International Journal of Mathematics最新文献

Let $X$ be a smooth complex projective curve and let $E$ be a vector bundle on $X$ which is not semistable. We consider a flag bundle $\pi :\mathrm{\text{Fl}}(E)\to X$ parametrizing certain flags of fibers of $E$. The dimensions of the successive quotients of the flags are determined by the ranks of vector bundles appearing in the Harder–Narasimhan filtration of $E$. We compute the Seshadri constants of nef line bundles on $\mathrm{\text{Fl}}(E)$.

For any almost-simple group $G$ over an algebraically closed field $k$ of characteristic zero, we describe the automorphism group of the moduli space of semistable $G$-bundles over a connected smooth projective curve $C$ of genus at least $4$. The result is achieved by studying the singular fibers of the Hitchin fibration. As a byproduct, we provide a description of the irreducible components of two natural closed subsets in the Hitchin basis: the divisor of singular cameral curves and the divisor of singular Hitchin fibers.

We consider spin manifolds with an Einstein metric, either Riemannian or indefinite, for which there exists a Killing spinor. We describe the intrinsic geometry of nondegenerate hypersurfaces in terms of a PDE satisfied by a pair of induced spinors, akin to the generalized Killing spinor equation.

Conversely, we prove an embedding result for real analytic pseudo-Riemannian manifolds carrying a pair of spinors satisfying this condition.

We give a geometric characterization of plus-one generated projective line arrangements that are next-to-free. We present new succinct proofs, via associated line bundles, for some properties of plus-one generated projective line arrangements.

We investigate stability conditions related to the existence of solutions of the Hull–Strominger system with prescribed balanced class. We build on recent work by the authors, where the Hull–Strominger system is recasted using non-Hermitian Yang–Mills connections and holomorphic Courant algebroids. Our main development is a notion of harmonic metric for the Hull–Strominger system, motivated by an infinite-dimensional hyperKähler moment map and related to a numerical stability condition, which we expect to exist for families of solutions. We illustrate our theory with an infinite number of continuous families of examples on the Iwasawa manifold.

In this paper, we establish nonlinear ellipticity of the equation of contact instantons with Legendrian boundary condition on punctured Riemann surfaces by proving the a priori elliptic coercive estimates for the contact instantons with Legendrian boundary condition, and prove an asymptotic exponential $C^{\infty }$-convergence result at a puncture under the uniform $C^1$ bound. We prove that the asymptotic charge of contact instantons at the punctures under the Legendrian boundary condition vanishes. This eliminates the phenomenon of the appearance of spiraling cusp instanton along a Reeb core, which removes the only remaining obstacle towards the compactification and the Fredholm theory of the moduli space of contact instantons in the open string case, which plagues the closed string case. Leaving the study of $C^1$-estimates and details of Gromov-Floer-Hofer style compactification of contact instantons to [27], we also derive an index formula which computes the virtual dimension of the moduli space. These results are the analytic basis for the sequels [27]–[29] and [36] containing applications to contact topology and contact Hamiltonian dynamics.

Let $X$ be either a general hypersurface of degree $n+1$ in ${\mathbb{P}}^n$ or a general $(2,n)$ complete intersection in ${\mathbb{P}}^{n+1},n\ge 4$. We construct balanced rational curves on $X$ of all high enough degrees. If $n=4$ or $g=1$, we construct rigid curves of genus $g$ on $X$ of all high enough degrees. As an application we construct some rigid bundles on Calabi–Yau threefolds. In addition, we construct some low-degree balanced rational curves on hypersurfaces of degree $n+2$ in ${\mathbb{P}}^n$.

In this paper, we prove two separate lower bounds — one for nondegenerate convex domains and the other for nondegenerate $ℂ$-convex (but not necessarily convex) domains — for the squeezing function that hold true for all domains in ${ℂ}^n$, for a fixed $n\ge 2$, of the stated class. We provide explicit expressions in terms of $n$ for these estimates.

We construct a new family of knot concordance invariants ${𝜃}^{(q)}(K)$, where $q$ is a prime number. Our invariants are obtained from the equivariant Seiberg–Witten–Floer cohomology, constructed by the author and Hekmati, applied to the degree $q$ cyclic cover of $S^3$ branched over $K$. In the case $q=2$, our invariant ${𝜃}^{(2)}(K)$ shares many similarities with the knot Floer homology invariant ${\nu }^{+}(K)$ defined by Hom and Wu. Our invariants ${𝜃}^{(q)}(K)$ give lower bounds on the genus of any smooth, properly embedded, homologically trivial surface bounding $K$ in a definite $4$-manifold with boundary $S^3$.

It is known that the LMO invariant of 3-manifolds with positive first Betti numbers is relatively weak and can be determined by “(semi-)classical” invariants such as the cohomology ring, the Alexander polynomial, and the Casson–Walker–Lescop invariant.

In this paper, we formulate a refinement of the LMO invariant for 3-manifolds with the first Betti number 1. It dominates the perturbative SO(3) invariant of such 3-manifolds, which is the power series invariant formulated by the arithmetic perturbative expansion of the quantum SO(3) invariants of such 3-manifolds. As the 2-loop part of the refinement of the LMO invariant, we define the 2-loop polynomial of such 3-manifolds. Further, as the $𝔰{𝔩}_m$ reduction at large $m$ limit of the $\ell$-loop part of the refinement of the LMO invariant for $\ell \le 5$, we formulate an $\ell$-variable polynomial invariant of such 3-manifolds whose Alexander polynomial is constant.