 So now the floor is to Sumitroy from the Tata Institute and he's going to speak about hitching vibration and prem varieties. Please. Thanks to Organizer for giving me the opportunity to talk about my research. Okay, so I will talk about the hitching vibration, which is amorphism from a modular space of certain kind of Higgs bundles to a vector space. And we'll see that the generic fibers of that map is given by some prem varieties. Okay, so let X be a compact prem surface of genus greater than equal to two and let K be the cotangent bundle that is the canonical bundle in case of curves. A Higgs bundle over X is a pair where E is a holomorphic vector bundle and Phi is amorphism from E to tensor K, which is called a Higgs field. Also, you can think about this Phi as a section of the endomorphism of E with values in K. And the Higgs bundle over the curve, it was introduced by Hitching in 1987. A sub bundle is called Phi invariant if it is preserved under Phi. So a Higgs bundle is called stable, respectively semi-stable. If every non-zero proper Phi invariant sub bundle satisfies this slope condition, strict inequality for stable and less than equal for semi-stable. Degree of a vector bundle means degree of the determinant bundle, which is a line bundle. And if degree and rank are co-prime, in that case, semi-stable implies stable. Now, if we impose the stability condition, then we have the modular space. So let M Higgs denote the modular space of stable Higgs bundles of fixed rank and fixed degree. And this M here is the modular space of stable vector bundles. And the cotangent bundle of that is an open sub variety of the modular space of Higgs bundles. Since the cotangent bundle has a symplectic structure, therefore it induced a symplectic structure on the modular space of Higgs bundles. And in fact, this modular space of Higgs bundles is hyperkeler. It satisfies a guess theoretic equation. So let us consider the characteristic polynomial of Phi, whose coefficients S i are basically trace of O H i Phi, which are sections of k power i. So we have a morphism from the modular space of Higgs bundles to vector space A, which is direct sum of H naught k power i, which sends a Higgs field phi to this coefficients S i. And this morphism is surjective. And in fact, it is proper. So, therefore, the fibers are compact. And the fibers are Avalian varieties. So the question is, what are exactly the generic fibers of this morphism? So let us consider an element in the base A. Then we have a spectral curve XS together with a covering map Pi from XS to X, how you degree R covering map. And this spectral curve is given by this equation, which is basically the zeros of the characteristic polynomial. But this eta here is a tautological section of the cotangent bundle. So and also for a generic point is the spectral curve is smooth. So if we consider line bundle over XS and take the push forward Pi star L, that's a rank curve vector bundle over X. Also, it has a Higgs field, which is given by the multiplication by this tautological section eta. The reverse of this map is exactly the isomorphic Jacobian of the spectral curve XS. Also, the base A has dimension half the dimension of this modular space. And therefore, this modular space of Higgs bundles, this is an algebraically completely integrable system. So now we're going to talk about the Hitchin vibration for SLRC Higgs bundles. So SLRC Higgs bundles is basically a Higgs bundle of rank R, which reveal determinant and whose trace of the Higgs field is zero. Therefore, the Hitchin map here, this is modular space of SLRC Higgs bundles. The base here, this first coefficient is zero, because the trace is zero. Therefore, I runs from two to R. So it sends the Higgs field to the coefficients of its characteristic polynomial. And in this case, the generic fiber is isomorphic to this subset of Jacobian, basically line bundles over XS for which determinant of the push forward bundle that is trivial. So for the map pi from XS to X, that's the occurring map, we have an associated norm map from the Picard group of XS to Picard group of X, which is defined by this, using the divisor, send summation Ni Pi Pi. And we have an isomorphism, the top exterior power of Pi star L, that is a determinant of Pi star L is isomorphic to norm of L, tensor k power minus R into R minus one by two. This is due to Beville, Norseman and Ramanan, this isomorphism. And so therefore, if its determinant is trivial, if and only if this norm of L is isomorphic to k power R into R minus one by two. And that is equivalent to saying that this L tensor Pi per star k power minus R minus one by two, that is in the kernel of this norm map. So kernel of this norm map is basically the premed variety for the morphism Pi. And it's also Higiene approved this that the genetic fibers for SLR6 bundles isomorphic to this premed variety. Okay, so now we are going to talk about the Higiene fibers in case of parabolic bundles. So let for that, let us first fix a set of finitely many points. So we fix this D throughout this talk. And a parabolic bundle basically means a vector bundle together with a weighted flag over D. So formally speaking, a parabolic vector bundle of rank R over X is a vector bundle of rank R together with the parabolic structure over D, which means that for all points P and D, the points in D are called parabolic points. For all parabolic points P and D, the fiber has a filtration of subspaces. And we have a sequence of real numbers between zero and one, which are called parabolic weights, satisfying this inequality. So we're assigning each subspace to a real number. So alpha one P is assigned to the subspace EP comma one. Similarly, alpha IP is assigned to EP comma I. And collection of all such weights, parabolic weights for all points in D, it is denoted by FA. And we say that alpha is full flag if the dimension of the successive cosines equal to one, which means that this filtration here is complete. So there are many subspaces here. And the parabolic degree is defined by the degree of the underlying vector bundle plus this additional part. So these parabolic weights times the dimension of these cosines. So in case of full flag, these dimensions are one. Therefore, this is just the summation of all parabolic weights. So one thing to note that the parabolic degree may not be an integer. So this is alpha IP. They are real numbers. So parabolic degree is an integer only when all parabolic weights are zero. And in that case, we will say that the parabolic structure is trivial. We also have the notion of parabolic tensor product parabolic dual and parabolic homomorphism between two parabolic bundles. So a parabolic Higgs bundle is a pair where E is a parabolic bundle and phi is a Higgs field from E to E tensor KD. So KD is the K tensor OD. And also assuming that the Higgs field is strongly parabolic, what does that mean? It means that for all points in D, the Higgs field sends the subspace EP comma I to the next one, EP comma I plus one. So this means that the Higgs field at each parabolic point P is nilpotent. So strongly parabolic means the Higgs field is nilpotent over D. Therefore, the coefficients of its characteristic polynomial all vanishes over each point of D. So therefore, trace of OAs, trace of OAs if I, they're basically sections of k power i d i minus one because they vanishes at each parabolic point D. So let M parabolic Higgs, it denote the motorized space of stable parabolic Higgs bundles of fixed rank and fixed degree and fixed parabolic structure alpha. Then we have the parabolic version of the Hitchin map which sends the Higgs field to these coefficients. And here the base is the direct sum of H naught k power i d power i minus one because this traces elements of H naught k power i d power i minus one. And this morphism is subjective also and it is proper due to Markman and genetic fibers also this isomorphic Jacobian of the spectral curve. Here the spectral curve we need to consider the tautological section of k d. So now let us define the simplistic and orthogonal parabolic Higgs bundle. So fix a parabolic line bundle and tau be a homomorphism parabolic bundles from E tensor E to L and tau tilde from E to L tensor E dual which is defined by this composition. Here the OX is a trivial parabolic bundle. It basically means all parabolic weights are zero and that's a sub bundle of E tensor E dual. And we say that a simplistic or respectively orthogonal parabolic bundle is a pair where this tau is anti-symmetric respectively symmetric and this tau tilde is an isomorphism. And for orthogonal or simplistic parabolic Higgs bundles the Higgs field need to be compatible with this form tau. It's a bilinear form. So when parabolic weights are all rational in that case the notion of simplistic or orthogonal parabolic bundles coincides with the notion of parabolic principle G bundles where G is a simplistic orthogonal group. And a sub bundle is called isotropic if tau the bilinear form that vanishes at on F. And the stability condition which we need to consider the isotropic sub bundles here and that induces a modular space of the simplistic orthogonal parabolic Higgs bundles. So let this denote the modular space of stable simplistic or orthogonal parabolic Higgs bundle. So G is simplistic orthogonal group. So when the parabolic structure alpha have full flags in the dimension of this modular space is twice the dimension of G minus 2 into G minus 1 dimension of G plus 2N into the dimension of G mod B. So where N is the number of points in D parabolic points and D here is a Borel subgroup of G. This is due to Bhosle and Ramanathan. Let E tau phi in element in this modular space then when the characteristic polynomial is of this form because the compatibility of the Higgs field with the bilinear form and the non-degeneracy of the bilinear form will imply that if lambda is an eigenvalue of phi then so is minus lambda. So eigenvalues comes in a pair. So if we assume that all eigenvalues are distinct in that case the characteristic polynomial is of this form. There are no odd terms. So this H2i is the sections of K2i D2i minus 1. And since the spectral curve here which is given by the zeros of this characteristic polynomial it possesses an involution because all exponents are even. So it possesses an involution sigma which ends lambda to minus lambda and therefore we have a double curve from the spectral curve Xs to Xs mod sigma. So we have a norm map for this covering map from the Jacobian of Xs to Jacobian of Xs mod sigma. And the kernel of that morphism is the prem variety. So in our case the prem variety for this covering map is isomorphic to subset. It consists of line bundles over Xs for which sigma star L is isomorphic to L dual. So the sigma also acts on the Jacobian. It sends a degree zero line bundles to degrees zero line bundles. So for that sigma star L is isomorphic to L dual because a norm map here is given by it sends the X to X plus sigma X. Therefore this is the kernel of this norm map which is the prem variety. And so the h in map here is given by this the base is now h naught k power 2i d2i minus 1 because these are the coefficients of the characteristic polynomial. And the dimension of the prem variety is basically genus of Xs minus genus of Xs mod sigma. And that is equal to dimension of the base a and which is half the dimension of this modular space. And the result in this direction is the generic fibers for parabolic sp2 mcx bundles. It is isomorphic to the prem variety. So let's keep an outline of the proof. So let E be an element in the fiber I mean the generic fiber which is a symplectic parabolic X bundle. Then since we are considering generic fiber the spectral curve is smooth. Therefore we have an eigen space bundle L. And then there is an isomorphism here which we need to have the same degree these the left hand side and right hand side. And also in the relation has section whose zero side is basically the zeros are the ramification locus of that spectral curve. And if we consider this line bundle U which is up there so we can we can take the square root of this bundle because this degree is even and then this line bundle U is a is an element in the prem variety. And so which means that sigma star U is isomorphic to U dual which comes from this isomorphism. Starting from an symplectic parabolic X bundle we obtain an element in the prem variety. Now conversely let U be an element in the prem variety then take L to be U tensor k x s tensor pi star k dual tensor pi star m dual minus half. And then consider the push forward of that line bundle. We have already seen that this push forward line bundle it has a Higgs field which is given by the tautological section of the cotangent bundle. I mean in case of parabolic bundles is tautological section of KD. So the parabolic structure is for the parabolic structure. So for all points in D one can show that there is an open subset V for which there is fiber E over V that is isomorphic as OV module to OV x mod x power 2 m plus H to x power 2 m minus 2 plus H to m. So this is H2i that basically components of S. And since this H2i vanishes at each parabolic point therefore E is shifted to P, E over P that is isomorphic to Cx mod x power 2 m. And also remember that this the Higgs field is given by the multiplication by X. So the Higgs field basically gives the parabolic structure on this push forward bundle. And for the symplectic structure it is considered two sections U and V over L. And the symplectic form is given by this I mean so this this form is non-regenerate and it is skew because this sigma star is square equal to minus 1. And this gives us a symplectic form on this push forward bundle E. For the orthogonal case the spectral curve basically the characteristic polynomial the last coefficient is square of a section. And so the spectral curve is given by this this equation where Pm is a section of k power m d power m. So therefore the spectral curve always has a singularity whenever this Pm is 0. So we need to consider the desingularized curve is called Xs hat. And also this spectral curve Xs has it has an involution sigma. So we are considering now the extended involution sigma hat on Xs hat. And yeah and for that we have a prime variety basically that we have a double cover for Xs hat and similarly as before. And the kernel of that on a norm map is the prime variety. And this and this Hitching map in this case the base is given by this direct sum of H not k power 2i d2i minus 1. I run from i to m minus 1. And the last coefficient is this because because this it sends the Hicks field phi to h2 comma or h2m minus 2 or Pm. And the result is that in the generic fiber isomorphic to the prime variety for this desingularized curve. And for the odd orthogonal group the characteristic polynomial is reducible. It always has a zero eigenvalue. So therefore therefore we consider the spectral curve with this component which is zero of this component which is similar to the case of symplectic group. In fact they have the same base here the symplectic group and the odd orthogonal group. Because that is quite that is because when they these two groups the symplectic group and odd orthogonal group they are they are Lagrange dual okay. Therefore they they need to have some similarities. And and this result for this case is the generic fibers are isomorphic to this prime variety for this spectral curve axis. Okay thank you. Thank you so much. Thank you. Thank you. Are there any questions? I have a question. Please. So your description of the fibers was in the stable locus of Hicks bundles. Is that right? Well which locus? The stable locus. So you've described like a generic fiber in the in the vibration. Is that right? Yes. Yes. Have you have you thought about the describing the fibers outside the stable locus? Okay okay. So yeah you and me and singular I mean okay. So I have not thought about that but yeah I mean there are really not enough description for. So you're saying that for a singular locus right? Yes. Yes. Yeah I mean a singular k a singular locus they are not enough good description. I mean there are some results by I guess so that is for for not for parabolic scenario for for Hicks bundles. There are some results the heart of the genetic fibers and not a genetic fiber but a singular a singular fibers for the Hitchin map and it is by I guess Oliveira and Oliveira maybe I don't remember their name okay. There are some results for Hicks bundles. There are no results for parabolic Hicks bundles or anything for heart of the singular locus of that Hitchin map yeah I have not thought about that also yeah. Thank you. I was also wondering so you had this involution on the desingularization. Yes. So how does it act on like the exceptional locus in the desingularization? Okay so this one? Yeah yeah. Okay so this I mean okay so the singular points here basically the fixed point of this curve has involution sigma and exactly whose fixed points are the the fixed point of that involution are the singular points of this spectral curve. Right okay and yes we are actually we are executing moving those points so from the from the spectral curve. Any more questions? Okay so thank you again Sumit. And so