A. Ramanathan

Bio: A. Ramanathan is an academic researcher from Tata Institute of Fundamental Research. The author has contributed to research in topics: Schubert variety & Fundamental group. The author has an hindex of 6, co-authored 6 publications receiving 991 citations.

Frobenius splitting and cohomology vanishing for Schubert varieties

Abstract: Let k be an algebraically closed field of characteristic p > 0 and X be a projective variety over k. We then have the absolute Frobenius F: X -> X and an injection Ox -* F *6x given by f fP, f e O9x. If this makes Ox a direct summand in F* Ox (as an (x-module) we call X a Frobenius split variety. For such a variety the vanishing theorem for ample line bundles follows trivially from Serre's vanishing theorem. For, tensoring Ox -F*Ox by an ample line bundle L and noting that L ? F * (x = F * F *L = F * LP (projection formula) we get that the map in cohomology H'(X, L) -+ H'(X, F*LP) = H'(X, LP) is an injection. Iterating this we see that H'(X, L) injects into H'(X, LP) for every P. But, for large v the latter is zero! Thus Frobenius split varieties have quite pleasant properties. It also turns out that using duality for the Frobenius morphism, or equivalently, the Cartier operator, one can give a manageable criterion for Frobenius splitting. The point here is the local nature of duality. The relevance of the compatibility of local and global duality was suggested to us by Grothendieck's proof of H'(X, Y) = 0 for a noncomplete variety X of dimension n ([4], Theorem 6.9) and by Kempf's paper [10]. By this criterion and the Bott-Samelson-Demazure desingularisation of Schubert varieties, it follows very easily that Schubert varieties in characteristic p are Frobenius split. The following vanishing theorem is then an immediate consequence. Let G be a reductive group over the field k (of arbitrary characteristic, zero or positive). Let Q be a parabolic subgroup and X c G/Q a Schubert variety. Let L be an ample line bundle on G/Q. Then HI(X, L) = 0 for i > 0 and the restriction map H0(G/Q, L) -+ H0(X, L) is surjective. If char k > 0 this is a consequence of the compatible Frobenius splitting of X in G/Q and the char k = 0 case is handled by semicontinuity. The above result for special Schubert varieties has been proved by SeshadriMusili-Lakshmibai [12], [13] and by Kempf [9] by using characteristic free methods. For X = G/Q, Andersen [1] and Haboush [5] have given simple proofs using characteristic p methods.

Restriction of stable sheaves and representations of the fundamental group

Abstract: Let X be a projective smooth variety of dimension n over an algebraically closed field k. Let H be an ample line bundle on X. A torsion free sheaf V on X is said to be stable (respectively, semistable) with respect to the polarisation H if for every proper subsheaf W c V we have deg W/rk W < deg V / r k V (respectively <) where deg W= cl(W). H\"-1, c t(W ) the first chern class and r k = rank (see [7, 13]). In [10] we proved that the restriction of a semistable sheaf V on X to a complete intersection subvariety of X in general position and of high multidegree is again semistable. We prove here that the restriction of a stable sheaf also remains stable. This has some interesting consequences. When k=~2 and d imX =2 it follows from the recent results of Donaldson [2] and Kobayashi 1-6] that any stable vector bundle V with c l (V)=0 and c2(V)=0 on the surface X comes from an irreducible unitary representation of the fundamental group rcl(X ). It follows from this and our restriction theorem that the same result holds for higher dimensional varieties as well. This answers a question of Kobayashi [6, Sect. 4, p. 161].

The geometry of moduli spaces of sheaves

Abstract: Preface to the second edition Preface to the first edition Introduction Part I. General Theory: 1. Preliminaries 2. Families of sheaves 3. The Grauert-Mullich Theorem 4. Moduli spaces Part II. Sheaves on Surfaces: 5. Construction methods 6. Moduli spaces on K3 surfaces 7. Restriction of sheaves to curves 8. Line bundles on the moduli space 9. Irreducibility and smoothness 10. Symplectic structures 11. Birational properties Glossary of notations References Index.

Combinatorial Commutative Algebra

Abstract: Monomial Ideals.- Squarefree monomial ideals.- Borel-fixed monomial ideals.- Three-dimensional staircases.- Cellular resolutions.- Alexander duality.- Generic monomial ideals.- Toric Algebra.- Semigroup rings.- Multigraded polynomial rings.- Syzygies of lattice ideals.- Toric varieties.- Irreducible and injective resolutions.- Ehrhart polynomials.- Local cohomology.- Determinants.- Plucker coordinates.- Matrix Schubert varieties.- Antidiagonal initial ideals.- Minors in matrix products.- Hilbert schemes of points.

Higgs bundles and local systems

Abstract: © Publications mathématiques de l’I.H.É.S., 1992, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » ( http://www. ihes.fr/IHES/Publications/Publications.html), implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Moduli of representations of the fundamental group of a smooth projective variety I

Abstract: © Publications mathématiques de l’I.H.É.S., 1994, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization

Abstract: The fundamental group is one of the most basic topological invariants of a space. The aim of this paper is to present a method of constructing representations of fundamental groups in complex geometry, using techniques of partial differential equations. A representation of the fundamental group of a manifold is the same thing as a vector bundle over the manifold with a connection whose curvature vanishes, and this condition amounts to a differential equation. On the other hand, the natural objects of geometry over a complex manifold are the holomorphic vector bundles and holomorphic maps between them. We will adopt a philosophy based on algebraic geometry, that these holomorphic objects are understandable, and this leads us to try to produce flat connections starting from holomorphic data. Briefly, the results are as follows. We solve the Yang-Mills equations on holomorphic vector bundles with interaction terms, over compact and some noncompact complex Kahler manifolds, yielding flat connections when certain Chern numbers vanish. An application in the compact case gives necessary and sufficient conditions for a variety to be uniformized by any particular bounded symmetric domain. The first such construction was the theorem of Narasimhan and Seshadri relating holomorphic vector bundles and unitary connections on a curve. It was later extended to higher dimensions by Donaldson, Uhlenbeck, and Yau. Their work serves as a paradigm for what we will prove, so it is worth describing first. Let X be a compact complex manifold. One can produce unitary connections using holomorphic vector bundles as follows. There is a natural operator 8 which reflects the holomorphic structure of a bundle E. Given a metric on E , there is an operator a defined by the condition that the sum D = a + 8 is a connection which preserves the metric. The curvature of D is a two-form F = D2 with coefficients in the endomorphisms of E. The equation F = 0 is usually overdetermined, but there is a natural intermediate equation, itself of