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.

In studying Mellin transforms of multivariable polynomial maps, Gelfand defined the so-called Archimedean zeta function of a polynomial and conjectured that the archimedean zeta function has an meromorphic continuation on the whole complex plane in 1950s. Bernstein introduced the so-called Bernstein-Sato polynomial and solved the conjecture in 1970s. In this talk, I will discuss how we can extend the constructions for a finite union of polynomial maps by defining Bernstein-Sato ideals. Then I will discuss some conjectures about the algebro-geometric properties of such ideals and what we know about them.

I will present a compactification of the moduli space of maps from families of curves, to certain moduli spaces M, via the example of M being the GIT moduli space of binary forms of degree 2n. The main application of our results is the construction of certain moduli of fibered log Calabi-Yau pairs. This is a joint work with Andrea Di Lorenzo.

In this talk, we will introduce the concept of cluster type varieties. These are compactifications of an algebraic torus such that the volume form does not have any zeros on the compactification. In the first part of the talk, we will introduce several examples of cluster type varieties. In the second half of the talk, we will discuss about the property of being cluster type in families of Fano varieties.

A theorem of Popa and Schnell shows that if a smooth projective variety X admits a 1-form with no zeros it cannot be of general type. However, one expects far more stringent constraints on the geometry of those X actually admitting nonvanishing 1-forms. If X is not uniruled and assuming the conjectures of MMP, we show that X is birational to an isotrivial fibration over an abelian variety. This partially answers conjectures of Hao--Schreieder, Meng--Popa, and Chen--Church--Hao. The proof involves a decomposition result for families of Calabi-Yau varieties surjecting onto a fixed abelian variety. If X is uniruled, we also give a weak structure theorem that relies on using higher direct image Hodge modules in the method of Popa--Schnell.

I will survey recent joint work with Daniel Halpern-Leistner which constructs a partial compactification of (a quotient of) the space of Bridgeland stability conditions on a triangulated category. The partial compactification is modular in the sense that its boundary points parameterise a new type of homological structure: an augmented stability condition. I will introduce augmented stability conditions and explain some of the key properties of the space parameterising them. I will also explain how certain boundary strata of this space relate to braid group actions on semiorthogonal decompositions.

A branchvariety of a projective k-scheme X is a geometrically reduced scheme Y equipped with a finite map to X. Alexeev and Knutson showed the existence of a proper moduli space of branchvarieties with fixed Hilbert polynomial and degree sequence, but the projectivity of this space remained an open question. In this talk, we will discuss positivity results for some line bundles related to the determinant line bundle on the moduli space of equidimensional branchvarieties. As a consequence, we establish that this moduli space is projective.