In this paper, we show that for a given finitely presented group $G$, there exist integers $h_G \geq 0$ and $n_G \geq 4$ such that for all $h \geq h_G$ and $n \geq n_G$, and for all $0 \leq i \leq 2n - 2$, there exists a genus-$(2h + n - 1)$ Lefschetz fibration on a minimal symplectic 4-manifold with $(\chi, c_{1}^{2}) = (n, i)$ whose fundamental group is isomorphic to $G$. We also prove that such a fibration cannot be decomposed as a fiber sum for $1 \leq i \leq 2n - 2$ if $h > (5n - 3)/2$. In addition, we give a relation among the genus of the base space of a ruled surface admitting a Lefschetz fibration, the number of blow-ups and the genus of the Lefschetz fibration.

We discuss the ACC conjecture and the LSC conjecture for minimal log discrepancies of generalized pairs. We prove that some known results on these two conjectures for usual pairs are still valid for generalized pairs. We also discuss the theory of complements for generalized pairs.

Let $n$ be an integer such that $n = 5$ or $n \geq 7$. In this article, we introduce a recipe for a certain infinite family of non-singular plane curves of degree $n$ which violate the local-global principle. Moreover, each family contains infinitely many members which are not geometrically isomorphic to each other. Our construction is based on two arithmetic objects; that is, prime numbers of the form $X^{3} + NY^{3}$ due to Heath-Brown and Moroz and the Fermat type equation of the form $x^{3} + Ny^{3} = Lz^{n}$, where $N$ and $L$ are suitably chosen integers. In this sense, our construction is an extension of the family of odd degree $n$ which was previously found by Shimizu and the author. The previous construction works only if the given degree $n$ has a prime divisor $p$ for which the pure cubic fields $\mathbb{Q}(p^{1/3})$ or $\mathbb{Q}((2p)^{1/3})$ satisfy a certain indivisibility conjecture of Ankeny–Artin–Chowla–Mordell type. In this time, we focus on the complementary cases, namely the cases of even degrees and exceptional odd degrees. Consequently, our recipe works well as a whole. This means that we can unconditionally produce infinitely many explicit non-singular plane curves of every degree $n = 5$ or $n \geq 7$ which violate the local-global principle. This gives a conclusion of the classical story of searching explicit ternary forms violating the local-global principle, which was initiated by Selmer (1951) and extended by Fujiwara (1972) and others.

We prove the convergence of (solid) ellipsoids to a Gaussian space in Gromov's concentration/weak topology as the dimension diverges to infinity. This gives the first discovered example of an irreducible nontrivial convergent sequence in the concentration topology, where ‘irreducible nontrivial’ roughly means to be not constructed from Lévy families nor box convergent sequences.

In this paper, we study the parabolic BGG categories for graded Lie superalgebras of Cartan type over the field of complex numbers. The gradation of such a Lie superalgebra $\mathfrak{g}$ naturally arises, with the zero component $\mathfrak{g}_{0}$ being a reductive Lie algebra. We first show that there are only two proper parabolic subalgebras containing Levi subalgebra $\mathfrak{g}_{0}$: the “maximal one” $\mathsf{P}_{\max}$ and the “minimal one” $\mathsf{P}_{\min}$. Furthermore, the parabolic BGG category arising from $\mathsf{P}_{\max}$ essentially turns out to be a subcategory of the one arising from $\mathsf{P}_{\min}$. Such a priority of $\mathsf{P}_{\min}$ in the sense of representation theory reduces the question to the study of the “minimal parabolic” BGG category $\mathcal{O}^{\min}$ associated with $\mathsf{P}_{\min}$. We prove the existence of projective covers of simple objects in these categories, which enables us to establish a satisfactory block theory. Most notably, our main results are as follows.

(1) We classify and obtain a precise description of the blocks of $\mathcal{O}^{\min}$.

(2) We investigate indecomposable tilting and indecomposable projective modules in $\mathcal{O}^{\min}$, and compute their character formulas.

A stochastic fractionally dissipative quasi-geostrophic equation with stochastic damping is considered in this paper. First, we show that the null solution is exponentially stable in the sense of $q^{-}$-th moment of $\|\cdot\|_{L^{q}}$, where $q > 2/(2\alpha - 1)$ and $q^{-}$ denotes the number strictly less than $q$ but close to it, and from this fact we further prove that the sample paths of solutions converge to zero almost surely in $L^{q}$ as time goes to infinity. In particular, a simple example is used to interpret the intuition. Then the uniform boundedness of pathwise solutions in $H^{s}$ with $s \geq 2 - 2\alpha$ and $\alpha \in (1/2, 1)$ is established, which implies the existence of non-trivial invariant measures of the quasi-geostrophic equation driven by nonlinear multiplicative noise.

Time-dependent free surface problem for the incompressible Navier–Stokes equations which describes the motion of viscous incompressible fluid nearly half-space are considered. We obtain global well-posedness of the problem for a small initial data in scale invariant critical Besov spaces. Our proof is based on maximal $L^{1}$-regularity of the corresponding Stokes problem in the half-space and special structures of the quasi-linear term appearing from the Lagrangian transform of the coordinate.