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We study Chow groups and \'etale motivic cohomology groups of smooth complete intersections with Hodge structures of level one, classified by Deligne and Rapoport, with particular attention to fivefolds. We extend these results to an…
Let $\MS_g$ be the moduli space of stable holomorphic vector bundles of rank 2 and fixed determinant of odd degree over a smooth complex projective curve of genus $g$. This paper proves various properties of the rational cohomology ring…
We define the logarithmic tautological rings of the moduli spaces of Deligne-Mumford stable curves (together with a set of additive generators lifting the decorated strata classes of the standard tautological rings). While these algebras…
We study the singular homology (with field coefficients) of the moduli stack of stable n-pointed complex curves of genus g (the Deligne-Mumford compactification). Each of its irreducible boundary components determines via the…
We use a graph to define a new stability condition for algebraic moduli spaces of rational curves. We characterize when the tropical compactification of the moduli space agrees with the theory of geometric tropicalization. The…
We describe stability conditions for pairs consisting of a coherent sheaf and a homomorphism to a fixed coherent sheaf on a projective variety. The corresponding moduli spaces are constructed for pairs on curves and surfaces. We consider…
We show that the homology of modules for Hurwitz spaces stabilizes and compute its stable value. As one consequence, we compute the moments of Selmer groups in quadratic twist families of abelian varieties over suitably large function…
We propose a remarkably simple and explicit conjectural formula for a bihamiltonian structure of the double ramification hierarchy corresponding to an arbitrary homogeneous cohomological field theory. Various checks are presented to support…
In the present paper, we study a new kind of anabelian phenomenon concerning the smooth pointed stable curves in positive characteristic. It shows that the topological structures of moduli spaces of curves can be understood from the…
Let $\MC$ be the moduli space of stable holomorphic vector bundles of rank 2 and fixed determinant of odd degree, over a smooth projective curve $C$. This paper identifies the algebraic cohomology ring $\HA^*(\MC)$, i.e. the subring of the…
The moduli space of stable bundles of rank 2 and degree 1 on a Riemann surface has rational cohomology generated by the so-called universal classes. The work of Baranovsky, King-Newstead, Siebert-Tian and Zagier provided a complete set of…
The tadpole conjecture suggests that the complete stabilization of complex structure deformations in Type IIB and F-theory flux compactifications is severely obstructed by the tadpole bound on the fluxes. More precisely, it states that the…
We prove the so-called master relation in the tautological ring of the moduli space of curves that implies polynomial properties of the Dubrovin-Zhang hierarchies associated to different versions of cohomological field theories as well as…
The moduli space of regular stable maps with values in a complex manifold admits naturally the structure of a complex orbifold. Our proof uses the methods of differential geometry rather than algebraic geometry. It is based on Hardy…
After a short exposition of the basic properties of the tautological ring of the moduli space of genus g Deligne-Mumford stable curves with n markings, we explain three methods of detecting non-tautological classes in cohomology. The first…
In this paper we compute the intersection number of two double ramification cycles (with different ramification profiles) and the top Chern class of the Hodge bundle on the moduli space of stable curves of any genus. These quadratic double…
We compute the mod 2 homology of spin mapping class groups in the stable range. In earlier work we computed the stable mod p homology of the oriented mapping class group, and the methods and results here are very similar. The forgetful map…
This paper establishes an explicit obstruction to constructing algebraic cycles from automorphic cohomology classes on Shimura varieties. We produce a rational Hodge class $\Omega_E$ in the intersection cohomology of the Baily-Borel…
In a previous paper, arXiv:1206.5498, we introduced a new homological invariant $\e$ for the faithful action of a finite group G on an algebraic curve. We show here that the moduli space of curves admitting a faithful action of a finite…
This paper has the purpose of presenting in an organic way a new approach to integrable (1+1)-dimensional field systems and their systematic quantization emerging from intersection theory of the moduli space of stable algebraic curves and,…