arXiv subjects

Daniel S. Sage

Publications and source records attributed to Daniel S. Sage.

25 records · Page 2Linked to original sources

arXiv2011

Generalized Serre conditions and perverse coherent sheaves

In algebraic geometry, one often encounters the following problem: given a scheme X, find a proper birational morphism from Y to X where the geometry of Y is "nicer" than that of X. One version of this problem, first studied by Faltings, requires Y to be Cohen-Macaulay; in this case Y is called a Macaulayfication of X. In another variant, one requires Y to satisfy the Serre condition S_r. In this paper, the authors introduce generalized Serre conditions--these are local cohomology conditions which include S_r and the Cohen-Macaulay condition as special cases. To any generalized Serre condition S_rho, there exists an associated perverse t-structure on the derived category of coherent sheaves on a suitable scheme X. Under appropriate hypotheses, the authors characterize those schemes for which a canonical finite S_rho-ification exists in terms of the intermediate extension functor for the associated perversity. Similar results, including a universal property, are obtained for a more general morphism extension problem called S_rho-extension.

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arXiv2011

Isomonodromic deformations of connections with singularities of parahoric formal type

In previous work, the authors have developed a geometric theory of fundamental strata to study connections on the projective line with irregular singularities of parahoric formal type. In this paper, the moduli space of connections that contain regular fundamental strata with fixed combinatorics at each singular point is constructed as a smooth Poisson reduction. The authors then explicitly compute the isomonodromy equations as an integrable system. This result generalizes work of Jimbo, Miwa, and Ueno to connections whose singularities have parahoric formal type.

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arXiv2008

Perverse coherent sheaves and the geometry of special pieces in the unipotent variety

Let X be a scheme of finite type over a Noetherian base scheme S admitting a dualizing complex, and let U be an open subset whose complement has codimension at least 2. We extend the Deligne-Bezrukavnikov theory of perverse coherent sheaves by showing that a coherent middle extension (or intersection cohomology) functor from perverse sheaves on U to perverse sheaves on X may be defined for a much broader class of perversities than has previously been known. We also introduce a derived category version of the coherent middle extension functor. Under suitable hypotheses, we introduce a construction (called "S2-extension") in terms of perverse coherent sheaves of algebras on X that takes a finite morphism to U and extends it in a canonical way to a finite morphism to X. In particular, this construction gives a canonical "S2-ification" of appropriate X. The construction also has applications to the "Macaulayfication" problem, and it is particularly well-behaved when X is Gorenstein. Our main goal, however, is to address a conjecture of Lusztig on the geometry of special pieces (certain subvarieties of the unipotent variety of a reductive algebraic group). The conjecture asserts in part that each special piece is the quotient of some variety (previously unknown in the exceptional groups and in positive characteristic) by the action of a certain finite group. We use S2-extension to give a uniform construction of the desired variety.

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arXiv2007

Staggered sheaves on partial flag varieties

Staggered $t$-structures are a class of $t$-structures on derived categories of equivariant coherent sheaves. In this note, we show that the derived category of coherent sheaves on a partial flag variety, equivariant for a Borel subgroup, admits an artinian staggered $t$-structure. As a consequence, we obtain a basis for its equivariant $K$-theory consisting of simple staggered sheaves.

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arXiv2007

On special pieces, the Springer correspondence, and unipotent characters

Let G be a reductive algebraic group over the algebraic closure of a finite field F_q of good characteristic. In this paper, we demonstrate a remarkable compatibility between the Springer correspondence for G and the parametrization of unipotent characters of G(F_q). In particular, we show that in a suitable sense, "large" portions of these two assignments in fact coincide. This extends earlier work of Lusztig on Springer representations within special pieces of the unipotent variety.

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arXiv1999

Group Actions on Central Simple Algebras

Let $G$ be a group, $F$ a field, and $A$ a finite-dimensional central simple algebra over $F$ on which $G$ acts by $F$-algebra automorphisms. We study the ideals and subalgebras of $A$ which are preserved by the group action. Let $V$ be the unique simple module of $A$. We show that $V$ is a projective representation of $G$ and $A\cong\text{End}_D(V)$ makes $V$ into a projective representation. We then prove that there is a natural one-to-one correspondence between $G$-invariant $D$-submodules of $V$ and invariant left (and right) ideals of $A$. Under the assumption that $V$ is irreducible, we show that an invariant (unital) subalgebra must be a simply embedded semisimple subalgebra. We introduce induction of $G$-algebras. We show that each invariant subalgebras is induced from a simple $H$-algebra for some subgroup $H$ of finite index and obtain a parametrization of the set of invariant subalgebras in terms of induction data. We then describe invariant central simple subalgebras. For $F$ algebraically closed, we obtain an entirely explicit classification of the invariant subalgebras. Furthermore, we show that the set of invariant subalgebras is finite if $G$ is a finite group. Finally, we consider invariant subalgebras when $V$ is a continuous projective representation of a topological group $G$. We show that if the connected component of the identity acts irreducibly on $V$, then all invariant subalgebras are simple. We then apply our results to obtain a particularly nice solution to the classification problem when $G$ is a compact connected Lie group and $F=\mathbf C$.

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arXiv1999

The Geometry of Fixed Point Varieties on Affine Flag Manifolds

Let $G$ be a semisimple, simply connected, algebraic group over an algebraically closed field $k$ with Lie algebra $\frak g$. We study the spaces of parahoric subalgebras of a given type containing a fixed nil-elliptic element of $\frak g\otimes k((π))$, i.e. fixed point varieties on affine flag manifolds. We define a natural class of $k^*$-actions on affine flag manifolds, generalizing actions introduced by Lusztig and Smelt. We formulate a condition on a pair $(N,f)$ consisting of $N\in\frak{g}\otimes k((π))$ and a $k^*$-action $f$ of the specified type which guarantees that $f$ induces an action on the variety of parahoric subalgebras containing $N$. For the special linear and symplectic groups, we characterize all regular semisimple and nil-elliptic conjugacy classes containing a representative whose fixed point variety admits such an action. We then use these actions to find simple formulas for the Euler characteristics of those varieties for which the $k^*$-fixed points are finite. We also obtain a combinatorial description of the Euler characteristics of the spaces of parabolic subalgebras containing a given element of certain nilpotent conjugacy classes of $\frak g$.

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