We revisit Rozansky’s construction of Khovanov homology for links in ${\mathit{S}}^2\times {\mathit{S}}^1$, extending it to define the Khovanov homology $\mathrm{Kh}(\mathit{L})$ for links L in ${\mathit{M}}^{\mathit{r}}={\mathrm{\#}}^{\mathit{r}}({\mathit{S}}^2\times {\mathit{S}}^1)$ for any r. The graded Euler characteristic of $\mathrm{Kh}(\mathit{L})$ can be used to recover WRT invariants at certain roots of unity and also recovers the evaluation of L in the skein module $\mathcal{S}({\mathit{M}}^{\mathit{r}})$ of Hoste and Przytycki when L is null-homologous in ${\mathit{M}}^{\mathit{r}}$. The construction also allows for a clear path toward defining a Lee’s homology ${\mathrm{Kh}}^{\prime }(\mathit{L})$ and associated s-invariant for such L, which we will explore in an upcoming paper. We also give an equivalent construction for the Khovanov homology of the knotification of a link in ${\mathit{S}}^3$ and show directly that this is invariant under handle-slides, in the hope of lifting this version to give a stable homotopy type for such knotifications in a future paper.

In this paper, we study the Khovanov homology of an alternating virtual link L and show that it is supported on $\mathit{g}+2$ diagonal lines, where g equals the virtual genus of L. Specifically, we show that $\mathit{K}{\mathit{h}}^{\mathit{i},\mathit{j}}(\mathit{L})$ is supported on the lines $\mathit{j}=2\mathit{i}-{\mathit{\sigma }}_{\mathit{\xi }}+2\mathit{k}-1$ for $0\le \mathit{k}\le \mathit{g}+1$ where ${\mathit{\sigma }}_{{\mathit{\xi }}^{\ast }}(\mathit{L})+2\mathit{g}={\mathit{\sigma }}_{\mathit{\xi }}(\mathit{L})$ are the signatures of L for a checkerboard coloring ξ and its dual ${\mathit{\xi }}^{\ast }$. Of course, for classical links, the two signatures are equal, and this recovers Lee’s H-thinness result for $\mathit{K}{\mathit{h}}^{\ast ,\ast }(\mathit{L})$. Our result applies more generally to give an upper bound for the homological width of the Khovanov homology of any checkerboard virtual link L. The bound is given in terms of the alternating genus of L, which can be viewed as the virtual analogue of the Turaev genus. The proof rests on associating, with any checkerboard colorable link L, an alternating virtual link diagram with the same Khovanov homology as L.

In the process, we study the behavior of the signature invariants under vertical and horizontal mirror symmetry. We also compute the Khovanov homology and Rasmussen invariants in numerous cases and apply them to show nonsliceness and determine the slice genus for several virtual knots. ??Table]ras-table at the end of the paper lists the signatures, Khovanov polynomial, and Rasmussen invariant for alternating virtual knots up to six crossings.

We prove that every projectively normal Fano manifold in ${\mathbb{P}}^{\mathit{n}+\mathit{r}}$ of index 1, codimension r, and dimension $\mathit{n}\ge 10\mathit{r}$ is birationally superrigid and K-stable. This result was previously proved by Zhuang under the complete intersection assumption.

In characteristic zero, quotient singularities are log terminal. Moreover, we can check whether a quotient variety is canonical or not by using only the age of each element of the relevant finite group if the group does not have pseudoreflections. In positive characteristic, a quotient variety is not log terminal in general. In this paper, we give an example of a quotient variety that is not log terminal such that the quotient varieties associated with any proper subgroups are canonical. In particular, we cannot determine whether a given quotient singularity is canonical by looking at proper subgroups.

We propose a general definition of mathematical instanton bundle with given charge on any Fano threefold extending the classical definitions on ${\mathbb{P}}^3$ and on Fano threefold with cyclic Picard group. Then we deal with the case of the blowup of ${\mathbb{P}}^3$ at a point, giving an explicit construction of instanton bundles satisfying some important extra properties. Moreover, we show that they correspond to smooth points of a component of the moduli space.

We explore the equimultiplicity theory of the F-invariants Hilbert–Kunz multiplicity, F-signature, Frobenius Betti numbers, and Frobenius Euler characteristic in strongly F-regular rings. Techniques introduced in this paper provide a unified approach to the study of localization of these invariants and detection of singularities.

We generalize some results of Coray on closed points on cubic hypersurfaces. We show that certain symmetric products of cubic hypersurfaces are stably birational.

This paper introduces techniques for computing a variety of numerical invariants associated with a Legendrian knot in a contact manifold presented by an open book with a Morse structure. Such a Legendrian knot admits a front projection to the boundary of a regular neighborhood of the binding. From this front projection, we compute the rotation number for any null-homologous Legendrian knot as a count of oriented cusps and linking or intersection numbers; in the case that the manifold has nontrivial second homology, we can recover the rotation number with respect to a Seifert surface in any homology class. We also provide explicit formulas for computing the necessary intersection numbers from the front projection, and we compute the Euler class of the contact structure supported by the open book. Finally, we introduce a notion of Lagrangian projection and compute the classical invariants of a null-homologous Legendrian knot from its projection to a fixed page.