Boston College Algebraic Geometry Seminar

Given a fibration of complex projective manifolds $f\colon X\to Y$ with general fiber $F$, the famous Iitaka conjecture predicts the inequality $\kappa(K_X)\geq \kappa(K_F)+\kappa(K_Y)$. Recently Chang has shown that, when the stable base locus of -K_X is vertical over Y, a similar statement holds for the anticanonical divisor: $\kappa(-K_X)\leq \kappa(-K_F)+\kappa(-K_Y)$. Both Iitaka's conjecture and Chang's theorem are known to fail in positive characteristic. In this talk I will introduce a special class of positive characteristic varieties with negative canonical bundle, and show that Chang's theorem can be recovered when the general fiber $F$ belongs to this class. Based on joint work with Marta Benozzo and Chi-Kang Chang.

Let X be a Fano variety with G action. The quantum GIT conjecture predicts a formula for the quantum cohomology of "anti-canonical" GIT quotients X//G in terms of the equivariant quantum cohomology of X. The formula is motivated by ideas from 3- dimensional gauge theory ("Coulomb branches") and provides a vast generalization of Batyrev's formula for the quantum cohomology of a toric Fano variety. I will explain describe our ongoing work with C. Teleman proving this conjecture. The strategy of proof involves ideas from Hamiltonian Floer theory.

Symmetric products of Riemann surfaces play a crucial role in symplectic geometry and low-dimensional topology. They are essential ingredients for defining Heegaard Floer homology and serve as important examples of Liouville manifolds when the surfaces are open. In this talk, I will discuss ongoing work on the symplectic topology of these spaces through Liouville sectorial methods, along with examples as applications of this decomposition construction to homological mirror symmetry.

Let N_{g,d} be the locus of curves of genus g admitting a degree d cover of an elliptic curve. For fixed g, is conjectured that the classes of N_{g,d} on M_g are the Fourier coefficients of a cycle-valued quasi-modular form in d. A key difficulty is that these classes are often non-tautological, so lie outside the reach of many known techniques. Via the Torelli map, one can move the conjecture to one on certain Noether-Lefschetz loci on A_g, where one has access to different tools. I will explain some evidence for these conjectures, gathered from results of many people, some of which are joint with François Greer.

We prove that when even-degree hyperelliptic curves are ordered by the sizes of their coefficients, a positive proportion of them have no unexpected quadratic points --- i.e., no points defined over quadratic fields except for those that arise by pulling back rational points from P^1. To obtain this result, we combine a generalization of Selmer-group Chabauty (due to Poonen-Stoll) with new results on the average size of the 2-Selmer groups of Jacobians of even-degree hyperelliptic curves. This is joint work with Manjul Bhargava, Jef Laga, and Arul Shankar.

Motivated by gauging topological field theories, Teleman studied a Hamiltonian group action on a symplectic manifold and conjectured that its mirror is a holomorphic fibration.  In this talk, I will explain how such a fibration comes up from mirror construction via equivariant Lagrangian Floer theory, and how mirror maps come up from the equivariant obstruction of Lagrangian correspondence for symplectic quotient. If time is allowed, I will also explain some ongoing works on the relation with Seidel elements and degenerations. This is a joint work with Nai-Chung Conan Leung and Yan-Lung Leon Li.

Gromov-Witten invariants are constructed from virtual fundamental classes of moduli spaces of stable maps. It has been speculated since the beginning of the subject that such classes should admit refinements in complex cobordism. I will present joint work (in progress) with Abouzaid which provides a solution to this problem. The key insight is the application of various methods of resolution of singularities, including the recent approach by Abramovich-Temkin-Wlodarczyk.

Manin’s Conjecture predicts an asymptotic formula for the number of rational points of bounded height on a smooth Fano variety defined over a number field.  This formula emerged from heuristic arguments by Batyrev; however, these heuristics erroneously assumed certain moduli spaces of embedded rational curves are irreducible.

Geometric Manin’s Conjecture (GMC) refines Batyrev’s heuristics into a conjecture about free rational curves on Fano varieties.  In this talk, I will first review this refinement and motivate the framework of GMC with concrete examples.  I will then describe a recent proof of GMC for smooth coindex 3 Fano varieties over the complex numbers.  This is joint work with Andrew Burke and with Fumiya Okamura.

A celebrated 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.