Semiorthogonal Decompositions on Total Spaces of Tautological Bundles

Let $U$ be the tautological subbundle on the Grassmannian $\operatorname{Gr}(k, n)$. There is a natural morphism $\textrm{Tot}(U) \to{\mathbb{A}}^n$. Using it, we give a semiorthogonal decomposition for the bounded derived category $D^b_{\!\textrm{coh}}(\textrm{Tot}(U))$ into several exceptional objects and several copies of $D^b_{\!\textrm{coh}}({\mathbb{A}}^n)$. We also prove a global version of this result: given a vector bundle $E$ with a regular section $s$, consider a subvariety of the relative Grassmannian $\operatorname{Gr}(k, E)$ of those subspaces that contain the value of $s$. The derived category of this subvariety admits a similar decomposition into copies of the base and the zero locus of $s$. This may be viewed as a generalization of the blow-up formula of Orlov, which is the case $k = 1$.

CATEGORICAL PROOF OF HOLOMORPHIC ATIYAH–BOTT FORMULA

Given a $2$-commutative diagramin a symmetric monoidal $(\infty ,2)$-category $\mathscr{E}$ where $X,Y\in \mathscr{E}$ are dualizable objects and $\unicode[STIX]{x1D711}$ admits a right adjoint we construct a natural morphism $\mathsf{Tr}_{\mathscr{E}}(F_{X})\xrightarrow[{}]{~~~~~}\mathsf{Tr}_{\mathscr{E}}(F_{Y})$ between the traces of $F_{X}$ and $F_{Y}$, respectively. We then apply this formalism to the case when $\mathscr{E}$ is the $(\infty ,2)$-category of $k$-linear presentable categories which in combination of various calculations in the setting of derived algebraic geometry gives a categorical proof of the classical Atiyah–Bott formula (also known as the Holomorphic Lefschetz fixed point formula).

Towards a symplectic version of the Chevalley restriction theorem

If $(G,V)$ is a polar representation with Cartan subspace $\mathfrak{c}$ and Weyl group $W$, it is shown that there is a natural morphism of Poisson schemes $\mathfrak{c}\oplus \mathfrak{c}^{\ast }/W\rightarrow V\oplus V^{\ast }/\!\!/\!\!/G$. This morphism is conjectured to be an isomorphism of the underlying reduced varieties if$(G,V)$ is visible. The conjecture is proved for visible stable locally free polar representations and some other examples.

MODELS OF TORSORS AND THE FUNDAMENTAL GROUP SCHEME

Given a relative faithfully flat pointed scheme over the spectrum of a discrete valuation ring $X\rightarrow S$, this paper is motivated by the study of the natural morphism from the fundamental group scheme of the generic fiber $X_{\unicode[STIX]{x1D702}}$ to the generic fiber of the fundamental group scheme of $X$. Given a torsor $T\rightarrow X_{\unicode[STIX]{x1D702}}$ under an affine group scheme $G$ over the generic fiber of $X$, we address the question of finding a model of this torsor over $X$, focusing in particular on the case where $G$ is finite. We provide several answers to this question, showing for instance that, when $X$ is integral and regular of relative dimension 1, such a model exists on some model $X^{\prime }$ of $X_{\unicode[STIX]{x1D702}}$ obtained by performing a finite number of Néron blowups along a closed subset of the special fiber of $X$. Furthermore, we show that when $G$ is étale, then we can find a model of $T\rightarrow X_{\unicode[STIX]{x1D702}}$ under the action of some smooth group scheme. In the first part of the paper, we show that the relative fundamental group scheme of $X$ has an interpretation as the Tannaka Galois group of a Tannakian category constructed starting from the universal torsor.

Moduli of vector bundles on higher-dimensional base manifolds — Construction and variation

We survey recent progress in the study of moduli of vector bundles on higher-dimensional base manifolds. In particular, we discuss an algebro-geometric construction of an analogue for the Donaldson–Uhlenbeck compactification and explain how to use moduli spaces of quiver representations to show that Gieseker–Maruyama moduli spaces with respect to two different chosen polarizations are related via Thaddeus-flips through other “multi-Gieseker”-moduli spaces of sheaves. Moreover, as a new result, we show the existence of a natural morphism from a multi-Gieseker moduli space to the corresponding Donaldson–Uhlenbeck moduli space.