Algebraic Geometry Seminar

Traditionally, we use mirror symmetry to map a difficult problem (A-model) to an easier problem (B-model). Recently, there is a great deal of activities in mathematics to understand the modularity properties of Gromov-Witten theory, a phenomenon suggested by BCOV almost twenty years ago. Mirror symmetry is again used in a crucial way. However, the new usage of mirror does not map a difficult problem to easy problem. Instead, we make both side of mirror symmetry to work together in a deep way. I will explain this interesting phenomenon in the talk.

We present some recent results concerning the birational geometry of varieties of maximal Albanese dimension obtained by the means of Pareschi--Popa M-regularity theory. Many of these result had been obtained in collaboration with Z. Jiang and M. Lahoz.

Over perfect fields, the geometry of regular del Pezzo surfaces has been classified, but over imperfect fields, the problem remains largely open. We construct the first examples of regular del Pezzo surfaces X that have positive irregularity h^1(X, O_X ) > 0. Our construction is by quotienting a regular, quasi-linear surface (i.e. a regular variety that is geometrically a non-reduced first-order neighborhood of a plane) by explicit rank 1 foliations. We also find a restriction on the integer pairs that are possible as the anti-canonical degree and irregularity of such surfaces.

This talk is a continuation of a talk given by Karl Schwede during the Fall semester. It is based on a joint project with Karl Schwede and Wenliang Zhang aiming to interpret many notions of the F-singularity theory in the relative setting. One of the surprising results is the definition of a relative test ideal that restricts to the test ideal on each fiber. The focus of the current talk will be applications to global geometry, such as: behavior of the canonical adjoint linear sub-system S^0 in flat families, definition of canonical subsheaves of pushforards of adjoint line bundles, global generation result for the latest, etc.

Hironaka's theorem on resolution of singularities allows to study the geometry of a singular variety by associating to it a smooth birational model. A more intrinsic approach to study singularities was proposed by Nash. The idea is to look at the space of arcs (i.e. analytic germs of curves) passing through the singular points. This space decomposes into finitely many irreducible families, and carries much of the information encoded in a resolution. The Nash problem gives a precise formulation of how such families of arcs should relate to resolutions of singularities. In this talk I will give an overview of the history and solution of the problem.

In this talk I will explain how some familiar constructions in birational geometry can be studied using tensor triangular geometry.

I will report on recent work with C. Schnell, in which we prove that every holomorphic one-form on a variety of general type must vanish at some point (together with a suitable generalization to arbitrary Kodaira dimension). The proof makes use of generic vanishing theory for Hodge D-modules on abelian varieties.

The title refers to a class of homogeneous complex manifolds which includes the period domains for Hodge structures of weight at least two. The absence of automorphic forms for higher weight variations of Hodge structure suggests the theorem in the title. This talk will present a proof of this theorem, which requires some interesting geometry of non-classical domains. This is joint work with Phillip Griffiths and Colleen Robles.

The moduli spaces of stable quasimaps unify various moduli appearing in the study of Gromov-Witten Theory. We introduce big I-functions as the quasimap version of J-functions, generalizing Givental's small I-functions of smooth toric complete intersections. The J-functions are the GW counterparts of periods of mirror families. We discuss some advantages of I-functions, in particular an explanation of mirror maps. This is joint work with I. Ciocan-Fontanine. If time permitted, I will also report on the stable quasimaps for orbifold targets and I-functions of toric orbifolds. This is joint work with I. Ciocan-Fontanine and D. Cheong.

This talk is aimed to give a classification of the pair (S, G), where S is a smooth minimal surface of general type with p_g=0 and K^2=7, and G is a subgroup of Aut(S), and G is isomorphic to (Z/2Z)^2. From the classification, a new family of surfaces is constructed. These new surfaces are the first known surfaces with p_g=0, K^2=7 and with birational bicanonical maps.

The space of stable maps to a smooth variety relative to a smooth divisor, defined by Jun Li, is an important moduli space in Gromov-Witten theory, which arises naturally when one tries to study Gromov-Witten invariants by degeneration. It's deformation theory however is complicated. A related space is the space of logarithmic stable maps, defined by B.Kim. This space has the same Gromov-Witten invariants as the space of relative stable maps, but its deformation theory has better formal properties. In this talk I will briefly describe these moduli spaces and explain how to do localization in the space of logarithmic stable maps when the targets admit a torus action. This is joint work with E. Routis.

I report on joint work with Luca Migliorini at Bologna. If you have a map of complex projective manifolds, then the rational cohomology of the domain splits into a direct sum of pieces in a way dictated by the singularities of the map. By Poincare' duality, the corresponding projections can be viewed as cohomology classes (projectors) on the self-product of the domain. These projectors are Hodge classes, i.e. rational and of type (p,p) for the Hodge decomposition. Take the same situation after application of an automorphism of the ground field of complex numbers. The new projectors are of course Hodge classes. On the other hand, you can also transplant, using the field automorphism, the old projectors into the new situation and it is not clear that the new projectors and the transplants of the old projectors coincide. We prove they do, thus proving that the projectors are absolute Hodge classes, i.e. their being of Hodge type survives the totally discontinuous process of a field automorphism. We also prove that these projectors are motivated in the sense of Andre.

The question that the Crepant Resolution Conjecture (CRC) wants to address is: given an orbifold X that admits a crepant resolution Y, can we systematically compare the Gromov-Witten theories of the two spaces? That this should happen was first observed by physicists and the question was imported into mathematics by Y.Ruan, who posited as the search for an isomorphism in the quantum cohomologies of the two spaces. In the last fifteen years this question has evolved and found different formulations which various degree of generality and validity. Perhaps the most powerful approach to the CRC is through Givental's formalism. In this case, Coates, Corti, Iritani and Tseng propose that the CRC should consist of the natural comparison of geometric objects constructed from the GW potential fo the space. We explore this approach in the setting of open GW invariants. We formulate an open version of the CRC using this formalism, and verify it for the family of A_n singularities. Our approach is well tuned with Iritani's approach to the CRC via integral structures

A rational curve on a smooth algebraic variety is very free if the pullback of the tangent bundle is positive. This notion is the key to the theory of rational curves and has various applications both in geometry and in arithmetics. In characteristic zero, the general theory of Kollar-Miyaoka-Mori predicts the existence of such curves on Fano varieties (or more generally, on rationally connected varieties). However, constructing very free curves is usually difficult, especially in arbitrary characteristic. In this talk, first I will survey basic results which motivate the search of very free curves. Then I will focus on several examples where we can construct them concretely.

Tropicalization is a procedure for associating a polyhedral complex to a subvariety of an algebraic torus. We explain the method of tropicalization and study the question of which graphs arise from tropicalizing algebraic curves. By applying Baker's technique of specialization of linear systems from curves to graphs, we are able to give a necessary condition for a balanced weighted graph to be the tropicalization of a curve. Our condition is phrased in terms of the harmonic theory of graphs, reproduces the known necessary conditions, and also gives new conditions. Moreover, our method gives a combinatorial way of thinking about the deformation theory of algebraic varieties.

Suppose that f : X \to S is a flat family over a smooth variety of characteristic p > 0. In the same setting in characteristic zero, the multiplier ideal of X restricts to the multipier ideal of the fibers for most fibers X_s. There is no hope that the same statement can hold in characteristic p > 0 (either for multiplier ideals, or their analogs test ideals) because it might happen that even if X is smooth, the fibers of f might not even be reduced, and even if they are reduced, might have horrendus singularities. In this talk I will discuss a way to correct for this phenomenon, and thus obtain a generic restriction theorem for test ideals in positive characteristic. This is joint work with Zsolt Patakfalvi and Wenliang Zhang.

In this joint work with Jun Li, we prove by Hilbert-Mumford criterion that a slope stable polarized weighted pointed nodal curve is Chow asymptotic stable.

In joint work with Mirel Caibar we show that the classical height pairing between algebraic (n-1)-cycles on a (2n-1)-dimensional complex projective manifold X is the imaginary part of a natural (multivalued) bi-holomorphic function on components of the Hilbert scheme of X. This pairing is intimately related to the Abel-Jacobi image of the respective cycles. Furthermore this pairing can be extended to integral currents whose support is a real (2n-2)-dimensional oriented submanifold of X. Properties and potential applications of the extended pairing will be presented.

In this talk I will describe the behavior under derived equivalence of a twisted version of Hochschild homology. This result is then applied to study the derived invariance of cohomological support loci, fibrations onto curves, the Albanese dimension, and certain Hodge numbers of special classes of irregular varieties.

If X is a smooth variety with a proper action of an algebraic group G, its equivariant K-theory ring $K_G(X)$ possesses power (or Adams) operations (and associated lambda ring operations) which are compatible with the Chern character and Chern classes. But $K_G(X)$ is a subring of the equivariant K-theory $K_G(IX)$ of the inertial variety IX endowed with a so-called inertial product. The prototypical example of such an inertial K-theory ring is the K-theoretic version of the Chen-Ruan orbifold cohomology. We show that under certain conditions, the inertial K-theory ring admits inertial generalizations of power operations, Chern classes, and the Chern character. The power operations are then used to compare the virtual K-theory ring of P(1,2) and P(1,3) with the K-theory of a crepant resolution. This is joint work with with D. Edidin and T. Jarvis.

In this talk, I will consider the moduli problem for smooth complete intersections in projective space. I will show the existence of a separated coarse moduli space. This coarse moduli space will be shown to be quasi-projective in some particular cases (for instance for codimension 2 Fano complete intersections).

We complete the explicit description of 3-fold flips studied by Kollár and Mori. The new cases occur in the minimal model program for a one parameter family of surfaces, which is used to describe compact moduli spaces of surfaces. The classification is understood in terms of a universal family of K-negative surfaces with an interesting combinatorial structure. This is joint work with Jenia Tevelev and Giancarlo Urzua.

This work is motivated by Kollár's conjecture on symplectic deformation invariance of rational connectedness and "symplectic birational geometry". I will discuss some simple observations which seem to suggest that extremal rays of the Mori cone deform together with the deformation of the variety as a symplectic manifold. In some special cases, one can use these observations to prove that the maximal rationally connected quotient of a variety is a symplectic deformation invariant, a stronger version of Kollár's original conjecture.

When X is a smooth complex variety, it was shown by Ein, Lazarsfeld and Mustata that there is a general correspondence between cylinders in the space of arcs of X and divisorial valuations of the function field of X. Via this correspondence, the codimension of the cylinders corresponds to the log discrepancy of the divisorial valuation. The use of log resolutions in their proof restricted the result to ground field of characteristic zero. In this talk, I'll show this correspondence holds in arbitrary characteristic. In particular, we have Mustata's formula, relating the log canonical threshold of a pair to the asymptotic behavior of the dimensions of the jet schemes, also available in positive characteristic. This has interesting applications, for example, to an inequality between log canonical threshold and F-pure threshold and to a version of inversion of adjunction in positivity characteristic.

The Schubert subvarieties of a rational homogeneous variety X are distinguished by the fact that their homology classes form an additive basis of the integer homology of X. It is then natural to ask: does a Schubert classes admit any other algebraic representatives (aside from the Schubert variety)? It is a remarkable consequence of Kostant's work on Lie algebra homology that, in the the case that X is cominuscule -- equivalently, X admits the structure of a compact Hermitian symmetric space (eg. X is a complex Grassmannian) -- the algebraic representatives of a Schubert class are solutions of a system of differential equations. This allows us to apply differential geometric techniques to the problem. I will discuss this, and related questions, and the essential role played by representation theory in determining both rigidity (the Schubert varieties are the only algebraic representatives) and flexibility.

This is joint work with Chenyang Xu. Combining work of Esnault with work of de Jong - He - Starr on "rational simple connectedness", we prove the existence of rational points of many varieties defined over global function fields, i.e., function fields of curves over finite fields. In this way we get uniform proofs and extensions of (1) Lang's theorem that every global function field $K$ is $C_2$, (2) a theorem of Brauer - Hasse - Noether that the period equals the index for every division algebra over $K$, and (3) the "split case" of Harder's proof of Serre's Conjecture II over $K$. We also get upper bounds on the heights of our rational points, independent of the characteristic.

A basic theorem in Hodge theory is the isomorphism between de Rham and Betti cohomology for complex manifolds; this follows directly from the Poincare lemma. The p-adic analogue of this comparison lies deeper, and was the subject of a series of extremely influential conjectures made by Fontaine in the early 80s (which have since been established by various mathematicians). In my talk, I will first discuss the geometric motivation behind Fontaine’s conjectures, and then explain a simple new proof based on general principles in derived algebraic geometry — specifically, derived de Rham cohomology — and some classical geometry with curve fibrations. This work builds on ideas of Beilinson who proved the de Rham comparison conjecture this way.

The celebrated Mirror Theorem of Givental and Lian-Liu-Yau states that the A model (quantum cohomology, rational curve counting) of the Fermat quintic threefold is equivalent to the B model (complex deformations, period integrals) of its mirror dual, the mirror quintic orbifold. In order for mirror symmetry to be a true duality however, one must also show that the B model of the Fermat quintic is equivalent to the A model of the mirror quintic. We prove such an equivalence by relating the orbifold Gromov-Witten theory of the mirror quintic to period integrals over a one parameter deformation of the Fermat quintic. This involves new calculations in orbifold Gromov-Witten theory.