Algebraic Geometry Seminar

In this work, we study étale cohomology of p-adic rigid-analytic spaces with F_p coefficients. For general (smooth) spaces, this cohomology theory does not behave so well. For example, F_p-cohomology groups of the 1-dimensional closed unit ball are infinite. Nevertheless, Scholze showed that H^i(X, F_p) are finite-dimensional for proper X. He further conjectured that these cohomology groups should satisfy Poincaré Duality when X is both smooth and proper. I will explain the proof of this conjecture using the concept of almost coherent sheaves that provides tools to "localize" the question in an appropriate sense and eventually reduce it to computations in group cohomology.

Log Gromov-Witten invariants, introduced by Abramovich-Chen-Gross-Siebert, are counts of curves in pairs (X,D) consisting of a smooth projective variety X together with a normal crossing divisor D, with prescribed tangency conditions along D. These invariants play a key role in mirror symmetry for log Calabi-Yau pairs (X,D), in which case D is an anticanonical divisor. After briefly reviewing log Gromov-Witten theory, I will explain a combinatorial recipe based on tropical geometry and wall-crossing algorithms to calculate such curve counts when (X,D) is obtained as a blow-up of a toric variety along hypersurfaces in the toric boundary divisor. This is based on joint work with Mark Gross.

In this talk, I will discuss my joint work with Sam Raskin on the derived geometric Satake equivalence. As time permits, I will explain how this work has been applied in the proof of the geometric Langlands conjecture and in my work with Hayash. The main point is that when combined with local-to-global methods, my results with Raskin give information about deformations of reducible local systems on a compact Riemann surface.

A fundamental problem at the confluence of algebraic geometry, commutative algebra, and representation theory is to understand the structure and vanishing behavior of the cohomology of line bundles on (partial) flag varieties. I will describe an answer in the case of the incidence correspondence (the partial flag variety consisting of pairs of a point in projective space and a hyperplane containing it), and highlight surprising connections to other questions of interest: the splitting of jet bundles on the projective line, the Han-Monsky representation ring, or Lefschetz properties for Artinian monomial complete intersections. This is based on joint work with Annet Kyomuhangi, Emanuela Marangone, and Ethan Reed.

Using tropical geometry, Block-Göttsche defined polynomials with the remarkable property to interpolate between Gromov-Witten counts of complex curves and Welschinger counts of real curves in toric del Pezzo surfaces. I will describe a generalization of Block-Göttsche polynomials to arbitrary, not-necessarily toric, rational surfaces and propose a conjectural relation with refined Donaldson-Thomas invariants. This is joint work in progress with Hulya Arguz.

In his 1998 ICM talk, Dubrovin conjectured that for a Fano manifold X, the derived category of X possesses a full exceptional collection if and only if the quantum cohomology of X is generically semi-simple, suggesting a deep connection between the derived category of X and its Gromov--Witten theory. This relationship has been clarified in recent years, with a prominent role being played by the quantum spectrum, that is, the eigenvalues of the operator c_1(X) *q - on H^*(X) where c_1(X) is the first Chern class of TX and *q denotes the quantum product at q. Kontsevich has conjectured that the quantum spectrum of X is closely related to semi-orthogonal decompositions of D^b(X). I will describe how, in the case of projective bundles, blow-ups, and standard flips, known semi-orthogonal decompositions of the derived category indeed correspond to the quantum spectrum at a special value of q. This is based on joint work with Yefeng Shen.