Algebraic Geometry Seminar

The Kodaira problem asks whether every compact Kähler manifold can be deformed to a projective one. While Voisin gave counterexamples in 2004, a modified version for non-uniruled spaces remains open, and in fact has been established in dimension at most three by Claudon, Höring, Lin and myself. I will review these results and then talk about the (im)possibility of extending the conjecture to uniruled spaces. If time permits, I will also outline a current project dealing with higher dimensions. The latter two works are joint with Martin Schwald (Essen).

In order to have a well behaved moduli functor for Fano varieties, it seems natural to restrict oneself to Fano varieties that are K-polystable. Recall, K-stability is an algebraic notion that characterizes when a smooth Fano variety admits a Kahler-Einstein metric. In this talk, we consider the behavior of uniform K-stability (a strengthening of K-stability) in families. We will explain that uniform K-stability is an open condition in Q-Fano families and the moduli functor of uniformly K-stable Q-Fano varieties is separated. Together with a boundedness result of C. Jiang, these results yield a separated DM stack paratmerizing uniformly K-stable Fano varieties of fixed dimension and volume. This is joint work with Yuchen Liu and Chenyang Xu.

I’ll begin with a discussion of the classification of vector bundles on P1 and explain what natural cohomology means in this context. Then I’ll consider the case of vector bundles on P1 x P1. In general vector bundles on surfaces are more complicated but a useful tool allows one to reduce many problems about vector bundles to questions of linear algebra. This is the theory of monads. I’ll discuss monads and show how they are used to prove a conjecture of Eisenbud and Scheryer about vector bundles on P1 x P1 with natural cohomology.

I will discuss an emerging theory where geometrical objects arising in representation theory are themselves viewed as "representations of a higher group". More general "higher groups" should arise as invariants of algebraic varieties and I will speculate on connections with higher algebraic K-theory and spaces of stability conditions. I will describe some aspects of the representation theory of the "higher group" associated to a point.

Birational superrigidity and K-stability are properties of Fano varieties that have many interesting geometric implications. For instance, birational superrigidity implies non-rationality and K-stability is related to the existence of Kähler-Einstein metrics. Nonetheless, both properties are hard to verify in general. In this talk, I will first explain how to relate birational superrigidity to K-stability using alpha invariants; I will then outline a method of proving birational superrigidity that works quite well with most families of index one Fano complete intersections and thereby also verify their K-stability. This is partly based on joint work with Charlie Stibitz and Yuchen Liu.

Birkar and Zhang recently introduced the notion of generalized pair. These pairs are closely related to the canonical bundle formula and have been a fruitful tool for recent developments in birational geometry. In this talk, I will introduce a version of the canonical bundle formula for generalized pairs. This machinery allows us to develop a theory of adjunction and inversion thereof for generalized pairs. I will conclude by discussing some applications to a conjecture of Prokhorov and Shokurov.

The minimal log discrepancy of an algebraic variety is an invariant which measures the singularites of the variety. For mild singularities the minimal log discrepancy is a non-negative real value; the closer to zero this value is, the more singular the variety. It is conjectured that in a fixed dimension, this invariant satisfies the ascending chain condition. In this talk we will show how boundedness of Fano varieties imply some local statements about the minimal log discrepancies of klt singularities. In particular, we will prove that the minimal log discrepancies of klt singularities which admit an e-plt blow-up can take only finitely many possible values in a fixed dimension. This result gives a natural geometric stratification of the possible mld's on a fixed dimension by finite sets. As an application, we will prove the ascending chain condition for minimal log discrepancies of exceptional singularities in arbitrary dimension.

People expect moduli spaces to be hyperbolic at least in the sense of stack. I will discuss known results about both analytic and algebraic hyperbolicity of certain moduli of varieties, for instance M_g and families of general type varieties. Then I will talk about hyperbolicity for base spaces of log smooth families of log general type pairs. I will use it to prove hyperbolicity of the moduli stack of Riemann surfaces of genus g with n marked points. This is joint work with Chuanhao Wei.

Kappa classes were introduced by Mumford, as a tool to explore the intersection theory of the moduli space of curves. Iterated use of the projection formula shows there is a close connection between the intersection theory of kappa classes on the moduli space of unpointed curves, and the intersection theory of psi classes on all moduli spaces. In terms of generating functions, we show that the potential for kappa classes is related to the Gromov-Witten potential of a point via a change of variables essentially given by complete symmetric polynomials, rediscovering a theorem of Manin and Zokgraf from '99. Surprisingly, the starting point of our story is a combinatorial formula that relates intersections of kappa classes and psi classes via a graph theoretic algorithm (the relevant graphs being dual graphs to stable curves). Further, this story is part of a large wall-crossing picture for the intersection theory of Hassett spaces, a family of birational models of the moduli space of curves. This is joint work with Vance Blankers (arXiv:1810.11443) .

Recent work of Kento Fujita, Yuji Odaka and Chen Jiang shows that among K-semistable Fano manifolds, the projective space can be characterized in terms of either the alpha invariant or the volume. In this talk, I will generalize the definition of alpha invariant to arbitrary codimension, and show that we can characterize projective spaces among all K-semistable Fano manifolds in terms of higher codimensional alpha invariants. This result also demonstrates the relation between alpha invariants and volumes in the characterization problem of projective spaces among K-semistable Fano manifolds.

Any central simple algebra A over a field K is a form of a matrix algebra. Further A/K comes equipped with a reduced norm map which is obtained by twisting the determinant function. Every element in the commutator subgroup [A*, A*] has reduced norm 1 and hence lies in SL_1(A), the group of reduced norm one elements of A. Whether the reverse inclusion holds was formulated as a question in 1943 by Tannaka and Artin in terms of the triviality of the reduced Whitehead group SK_1(A) := SL_1(A)/[A*,A*]. Platonov negatively settled the Tannaka-Artin question by giving a counter example over a cohomological dimension (cd) 4 base field. In the same paper however, the triviality of SK_1(A) was shown for all algebras over cd at most 2 fields. In this talk, we investigate the situation for l-torsion algebras over a class of cd 3 fields of some arithmetic flavour, namely function fields of p-adic curves where l is any prime not equal to p. We partially answer a question of Suslin by proving the triviality of the reduced Whitehead group for these algebras. The proof relies on the techniques of patching as developed by Harbater-Hartmann-Krashen and exploits the arithmetic of these fields.