We compare the smooth and deformation equivalence of actions of finite groups on $K3$-surfaces by holomorphic and antiholomorphic transformations. We prove that the number of deformation classes is finite and, in a number of cases, establish the expected coincidence of the two equivalence relations. More precisely, in these cases we show that an action is determined by the induced action in the homology. On the other hand, we construct two examples to show first that, in general, the homological type of an action does not even determine its topological type, and second that $K3$-surfaces $X$ and $\overline{X}$ with the same Klein action do not need to be equivariantly deformation equivalent even if the induced action on $H^{2,0}(X)$ is real, that is, reduces to multiplication by $\pm 1$.

We investigate trace functions of modules for vertex operator algebras (VOA) satisfying $C_2$-cofiniteness. For the modular invariance property, Zhu assumed two conditions in [Z]: (1) $A(V)$ is semisimple and (2) $C_2$-cofiniteness. We show that $C_2$-cofiniteness is enough to prove a modular invariance property. For example, if a VOA $V={\oplus }_{m=0}^{\infty }{V}_m$ is $C_2$-cofinite, then the space spanned by generalized characters (pseudotrace functions of the vacuum element) of $V$-modules is a finite-dimensional $\SL_2(\mathbb{Z})$${\text{SL}}_{\text{2}}(Z)$-invariant space and the central charge and conformal weights are all rational numbers. Namely, we show that $C_2$-cofiniteness implies "rational conformal field theory" in a sense as expected in Gaberdiel and Neitzke [GN]. Viewing a trace map as a symmetric linear map and using a result of symmetric algebras, we introduce "pseudotraces" and pseudotrace functions and then show that the space spanned by such pseudotrace functions has a modular invariance property. We also show that $C_2$-cofiniteness is equivalent to the condition that every weak module is an $N$-graded weak module that is a direct sum of generalized eigenspaces of $L(0)$.

We prove that in characteristic zero the multiplication of sections of line bundles generated by global sections on a complete symmetric variety $X=\overline{G/H}$ is a surjective map. As a consequence, the cone defined by a complete linear system over $X$ or over a closed $G$-stable subvariety of $X$ is normal. This gives an affirmative answer to a question raised by Faltings in [11]. A crucial point of the proof is a combinatorial property of root systems.

Fulton's universal Schubert polynomials [F3] represent degeneracy loci for morphisms of vector bundles with rank conditions coming from a permutation. The quiver formula of Buch and Fulton [BF] expresses these polynomials as an integer linear combination of products of Schur determinants. We present a positive, nonrecursive combinatorial formula for the coefficients. Our result is applied to obtain new expansions for the Schubert polynomials of Lascoux and Schützenberger [LS1] and explicit Giambelli formulas in the classical and quantum cohomology ring of any partial flag variety.

We give a topological interpretation of the space of $L^2$-harmonic forms of finite-volume manifolds with sufficiently pinched negative curvature. We give examples showing that this interpretation fails if the curvature is not sufficiently pinched and that our result is sharp with respect to the pinching constants. The method consists first in comparing $L^2$-cohomology with weighted $L^2$-cohomology thanks to previous works done by T. Ohsawa, and then in identifying these weighted spaces.

Let $E$ be an elliptic curve over $Q$ attached to a newform $f$ of weight 2 on ${\Gamma }_0(N)$, and let $K$ be a real quadratic field in which all the primes dividing $N$ are split. This paper relates the canonical $ℝ/\mathbb{Z}$-valued "circle pairing" on $E(K)$ defined by Mazur and Tate [MT1] to a period integral $I^{\prime }(f,k)$ defined in terms of $f$ and $k$. The resulting conjecture can be viewed as an analogue of the classical Birch and Swinnerton-Dyer conjecture, in which $I^{\prime }(f,k)$ replaces the derivative of the complex $L$-series $L(f,K,s)$ and the circle pairing replaces the Néron-Tate height. It emerges naturally as an archimedean fragment of the theory of anticyclotomic p-adic L-functions developed in [BD], and has been tested numerically in a variety of situations. The last section formulates a conjectural variant of a formula of Gross, Kohnen, and Zagier [GKZ] for the Mazur-Tate circle pairing.

In this note we improve the known $L^p$-bounds for Bochner-Riesz operators and their maximal operators.