's bulk-deformed potential function, we prove that every complete flag manifold $\mathrm{Fl}(n)$ ($n \geq 3$) with a…

We study Euclidean M5-branes wrapping vertical divisors in elliptic Calabi-Yau fourfold compactifications of M/F-theory that admit a Sen limit. We construct these Calabi-Yau fourfolds as elliptic fibrations over coordinate flip O3/O7…

Let X be a compact K\"ahler manifold such that the anticanonical bundle $-K_X$ is nef. A classical conjecture claims that the Albanese map is submersive. We prove this conjecture if the general fibre is a weak Fano manifold. If X is…

This is the third paper in a series of papers studying intersection Floer theory of Lagrangians in the complement of a smooth divisor. We complete the construction of Floer homology for such Lagrangians.

In a recent paper we defined a new filtration of the mapping class group--the "Lagrangian" filtration. We here determine the successive quotients of this filtration, up to finite index. As an application we show that, for any additive…

This is a survey paper about a selection of recent results on the geometry of a special class of Fano varieties, which are called of K3 type. The focus is mostly Hodge-theoretical, with an eye towards the multiple connections between Fano…

The first part of this paper is a review of the Strominger-Yau-Zaslow conjecture in various settings. In particular, we summarize how, given a pair (X,D) consisting of a Kahler manifold and an anticanonical divisor, families of special…

We classify birationally rigid orbifold Fano 3-folds of index one defined by $5 \times 5$ Pfaffians. We give a sharp criterion for birational rigidity of these families based on the type of singularities that the varieties admit. Various…

We show that any Fano fivefold with canonical Gorenstein singularities has an effective anticanonical divisor. Moreover,if a general element of the anticanonical system is reduced, then it has canonical singularities. We also prove…

Iterating the procedure of making a double cover over a given variety, we construct large families of smooth higher-dimensional Fano varieties of index 1. These varieties can be realized as complete intersections in various weighted…